I Know What I Don't Know: Latent Posterior Factor Models for Multi-Evidence Probabilistic Reasoning

I Kno w What I Don’t Kno w: Laten t P osterior F actor Mo dels for Multi-Evidence Probabilistic Reasoning Alege Aliyu Agb oola Epalea aaa@epalea.com Marc h 19, 2026 Abstract The m ulti-evidence aggregation challenge. Real-world decision-making—from tax compliance assessmen t to medical diagnosis—requires aggregating multiple noisy and potentially con tradictory evidence sources. Existing approaches either lack explicit uncertain ty quantiﬁcation (neural aggregation methods) or rely on man ually engineered discrete predicates (probabilistic logic framew orks), limiting scalability to unstructured data. Our approach: Laten t P osterior F actors (LPF). W e in tro duce LPF, a framework that transforms V ariational Auto encoder (V AE) latent p osteriors in to soft likelihoo d factors for Sum-Product Netw ork (SPN) inference. This enables tractable probabilistic reasoning o v er unstructured evidence while preserving calibrated uncertain t y estimates. T w o complementary architectures. W e instantiate LPF in t wo forms: LPF-SPN , whic h p erforms structured factor-based inference, and LPF-Learned , which learns evidence aggregation end-to-end. This design enables a principled comparison b et ween explicit probabilistic reasoning and learned aggregation under a shared uncertaint y represen tation. Comprehensiv e ev aluation. A cross eight domains (seven syn thetic and the FEVER b enc hmark), LPF-SPN ac hieves high accuracy (up to 97.8%), lo w calibration error (ECE 1.4%), and strong probabilistic ﬁt as measured by negativ e log-lik eliho od, substantially outperforming eviden tial deep learning and graph-based baselines. Results are a veraged o v er 15 random seeds to ensure statistical reliabilit y . Key contributions: 1. First general framew ork bridging latent uncertain ty represen tations with structured proba- bilistic reasoning 2. Dual arc hitectures enabling controlled comparison of reasoning paradigms 3. A repro ducible training methodology with seed selection 4. Extensiv e ev aluation against strong baselines including EDL, BER T, R-GCN, and large language models 5. Cross-domain v alidation demonstrating broad applicabilit y 6. F ormal guarantees (presen ted in companion paper [ Alege , 2026 ]) Con ten ts 1 In tro duction 10 1.1 The Multi-Evidence Reasoning Challenge . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2 Limitations of Existing Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Our Approach: Latent Posterior F actors (LPF) . . . . . . . . . . . . . . . . . . . . . 11 1 1.4 Con tributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 P ap er Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2 Glossary of Symbols 15 3 Bac kground 16 3.1 V ariational Auto encoders (V AEs) Kingma and W elling [2014] . . . . . . . . . . . . . 16 3.2 Sum-Pro duct Net works (SPNs) Poon and Domingos [2011] . . . . . . . . . . . . . . . 16 3.3 The Multi-Evidence Aggregation Problem . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Metho d: Laten t P osterior F actors (LPF) 18 4.1 Problem F ormulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Arc hitecture Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3 Phase 1: Evidence Retriev al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3.1 System Arc hitecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3.2 T w o-Tier Indexing Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.3.3 F AISS V ector Store . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3.4 Metadata Store . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.4 Phase 2: V AE Enco ding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.5 Deco der Net work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.6 Laten t-to-F actor Mapping (Monte Carlo Integration) . . . . . . . . . . . . . . . . . . 24 4.6.1 Theoretical F oundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.6.2 Mon te Carlo Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.6.3 T emp erature Scaling and Normalization . . . . . . . . . . . . . . . . . . . . . 26 4.6.4 Credibilit y W eighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.6.5 Con vergence and Sample Eﬃciency . . . . . . . . . . . . . . . . . . . . . . . . 27 4.7 LPF-SPN: Structured Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.7.1 SPN Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.7.2 Joint Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.7.3 Marginal Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.7.4 A dv an tages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.8 LPF-Learned: Neural Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.8.1 Wh y Learn Aggregation? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.8.2 Arc hitecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.8.3 Qualit y Netw ork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.8.4 Consistency Net work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.8.5 W eigh t Net work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.8.6 Laten t-Space Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.8.7 Gradien t Flow and End-to-End T raining . . . . . . . . . . . . . . . . . . . . . 33 4.8.8 A dv an tages and T rade-oﬀs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.9 Comparison: LPF-SPN vs. LPF-Learned . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.10 T raining Pro cedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.10.1 Stage 1: Enco der + Deco der T raining . . . . . . . . . . . . . . . . . . . . . . 34 4.10.2 Stage 2 (LPF-Learned only): Aggregator T raining . . . . . . . . . . . . . . . 35 4.10.3 Seed Search Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2 5 W orked Example: T ax Compliance Risk Assessmen t 36 5.1 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.2 Evidence Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.3 V AE Enco ding to Latent Posteriors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.4 LPF-SPN Architecture: F actor Con version via Monte Carlo . . . . . . . . . . . . . . 38 5.4.1 Step-b y-step conv ersion for evidence e 1 . . . . . . . . . . . . . . . . . . . . . . 38 5.4.2 F actors for all evidence items . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.5 LPF-SPN: Bay esian Inference with Sum-Pro duct Netw orks . . . . . . . . . . . . . . . 40 5.5.1 Prior Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.5.2 SPN Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.5.3 Marginal Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.5.4 LPF-SPN Final Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.6 LPF-Learned Architecture: Neural Evidence Aggregation . . . . . . . . . . . . . . . 41 5.6.1 Step 1: Qualit y Score Computation . . . . . . . . . . . . . . . . . . . . . . . . 41 5.6.2 Step 2: P airwise Consistency Matrix . . . . . . . . . . . . . . . . . . . . . . . 41 5.6.3 Step 3: Final W eight Computation . . . . . . . . . . . . . . . . . . . . . . . . 42 5.6.4 Step 4: Laten t Space Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.6.5 Step 5: Decode Aggregated Posterior . . . . . . . . . . . . . . . . . . . . . . . 43 5.6.6 LPF-Learned Final Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 5.7 Comparison: LPF-SPN vs. LPF-Learned . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.8 Design Choices Explained . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.8.1 Wh y exp onentiate the distribution b y weigh t? . . . . . . . . . . . . . . . . . . 44 5.8.2 Wh y use temp erature scaling? . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.8.3 Wh y multiply prior by pro duct of factors? . . . . . . . . . . . . . . . . . . . . 45 5.8.4 Wh y aggregate in latent space (LPF-Learned)? . . . . . . . . . . . . . . . . . 45 5.8.5 Wh y use quality and consistency netw orks? . . . . . . . . . . . . . . . . . . . 45 5.8.6 Wh y Monte Carlo with 16 samples? . . . . . . . . . . . . . . . . . . . . . . . 45 5.8.7 Wh y log-space computation in SPN? . . . . . . . . . . . . . . . . . . . . . . . 46 5.9 Mon te Carlo Sample Size Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.10 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.10.1 T emperature sensitivity ( α = 2 . 0 , M = 16 ) . . . . . . . . . . . . . . . . . . . . 47 5.10.2 W eight p enalt y sensitivity ( T = 1 . 0 , M = 16 ) . . . . . . . . . . . . . . . . . . 47 5.11 Theoretical Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.12 Conv ergence of Monte Carlo Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.13 Comparison Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.13.1 Quantitativ e Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.13.2 Qualitative Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.14 Summary and Key T ak ea wa ys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6 Algorithms 50 6.1 Algorithm 1: Conv ertLaten tT oF actors . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.2 Algorithm 2: V AEEncoder.Enco de . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.3 Algorithm 3: LearnedAggregator.Aggregate (LPF-Learned) . . . . . . . . . . . . . . 52 6.4 Algorithm 4: SPNMo dule.Query (LPF-SPN) . . . . . . . . . . . . . . . . . . . . . . . 53 6.5 Algorithm 5: Orchestrator.HandleQuery . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.6 Supp orting Pro cedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 6.7 Implemen tation Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3 7 System Arc hitecture 56 7.1 Comp onen t Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.2 Data Flow: LPF-SPN V arian t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.2.1 Step 1: Evidence Retriev al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 7.2.2 Step 2: V AE Enco ding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 7.2.3 Step 3: F actor Conv ersion (Mon te Carlo Integration) . . . . . . . . . . . . . . 58 7.2.4 Step 4: SPN Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.2.5 Step 5: Result F ormatting and Prov enance . . . . . . . . . . . . . . . . . . . 60 7.3 Data Flow: LPF-Learned V arian t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 7.3.1 Step 3: Learned Evidence Aggregation . . . . . . . . . . . . . . . . . . . . . . 61 7.3.2 Step 4: Decode Aggregated Posterior . . . . . . . . . . . . . . . . . . . . . . . 62 7.4 Arc hitectural Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.5 Implemen tation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.5.1 T ec hnology Stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 7.5.2 Mo del Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.5.3 Hyp erparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 7.5.4 Data Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 7.5.5 T raining Data Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.5.6 Inference P erformance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.5.7 Deplo yment Architecture (Suggestiv e) . . . . . . . . . . . . . . . . . . . . . . 68 7.6 Visual Architecture Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7.7 Numerical Flow Example Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 7.8 Key T akea wa ys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 8 T raining Metho dology 74 8.1 Enco der + Deco der T raining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.1.1 Dataset Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.1.2 T raining Ob jective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 8.1.3 Hyp erparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 8.1.4 Seed Searc h Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 8.2 T raining Results: Compliance Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 75 8.2.1 Seed-Lev el Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 8.2.2 Aggregate Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.2.3 Loss Decomp osition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.3 T raining Results: All Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 8.4 Learned Aggregator T raining (LPF-Learned Only) . . . . . . . . . . . . . . . . . . . 80 8.4.1 Dataset Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8.4.2 T raining Ob jective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8.4.3 Hyp erparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8.4.4 T raining Pro cedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 9 Hyp erparameters and Implementation Guidelines 82 9.1 Key Hyp erparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 9.2 Implemen tation Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 9.3 Quic k Reference: Key Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4 10 Related W ork 83 10.1 Neuro-Symbolic AI and Probabilistic Logic . . . . . . . . . . . . . . . . . . . . . . . 83 10.2 Probabilistic Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 10.3 Uncertaint y Quantiﬁcation in Deep Learning . . . . . . . . . . . . . . . . . . . . . . 84 10.4 Evidence Aggregation and Multi-Do cument Reasoning . . . . . . . . . . . . . . . . . 85 10.5 V ariational Auto enco ders and Latent Represen tations . . . . . . . . . . . . . . . . . 86 10.6 Knowledge Base Completion and Link Prediction . . . . . . . . . . . . . . . . . . . . 87 10.7 F act V eriﬁcation and T extual Entailmen t . . . . . . . . . . . . . . . . . . . . . . . . . 87 10.8 Calibration and Conﬁdence Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 88 10.9 T rust w orthy AI and Explainability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 10.10Multi-T ask and T ransfer Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 10.11Positioning LPF: Key Innov ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 10.11.1 No vel Architecture: Latent Posteriors as Soft F actors . . . . . . . . . . . . . . 90 10.11.2 Dual Arc hitecture Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 10.11.3 Purp ose-Built Multi-Evidence Aggregation . . . . . . . . . . . . . . . . . . . . 91 10.11.4 Sup erior Calibration by Design . . . . . . . . . . . . . . . . . . . . . . . . . . 91 10.11.5 Nativ e Prov enance and Auditability . . . . . . . . . . . . . . . . . . . . . . . 92 10.11.6 Cross-Domain Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 10.12Summary: Research Gaps Addressed . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 10.13Comparative Performance Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 11 Exp erimen tal Design 94 11.1 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 11.2 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 11.2.1 Primary Ev aluation Domain: Compliance . . . . . . . . . . . . . . . . . . . . 95 11.2.2 A dditional Ev aluation Domains . . . . . . . . . . . . . . . . . . . . . . . . . . 96 11.3 Ev aluation Proto col . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 11.3.1 Data Splits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 11.3.2 Statistical Rigor: Seed Search Proto col . . . . . . . . . . . . . . . . . . . . . . 98 11.3.3 Ev aluation Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 11.4 Baseline Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 11.4.1 LPF V ariants (Ours) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 11.4.2 Neural Baselines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 11.4.3 Uncertaint y Quan tiﬁcation Baselines . . . . . . . . . . . . . . . . . . . . . . . 100 11.4.4 Graph Neural Baseline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 11.4.5 Large Language Mo del Baselines . . . . . . . . . . . . . . . . . . . . . . . . . 100 11.4.6 Upp er Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 11.5 Ablation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 11.5.1 Monte Carlo Sample Count . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 11.5.2 T emperature Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 11.5.3 Uncertaint y P enalty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 11.5.4 Evidence Count . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 11.6 Implementation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5 12 Results 102 12.1 Main Results: Compliance Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 12.1.1 Best Seed Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 12.1.2 Statistical Analysis Across Seeds . . . . . . . . . . . . . . . . . . . . . . . . . 104 12.1.3 Conﬁdence and Uncertaint y Analysis . . . . . . . . . . . . . . . . . . . . . . . 107 12.1.4 Runtime Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . 109 12.2 Cross-Domain Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 12.2.1 Domain-Sp eciﬁc Observ ations . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 12.3 Ablation Study Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 12.3.1 Monte Carlo Sample Count . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 12.3.2 T emperature Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 12.3.3 Uncertaint y P enalty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 12.3.4 Evidence Count . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 12.3.5 Ablation Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 12.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 12.4.1 Overall Error Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 12.4.2 Confusion Matrix Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 12.4.3 High-Conﬁdence Errors (LPF-SPN) . . . . . . . . . . . . . . . . . . . . . . . 121 12.4.4 Error Distribution by T rue Lab el . . . . . . . . . . . . . . . . . . . . . . . . . 122 12.4.5 Conﬁdence vs. Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 12.4.6 Evidence Quality Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 12.5 Prov enance and Explainability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 12.5.1 Sample Prov enance Records (Compliance Domain) . . . . . . . . . . . . . . . 123 12.5.2 Prov enance Records for Other Domains . . . . . . . . . . . . . . . . . . . . . 125 12.5.3 Audit T rail Prop erties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 12.6 Comparison with Large Language Mo dels . . . . . . . . . . . . . . . . . . . . . . . . 126 12.7 Real-W orld V alidation: FEVER Benchmark . . . . . . . . . . . . . . . . . . . . . . . 126 12.8 Theoretical F oundations and F ormal Guarantees . . . . . . . . . . . . . . . . . . . . 127 12.8.1 Overview of Theoretical Guarantees . . . . . . . . . . . . . . . . . . . . . . . 128 12.8.2 Detailed Theorem Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 12.8.3 Comparative Analysis: LPF-SPN vs. LPF-Learned . . . . . . . . . . . . . . . 131 12.8.4 Comparison with Baselines: Theoretical Adv antages . . . . . . . . . . . . . . 131 12.8.5 Practical Implications of Theoretical Guarantees . . . . . . . . . . . . . . . . 131 12.8.6 Limitations of Current Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 132 12.8.7 Summary: Why Theoretical Guaran tees Matter . . . . . . . . . . . . . . . . . 133 13 Discussion 134 13.1 Why LPF W orks: Architectural Insigh ts . . . . . . . . . . . . . . . . . . . . . . . . . 134 13.1.1 Explicit Uncertaint y Propagation . . . . . . . . . . . . . . . . . . . . . . . . . 134 13.1.2 Multi-Evidence Architectural Design . . . . . . . . . . . . . . . . . . . . . . . 134 13.1.3 Calibration by Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 13.2 Arc hitectural Comparison: LPF-SPN vs. LPF-Learned . . . . . . . . . . . . . . . . . 135 13.2.1 Performance T radeoﬀs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 13.2.2 When to Use Each V ariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 13.2.3 Theoretical Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 13.3 The Multi-Evidence Paradigm Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 13.3.1 Contrast with Standard ML . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 13.3.2 Positioning Against Existing Paradigms . . . . . . . . . . . . . . . . . . . . . 137 6 13.4 Lessons from Cross-Domain Ev aluation . . . . . . . . . . . . . . . . . . . . . . . . . . 137 13.4.1 Domain Characteristics and Performance . . . . . . . . . . . . . . . . . . . . 137 13.4.2 Evidence Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 13.4.3 Hyp erparameter Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 13.5 Practical Deploymen t Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 13.5.1 When LPF is a Go o d Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 13.5.2 When LPF May Not Be Optimal . . . . . . . . . . . . . . . . . . . . . . . . . 139 13.5.3 Implementation Best Practices . . . . . . . . . . . . . . . . . . . . . . . . . . 139 13.5.4 Computational Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 13.6 Limitations and F ailure Mo des . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 13.6.1 Arc hitectural Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 13.6.2 Data Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 13.6.3 Scalability Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 13.6.4 Observed F ailure Mo des . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 13.7 Broader Impact and Ethical Considerations . . . . . . . . . . . . . . . . . . . . . . . 141 13.7.1 T rust w orthy AI Beneﬁts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 13.7.2 Poten tial Risks and Mitigation . . . . . . . . . . . . . . . . . . . . . . . . . . 141 13.7.3 So cietal Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 13.8 Key T ak ea wa ys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 14 F uture W ork 142 14.1 Immediate Extensions (6–12 Months) . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 14.1.1 Contin uous Predicate Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . 142 14.1.2 A ctiv e Evidence Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 14.1.3 Contrastiv e Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 14.2 Medium-T erm Researc h (1–2 Y ears) . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 14.2.1 Multi-Hop Reasoning and Chained Inference . . . . . . . . . . . . . . . . . . . 144 14.2.2 T emporal Dynamics and Evidence Decay . . . . . . . . . . . . . . . . . . . . . 144 14.2.3 Multi-Mo dal Evidence F usion . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 14.2.4 Hierarchical Predicate Structures . . . . . . . . . . . . . . . . . . . . . . . . . 145 14.3 Long-T erm Vision (3+ Y ears) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 14.3.1 F ederated Priv acy-Preserving LPF . . . . . . . . . . . . . . . . . . . . . . . . 145 14.3.2 Appro ximate Inference for Massive Evidence Sets . . . . . . . . . . . . . . . . 146 14.3.3 Curriculum Learning for Evidence Understanding . . . . . . . . . . . . . . . . 146 14.3.4 Interactiv e Evidence Reﬁnemen t with Human-in-the-Lo op . . . . . . . . . . . 146 14.4 Nov el Application Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 14.4.1 Scientiﬁc Literature Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 14.4.2 News V eriﬁcation and F act-Chec king at Scale . . . . . . . . . . . . . . . . . . 146 14.4.3 Climate Mo del Ensem bles and Uncertaint y Quan tiﬁcation . . . . . . . . . . . 147 14.4.4 Quantum State V eriﬁcation and Multi-Measurement F usion . . . . . . . . . . 147 14.5 Priority Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 A Complete T raining Results 148 A.1 Compliance Domain (Detailed) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 A.2 A cademic Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 A.3 Construction Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 A.4 Finance Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 A.5 Healthcare Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7 A.6 Legal Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 A.7 Materials Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 A.8 FEVER Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 A.9 Cross-Domain Insights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 A.10 Per-Domain Seed Visualizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 A.11 Loss Decomp osition Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 B Detailed Domain-by-Domain Results 167 B.1 Compliance Domain (Primary Ev aluation) . . . . . . . . . . . . . . . . . . . . . . . . 167 B.1.1 Complete Ablation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 B.1.2 All Seeds Detailed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 B.1.3 Complete Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 B.1.4 Complete Prov enance Records (Sample) . . . . . . . . . . . . . . . . . . . . . 170 B.2 Academic Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 B.2.1 Domain Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 B.2.2 Best Seed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 B.2.3 Ablation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 B.2.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 B.2.5 Prov enance Records (Sample) . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 B.3 Construction Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 B.3.1 Domain Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 B.3.2 Complete Ablation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 B.3.3 Best Seed Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 B.3.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 B.3.5 Prov enance Records (Sample) . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 B.4 FEVER Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 B.4.1 Domain Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 B.4.2 Best Seed Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 B.4.3 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 B.4.4 Prov enance Records (Sample) . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 B.5 Finance Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 B.5.1 Domain Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 B.5.2 Complete Ablation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 B.5.3 Best Seed Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 B.5.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 B.5.5 Prov enance Records (Sample) . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 B.6 Healthcare Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 B.6.1 Domain Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 B.6.2 Complete Ablation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 B.6.3 Best Seed Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 B.6.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 B.6.5 Prov enance Records (Sample) . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 B.7 Legal Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 B.7.1 Domain Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 B.7.2 Complete Ablation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 B.7.3 Best Seed Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 B.7.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 B.7.5 Prov enance Records (Sample) . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 8 B.8 Materials Science Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 B.8.1 Domain Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 B.8.2 Complete Ablation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 B.8.3 Best Seed Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 B.8.4 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 B.8.5 Prov enance Records (Sample) . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 B.9 Cross-Domain Ablation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 B.9.1 n_samples Across All Domains . . . . . . . . . . . . . . . . . . . . . . . . . . 191 B.9.2 T emp erature Across All Domains . . . . . . . . . . . . . . . . . . . . . . . . . 191 B.9.3 Alpha Across All Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 B.9.4 T op-K A cross All Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 B.10 LLM Ev aluation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 B.10.1 P rompt T emplate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 B.10.2 R esponse Parsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 B.10.3 D etailed LLM Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 B.10.4 Cost Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 B.11 Statistical Signiﬁcance T esting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 B.11.1 Paired t-tests (Compliance Domain) . . . . . . . . . . . . . . . . . . . . . . . 193 B.11.2 Calibration Qualit y T ests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 B.12 Computational Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 B.12.1 T raining Resource Requirements . . . . . . . . . . . . . . . . . . . . . . . . . 194 B.12.2 Inference Resource Requirements . . . . . . . . . . . . . . . . . . . . . . . . . 194 B.13 Data Generation Pro cess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 B.13.1 Synthetic Data Generation Parameters . . . . . . . . . . . . . . . . . . . . . . 194 B.13.2 F EVER Data Prepro cessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 B.14 Hyp erparameter Search Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 B.14.1 Search Space Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 B.14.2 B est Conﬁguration by Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 196 B.15 Repro ducibilit y Chec klist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 B.15.1 Co de and Data A v ailability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 B.15.2 Hardware Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 B.15.3 Softw are Dep endencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 B.16 Ac kno wledgments and Ethics Statemen t . . . . . . . . . . . . . . . . . . . . . . . . . 197 B.16.1 D ata Ethics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 B.16.2 Poten tial Misuse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 B.16.3 Intended Use Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 B.17 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 9 1 In tro duction 1.1 The Multi-Evidence Reasoning Challenge Real-w orld decision-making rarely relies o n single, deﬁnitiv e data points. Instead, it requires aggregating multiple pieces of evidence that may b e: • Noisy : Individual evidence items contain measurement errors • Con tradictory : Diﬀerent sources provide conﬂicting signals • V ariable quality : Evidence credibility v aries widely • Incomplete : Critical information may b e missing Example: T ax Compliance Assessment Company X has: - Evidence 1: "Filed taxes on time" (credibility: 0.95) - Evidence 2: "Audit found minor discrepancies" (credibility: 0.78) - Evidence 3: "Industry compliance issues reported" (credibility: 0.42) - Evidence 4: "Strong internal controls documented" (credibility: 0.88) - Evidence 5: "Previous violations 3 years ago" (credibility: 0.65) Question: What is the current compliance level? How confident should we be? This scenario demands: 1. Aggregation : Com bine heterogeneous evidence 2. Uncertain ty quan tiﬁcation : Express conﬁdence calibrated to evidence qualit y 3. Pro venance : T race predictions back to source evidence for audit trails 4. Robustness : Handle missing or contradictory information gracefully 1.2 Limitations of Existing Approac hes Neural metho ds (BER T, T ransformers, A ttention mec hanisms) [ Devlin et al. , 2019 , V aswani et al. , 2017 , Bahdanau et al. , 2015 ]: × No explicit uncertaint y quantiﬁcation × Poor calibration—often ov erconﬁden t × Black-box aggregation—no in terpretability × Require massive training data Probabilistic metho ds (Probabilistic Soft Logic, Marko v Logic Net works) [ Bach et al. , 2017 , Ric hardson and Domingos , 2006 ]: × Require manual rule engineering × Assume discrete symbolic predicates 10 × Limited handling of unstructured evidence × Intractable inference at scale Eviden tial Deep Learning (EDL) [ Sensoy et al. , 2018 ]: × Designed for single-input scenarios × T raining-inference mismatch in multi-evidence settings × No principled aggregation mec hanism • Our exp erimen ts sho w : 56.3% accuracy (EDL-Aggregated) vs 97.8% (LPF-SPN) The gap : No existing method com bines neural perception of unstructured evidence with structured probabilistic reasoning under uncertaint y . 1.3 Our Approac h: Laten t Posterior F actors (LPF) Core inno v ation : Evidence → V AE p osterior → Soft factor → Probabilistic reasoning Evidence e V AE Enco der q ( z | e ) ∼ N ( µ, σ 2 ) z (1) , z (2) , . . . , z ( M ) p ( y | z (1) ) , p ( y | z (2) ) , . . . , p ( y | z ( M ) ) Φ e ( y ) = soft factor LPF-SPN (SPN reasoning o ver factors) LPF-Learned (Neural aggregation) P ( y | all evidence ) Monte Carlo sampling De c o de e ach sample A ggr e gate Figure 1: Pip eline: Evidence through V AE to soft factors and probabilistic reasoning. 11 T w o complemen tary arc hitectures: 1. LPF-SPN : Structured probabilistic reasoning • Conv erts each p osterior to a soft factor Φ e ( y ) • Attac hes factors to Sum-Pro duct Netw ork • Performs exact marginal inference • A dv an tages : Principled, interpretable, b est calibration • Use case : High-stak es decisions requiring audit trails 2. LPF-Learned : Neural evidence aggregation • Computes quality and consistency scores for each p osterior • Learns aggregation weigh ts via neural netw orks • Aggregates in latent space, then deco des once • A dv an tages : Simpler architecture, comp etitiv e p erformance • Use case : Deplo ymen t scenarios prioritizing simplicity 1.4 Con tributions 1. No vel framew ork : First to bridge V AE uncertaint y quantiﬁcation with structured proba- bilistic reasoning via soft factors 2. Dual arc hitectures : Principled comparison of structured (SPN-based) v ersus learned (neural) aggregation mechanisms 3. Comprehensiv e ev aluation : • 8 diverse domains: Compliance, Healthcare, Finance, Legal, Academic, Materials, Con- struction, FEVER • 15 random seeds for statistical rigor • 10 baselines including EDL, BER T, R-GCN, large language mo dels 4. T raining metho dology : • Seed search strategy for repro ducible results • Enco der + deco der training proto col • Learned aggregator training (LPF-Learned only) 5. Empirical v alidation : • Sup erior accuracy: 97.8% (vs 56.3% EDL, 94.1% BER T) • Exceptional calibration: ECE 1.4% (vs 12.1% BER T) • Robust generalization: +2.4% ov er b est baselines across domains • Real-world transfer: 92.3% on FEVER b enc hmark 6. Detailed analysis : • Ablation studies: n_samples, temp erature, alpha, top_k • Robustness tests: Missing evidence, noise, contradictions • Error analysis: F ailure mo des and conﬁdence calibration 12 1.5 P ap er Organization • Section 2 : Notation and symbols • Section 3 : Bac kground on V AEs, SPNs, multi-evidence aggregation • Section 4 : LPF metho d with b oth architectures • Section 5 : W ork ed example with complete calculations • Section 6 : F ormal algorithms • Section 7 : System architecture and implementation • Section 8 : T raining metho dology and seed searc h • Section 9 : Hyperparameters and implementation guidelines • Section 10 : Related work and key diﬀeren tiators • Section 11 : Experimental design and proto cols • Section 12 : Results across all domains and baselines • Section 13 : Discussion and analysis • Section 14 : F uture work • Section 15 : Conclusion • Section 16 : A c knowledgmen ts • Section 17 : References • Section 18 : Appendices 13 14 2 Glossary of Symbols Sym b ol Meaning Evidenc e & Entities e or x Evidence item (text, do cument, sensor reading) E Set of all evidence items en tity_id Unique iden tiﬁer for an entit y (company , patient, case) L atent Sp ac e z ∈ R d Laten t co de (hidden representation) d Latent dimensionalit y (typically 64) q ϕ ( z | e ) V AE enco der p osterior distribution ov er z given evidence e µ ϕ ( e ) P osterior mean vector σ ϕ ( e ) P osterior standard deviation vector (diagonal cov ariance) De c o ding & Pr e diction p θ ( y | z ) Deco der mapping latent z to predicate distribution π θ Deco der netw ork parameters y Predicate v alue (e.g., “low”, “medium”, “high”) Y Domain of predicate v alues F actor Conversion Φ e ( y ) Laten t-p osterior factor: lik eliho od of y giv en evidence e M Num b er of Monte Carlo samples for factor conv ersion z ( m ) m -th sample: z ( m ) = µ + σ ⊙ ϵ ( m ) where ϵ ( m ) ∼ N (0 , I ) T T emp erature parameter for softmax calibration w ( e ) Credibilit y weigh t for evidence e α P enalty parameter for uncertaint y weigh ting A ggr e gation N Num b er of evidence items for an entit y F Set of soft factors { Φ e 1 , Φ e 2 , . . . , Φ e N } SPN (LPF-SPN only) V Set of structured v ariables in the SPN S SPN structure (sum and pro duct no des) P SPN ( V ) Join t distribution deﬁned by the SPN L e arne d A ggr e gation (LPF-L e arne d only) q i Qualit y score for p osterior i c ij Consistency score b etw een p osteriors i and j w i Final aggregation weigh t for p osterior i z agg Aggregated latent co de Gener al D train T raining dataset D v al V alidation dataset D test T est dataset β KL regularization weigh t in V AE training T able 1: Glossary of symbols used throughout the pap er. 15 Key F orm ula : Φ e ( y ) = Z p θ ( y | z ) q ϕ ( z | e ) dz ≈ 1 M M X m =1 p θ ( y | z ( m ) ) (1) 3 Bac kground 3.1 V ariational Auto enco ders (V AEs) Kingma and W elling [ 2014 ] Ob jective : Learn latent representations of data via v ariational inference. Enco der : q ϕ ( z | e ) maps evidence e to laten t p osterior • Typically parameterized as Gaussian: q ϕ ( z | e ) = N ( µ ϕ ( e ) , diag ( σ 2 ϕ ( e ))) • Neural netw ork outputs µ and log σ Deco der : p θ ( e | z ) reconstructs evidence from latent co de • In our case: p θ ( y | z ) predicts predicate v alues Reparameterization tric k : Enable backpropagation through sampling z = µ + σ ⊙ ϵ, ϵ ∼ N (0 , I ) (2) T raining ob jective (ELBO) : L ( ϕ, θ ; e, y ) = E q ϕ ( z | e ) [log p θ ( y | z )] − β · D KL ( q ϕ ( z | e ) ∥ p ( z )) (3) where: • First term: Reconstruction/prediction accuracy • Second term: KL regularization (prior p ( z ) = N (0 , I ) ) • β : Regularization weigh t (typically 0.01) Epistemic uncertain t y : P osterior v ariance σ 2 captures uncertain ty ab out the laten t represen- tation, reﬂecting evidence quality . 3.2 Sum-Pro duct Netw orks (SPNs) Poon and Domingos [ 2011 ] Deﬁnition : A directed acyclic graph representing a probability distribution through hierarc hical comp osition of sum and pro duct no des. Prop erties : • Completeness : All sum no de children ha ve the same scop e • Decomp osabilit y : Pro duct no de children ha ve disjoint scop es • T ractabilit y : If complete and decomp osable, exact marginals computable in time linear in net work size No des : • Leaf no des : Probability distributions ov er individual v ariables 16 • Pro duct no des : f ( V ) = Q i f i ( V i ) where V i are disjoint • Sum no des : f ( V ) = P i w i f i ( V ) where w i ≥ 0 and P i w i = 1 Inference : Bottom-up ev aluation P ( query | evidence ) = SPN ( query ∧ evidence ) SPN ( evidence ) (4) Ro ot (Sum) Pro duct ( w 1 ) Pro duct ( w 2 ) Leaf V 1 Leaf V 2 Leaf V 1 Leaf V 2 Figure 2: T ree structure showing Ro ot (Sum) no de splitting in to tw o Pro duct no des (w1, w2), eac h splitting into tw o Leaf no des ov er v ariables V1 and V2. 3.3 The Multi-Evidence Aggregation Problem F ormal problem statement : Giv en: • Entit y e id • Predicate p with domain Y = { y 1 , . . . , y K } • Evidence set E = { e 1 , e 2 , . . . , e N } where each e i is unstructured (text/images) Find: • Probability distribution P ( p = y | e id , E ) for all y ∈ Y • Conﬁdence measure reﬂecting evidence qualit y • Prov enance trace mapping prediction to source evidence Challenges : 1. Heterogeneous evidence : T ext, numerical, timestamps — diﬀerent mo dalities 2. V arying credibility : Some sources more reliable than others 3. Con tradictions : Evidence ma y supp ort conﬂicting conclusions 4. Incomplete information : Critical evidence may b e missing 17 Requiremen t Description T ractabilit y Inference in < 10ms for real-time applications Calibration Predicted probabilities match empirical frequencies Pro venanc e Ev ery prediction traceable to source evidence Robustness Graceful degradation with missing/noisy evidence Scalabilit y Handle 10–100 evidence items p er entit y T able 2: Requiremen ts for a multi-evidence aggregation solution. 5. Uncertain ty propagation : Ho w to combine individual uncertain ties? Requiremen ts for a solution : Wh y existing metho ds fail : • Neural aggregation (atten tion, p o oling): No explicit uncertaint y , p oor calibration • Naiv e av eraging : T reats all evidence equally , ignores quality • Ma jorit y v oting : Loses distributional information • Eviden tial DL : Designed for single inputs, not multi-evidence scenarios Our solution preview : LPF conv erts each evidence item to a calibrated lik eliho o d factor, then aggregates via structured reasoning (SPN) or learned weigh ting. 4 Metho d: Latent P osterior F actors (LPF) 4.1 Problem F ormulation Input : • Entit y identiﬁer: e id • Predicate: p with domain Y = { y 1 , . . . , y K } • Evidence set: E = { e 1 , e 2 , . . . , e N } Output : • Posterior distribution: P ( p = y | e id , E ) for all y ∈ Y • T op prediction: ˆ y = arg max y P ( p = y | e id , E ) • Conﬁdence: max y P ( p = y | e id , E ) • Prov enance: Evidence IDs and factor weigh ts Assumptions : 1. Evidence items are conditionally indep enden t giv en entit y and predicate 2. Eac h evidence provides partial information ab out the predicate 3. Evidence qualit y v aries and can b e estimated from enco der uncertaint y 18 4.2 Arc hitecture Ov erview LPF consists of four phases : Phase 1: Evidence Retriev al En tity + Predicate → Evidence Index → { e 1 , e 2 , . . . , e n } Phase 2: V AE Enco ding e i → V AE Enco der → q ( z | e i ) ∼ N ( µ i , σ 2 i ) Phase 3: F actor Conv ersion q ( z | e i ) → Monte Carlo Sampling → Φ e i ( y ) Phase 4: Aggregation { Φ e 1 , . . . , Φ e n } → ( LPF-SPN: SPN reasoning LPF-Learned: Neural aggregation Figure 3 summarizes the end-to-end execution of a query in LPF. Given an entit y-predicate pair, the system ﬁrst p erforms a fast canonical chec k and immediately returns a veriﬁed result when a v ailable. Otherwise, LPF retrieves relev an t evidence, enco des each item as a latent p osterior distribution, conv erts these p osteriors in to soft probabilistic factors, and combines them with any a v ailable hard conditionals. Aggregation is p erformed either via structured SPN-based inference or a neural fallbac k, pro ducing a p osterior distribution, conﬁdence estimate, and a complete prov enance record. 19 User Query Canonical DB Chec k Evidence Index Retriev er Return Canonical V AE Enco der Laten t P osteriors ( µ, σ ) convert_latent _to_factors LPF-SPN SPN reasoning o ver factors + hard conditionals LPF-Learned Neural aggregation Output + Pro venance Ledger Hit Miss emb e ddings dim [soft factors] [p osterior distribution] Figure 3: (System Overview) illustrates the complete pip eline from user query through canonical database chec k, evidence retriev al, V AE enco ding, factor conv ersion, and SPN reasoning to ﬁnal output with prov enance. 4.3 Phase 1: Evidence Retriev al Purp ose : F etc h relev an t evidence for entit y-predicate pair. The evidence retriev al system employs a h ybrid arc hitecture combining exact lo okup with seman tic searc h capabilities. At its core, the system maintains tw o complementary indexes: a hash-based en tity-predicate index for exact retriev al, and a F AISS vector store for semantic similarit y search. 20 4.3.1 System Arc hitecture En tity-Predicate Hash Index ( entity , pr e d ) ↓ { e 1 , e 2 , . . . , e n } F AISS V ector Store 384-dim em b eddings Metadata Store (JSONL) evidence_id → { entity_id, predicate, text_content, credibility, embedding_id, ... } Evidence Index Query Flow: 1. search(entity_id="ACME", predicate="compliance_level") 2. Hash lookup → candidate_ids = { e 1 , e 2 , e 3 } 3. (Optional) Seman tic rerank with query_text 4. Return top_k evidence IDs Figure 4: Evidence Index Architecture illustrating the t wo-tier indexing strategy: an En tity-Predicate Hash Index and F AISS V ector Store feeding into a central Metadata Store, with the query ﬂow from en tity lo okup through optional semantic reranking to top- k evidence retriev al. As sho wn in Figure 4 , the Evidence Index Architecture illustrates the t wo-tier indexing strategy . 4.3.2 T w o-Tier Indexing Strategy The primary index maps ( entity_id , predicate ) tuples directly to lists of evidence iden tiﬁers using hash-based lo okup, enabling O (1) retriev al of all evidence relev an t to a sp eciﬁc entit y-predicate pair. When a query includes additional text context, the system p erforms secondary semantic re-ranking: evidence em b eddings are retrieved from the F AISS index, similarity scores are computed against the query embedding, and results are reordered by semantic relev ance while maintaining the en tity-predicate constrain t. While there are many improv emen ts (for example, semantic evidence indexing and searc h without the need for en tity_ids) that can b e made to the evidence indexing strategy , the scop e of the current work has b een limited to the current v ersion for brevity . 21 4.3.3 F AISS V ector Store Wh y F AISS? F AISS (F aceb ook AI Similarity Search) provides eﬃcient appro ximate nearest neighbor searc h in high-dimensional spaces. F or our 384-dimensional sentence em b eddings, F AISS enables sub-millisecond retriev al even with millions of evidence items. W e use the IndexFlatL2 v arian t for exact searc h during developmen t (p erfect recall, slow er) and can switch to IndexIVFFlat for pro duction deploymen t (approximate search, 10–100 × faster with > 95% recall). Em b edding mo del : W e use sentence-BER T ( all-MiniLM-L6-v2 ) to enco de evidence text into 384-dimensional dense vectors. This mo del balances quality and sp eed, pro ducing seman tically meaningful embeddings that cluster similar evidence items. T rade-oﬀs : The exact Flat index guarantees ﬁnding all true nearest neighbors but scales linearly with corpus size (acceptable for datasets under 1M items). The approximate IVF (Inv erted File) index uses clustering to prune the search space, ac hieving sub-linear scaling at the cost of occasionally missing some neighbors. In practice, the hybrid approach — exact entit y-predicate lo okup with optional semantic reranking — provides b oth precision (w e nev er miss relev an t evidence for the en tity) and ﬂexibility (w e can reﬁne results semantically when needed). 4.3.4 Metadata Store The metadata store maintains a JSONL (JSON Lines) ﬁle mapping evidence IDs to rich metadata including: • text_con tent : Raw evidence text • en tity_id : Entit y this evidence relates to • predicate : T arget predicate • credibilit y : Evidence quality score [0 , 1] • timestamp : ISO timestamp for temp oral reasoning • evidence_t yp e : T yp e classiﬁcation (text, structured, hybrid) • source : Origin (SEC ﬁling, news article, internal rep ort) • em b edding_id : Index in F AISS vector store This dual storage strategy — v ectors in F AISS, metadata in JSONL — enables eﬃcien t similarity searc h while maintaining full evidence context for do wnstream pro cessing. Retriev al quality : The en tit y-predicate index ensures high precision by only returning evidence explicitly linked to the query en tity , a voiding the noisy results common in pure semantic searc h. The optional seman tic lay er adds recall by surfacing evidence that ma y b e relev an t despite not containing exact keyw ord matches, particularly v aluable for handling paraphrases and conceptual queries. 4.4 Phase 2: V AE Enco ding Arc hitecture : Evidence → [Embedding] → Enco der → ( µ, σ ) The V AE enco der transforms evidence embeddings into laten t p osterior distributions, learning to map the 384-dimensional embedding space in to a low er-dimensional latent space that captures seman tic conten t while quantifying uncertaint y . Enco der netw ork structure : 22 Input: 384-dim sentence-BERT embedding ↓ Layer 1: Linear(384 → 256) + ReLU + Dropout(0.2) ↓ Layer 2: Linear(256 → 128) + ReLU + Dropout(0.2) ↓ Split into two heads: ⊢ − → µ _head: Linear(128 → 64) ⌞ − → log_ σ _head: Linear(128 → 64) Wh y diagonal Gaussian p osteriors? The choice of diagonal (factorized) Gaussian p osteriors balances expressiveness with computational eﬃciency . While a full-cov ariance Gaussian could mo del correlations b et w een latent dimensions, it w ould require O ( d 2 ) parameters and O ( d 3 ) op erations for sampling. Diagonal cov ariance requires only O ( d ) parameters and O ( d ) sampling complexity while still capturing the magnitude of uncertaint y in each laten t dimension indep endently . Empirically , w e found that inter-dimensional correlations in the latent space contribute minimally to prediction qualit y , making the diagonal assumption a worth while simpliﬁcation. Role of drop out : Drop out ( p = 0 . 2 ) in the enco der serves dual purp oses. During training, it preven ts the enco der from ov erﬁtting to spurious patterns in evidence em b eddings by randomly zeroing 20% of activ ations, forcing the netw ork to learn robust representations. During inference (drop out disabled), the mo del pro duces deterministic enco dings. The drop out rate of 0.2 was chosen to regularize without ov er-constraining: lo w er rates (0.1) sho wed some ov erﬁtting on small datasets, while higher rates (0.3+) degraded evidence-level accuracy . What do es the laten t space learn? Through the reconstruction ob jective, the latent space learns to organize evidence by semantic similarit y — evidence supp orting the same predicate v alue clusters together. The v ariance parameters learn to reﬂect epistemic uncertaint y: ambiguous or con tradictory evidence pro duces high-v ariance p osteriors (broad distributions), while clear, deﬁnitiv e evidence pro duces low-v ariance p osteriors (p eak ed distributions). This learned uncertain ty quan tiﬁcation is what enables principled evidence weigh ting in later stages. Output : P osterior q ϕ ( z | e ) = N ( µ ϕ ( e ) , diag ( σ 2 ϕ ( e ))) The enco der learns to map evidence embeddings to latent representations that capture semantic con tent while quantifying uncertain ty through the v ariance parameters. Credibilit y computation : w ( e ) = 1 1 + mean ( σ ϕ ( e )) (5) Higher v ariance yields lo wer credibility . This simple formulation pro vides an in terpretable measure of evidence quality based directly on the p osterior’s uncertaint y . Evidence with broad, uncertain p osteriors (high σ ) receives low er weigh t, while evidence with tight, conﬁdent p osteriors (low σ ) receiv es higher weigh t. 4.5 Deco der Netw ork Arc hitecture : ( z , predicate ) → p ( y | z ) The deco der translates con tinuous latent represen tations into discrete probabilit y distributions o ver predicate v alues. Its conditional arc hitecture enables a single netw ork to handle multiple predicates without structural changes. Conditional deco der structure : 23 Input: [z (64-dim), predicate_embedding (32-dim)] ↓ Concatenate: 96-dim ↓ Layer 1: Linear(96 → 128) + ReLU + Dropout(0.2) ↓ Layer 2: Linear(128 → 64) + ReLU + Dropout(0.2) ↓ Output: Linear(64 → K) + Softmax Wh y predicate em b eddings? The predicate embedding allows the deco der to b e conditional on which attribute is b eing queried. Without this, we would need separate deco der netw orks for eac h predicate — an approach that scales p oorly and preven ts transfer learning across predicates. By em b edding the predicate name into a con tinuous 32-dimensional space, the netw ork learns shared structure across related predicates (e.g., compliance_level and regulatory_risk ma y share similar deco ding patterns) while maintaining predicate-sp eciﬁc output heads for the ﬁnal classiﬁcation. Multi-predicate support : Each predicate has a dedicated output head (ﬁnal linear lay er) mapping the 64-dimensional hidden representation to a K -dimensional logit v ector, where K is the domain size for that predicate. This design enables the system to handle predicates with diﬀerent domain sizes (e.g., binary y es/no vs. ordinal 5-level ratings) without architectural mo diﬁcations. A dding a new predicate requires only: (1) adding its name to the embedding vocabulary , and (2) initializing a new output head — the shared trunk (enco ding lay ers) transfers kno wledge from existing predicates. What inductiv e biases do es this enco de? The shared deco der trunk em b eds an assumption that predicates share underlying seman tic structure. F or instance, evidence indicating “high ﬁnancial stabilit y” may correlate with b oth “low bankruptcy risk” and “strong credit rating” — the deco der learns these cross-predicate patterns in its shared la yers. The predicate-sp eciﬁc output heads then sp ecialize this shared representation to each target distribution. This architecture p erforms well when predicates are seman tically related (common in knowledge bases) but can b e less eﬃcien t for completely unrelated predicates. The deco der π θ is trained to output the probability of predicate p taking v alues v S p giv en latent co de z . This formulation enables the system to translate contin uous laten t represen tations into discrete symbolic distributions. 4.6 Laten t-to-F actor Mapping (Monte Carlo In tegration) 4.6.1 Theoretical F oundation Let e denote observed evidence and z ∈ R d b e the latent v ariable inferred by the v ariational enco der q ϕ ( z | e ) with parameters ( µ ϕ ( e ) , Σ ϕ ( e )) . The goal is to translate each p osterior into a soft factor compatible with an SPN deﬁned ov er structured v ariables V = { V 1 , . . . , V m } . Eac h factor represents a lik eliho o d p oten tial ov er a subset of v ariables asso ciated with predicate p : f p ( v S p ) = E z ∼ q ϕ ( z | e )  π θ ( p, v S p | z )  (6) where π θ is the deco der netw ork. This exp ectation marginalizes ov er the laten t uncertaint y , pro ducing a distribution ov er predicate v alues that accoun ts for evidence ambiguit y . 24 4.6.2 Mon te Carlo Appro ximation W e appro ximate the integral using Monte Carlo sampling with M samples: Φ e ( y ) := Z p θ ( y | z ) q ϕ ( z | e ) dz (7) Algorithm : 1. Sample z (1) , . . . , z ( M ) ∼ q ϕ ( z | e ) using reparameterization: z ( m ) = µ ϕ ( e ) + σ ϕ ( e ) ⊙ ϵ ( m ) , ϵ ( m ) ∼ N (0 , I ) (8) 2. Deco de each sample: p θ ( y | z ( m ) ) for m = 1 , . . . , M 3. A v erage predictions: ˆ Φ e ( y ) = 1 M M X m =1 p θ ( y | z ( m ) ) (9) This is an unbiased estimator that conv erges almost surely as M → ∞ b y the la w of large n umbers. The reparameterization trick enables gradient ﬂow during training while maintaining sampling eﬃciency during inference. 25 Evidence Blob (text/do c) Enco der ( µ, σ ) ← L atent Posterior z 1 z 2 z 3 · · · z M R ep ar ameterization: z = µ + σ ⊙ ε Deco der ( z , pr e dic ate ) A : .6 B : .3 C : .1 A : .5 B : .4 C : .1 A : .55 B : .35 C : .10 · · · A : .6 B : .3 C : .1 Aggregate (mean) ← Monte Carlo aver aging T emp erature Scaling Final F actor { vars , Φ e ( y ) } ← Soft likeliho o d p ( y | z 1 ) p ( y | z 2 ) p ( y | z 3 ) p ( y | z M ) Figure 5: Monte Carlo Deco ding: evidence ﬂows through the enco der to pro duce ( µ, σ ) , multiple laten t co des are sampled via reparameterization, eac h is deco ded to a distribution, and the results are av eraged and temp erature-scaled to pro duce the ﬁnal soft factor. 4.6.3 T emp erature Scaling and Normalization T emperature scaling : Calibration adjustmen t Φ e ( y ) = softmax  logits ( y ) T  (10) where T is the temp erature parameter (typically 1.0). T emp erature scaling allo ws us to con trol the sharpness of the distribution: • T > 1 : Softer (reduces ov erconﬁdence) 26 • T < 1 : Sharp er (increases conﬁdence) • T = 1 : No adjustmen t Soft-factor normalization : T o maintain v alid probability p otentials, each aggregated distribu- tion is renormalized: ˜ f p ( v S p ) = f p ( v S p ) P v ′ S p f p ( v ′ S p ) (11) 4.6.4 Credibilit y W eigh ting A credibility weigh t w i ∈ [0 , 1] is attached to every factor, yielding a ﬁnal weigh ted p otential. W e use: w ( e ) = sigmoid ( − α · mean ( σ ϕ ( e ))) (12) where α > 0 controls the p enalt y strength (typically 2.0). This form ulation down w eights evidence when the p osterior exhibits high uncertaint y (broad v ariance), providing a principled wa y to assess evidence quality . The w eighted factor b ecomes: w ( e ) · ˜ f p ( v S p ) (13) These weigh ted factors b ecome leaf-no de lik eliho o ds in the SPN. 4.6.5 Con vergence and Sample Eﬃciency The Mon te Carlo estimator has standard error: SE ≈ r 0 . 25 M (14) Recommended sample sizes (from empirical v alidation): • M = 16 : SE ≈ 0.125 (fast, go o d for most applications) • M = 32 : SE ≈ 0.088 (balanced) • M = 64 : SE ≈ 0.063 (high precision) F or target error ϵ , the required n umber of samples is M = ⌈ 0 . 25 /ϵ 2 ⌉ . 4.7 LPF-SPN: Structured Aggregation 4.7.1 SPN Construction Sum-Pro duct Netw orks (SPNs) provide the probabilistic reasoning backbone of LPF-SPN. Unlike graphical mo dels that require approximate inference metho ds (MCMC, v ariational inference), SPNs enable exact marginal computation in time linear in the netw ork size through their structured decomp osition. Wh y SPNs are tractable : SPNs achiev e tractabilit y through t wo structural constrain ts. First, decomp osabilit y requires that pro duct no des combine children with disjoint v ariable scop es 27 — when computing P ( A, B ) = P ( A ) × P ( B ) , v ariables A and B m ust b e indep enden t. Second, completeness requires that all children of a sum no de range o ver the same v ariable scop e — when computing P ( X ) = w 1 P 1 ( X ) + w 2 P 2 ( X ) , b oth distributions m ust b e o ver the same v ariable X . These constrain ts enable a single b ottom-up pass through the net work: each node computes its v alue from its children without backtrac king, yielding O ( | E | ) complexity where | E | is the num ber of edges. Comparison to alternativ es : Ba yesian net w orks require v ariable elimination or b elief propaga- tion, which can b e exp onential in treewidth. Mark ov netw orks require partition function computation, whic h is #P-hard in general. V ariational methods trade exactness for appro ximation. SPNs pro vide a sw eet sp ot: exact inference with tractable complexity , at the cost of restrictions on net w ork structure. SPN structure for LPF : 1. V ariables : Create v ariable for predicate p with domain Y 2. Leaf no des : Attac h soft factor Φ e i for each evidence item as likelihoo d no des 3. Pro duct no des : Mo del indep endence assumptions b et ween factors 4. Sum no des : Create mixtures ov er evidence com binations Ro ot (Sum) Pro duct Pro duct Pro duct Φ e 1 Φ e 2 Φ e 1 Φ e 3 Φ e 2 Φ e 3 Figure 6: SPN tree with Root (Sum) branc hing in to three Product no des, eac h Product node branc hing into tw o soft factor leaf no des Φ e i . As sho wn in Figure 6 , the SPN tree illustrates how the Ro ot (Sum) no de branc hes into three Pro duct no des, each combining a pair of soft factor leav es. 4.7.2 Join t Distribution Giv en a collection of factors F = { f p j } J j =1 , the SPN deﬁnes a joint distribution ov er v ariables V as: P SPN ( V ) = X s ∈S Y f p j ∈ s f p j ( v S p j ) (15) where S is the set of v alid sum-pro duct decomp ositions satisfying decomp osability and complete- ness constraints. 4.7.3 Marginal Inference Marginal inference : P ( p = y | E ) = SPN.marginal ( p = y | { Φ e 1 , . . . , Φ e N } ) (16) 28 During inference, marginal queries are computed in time linear in the num ber of net work edges, enabling eﬃcien t exact inference o ver structured v ariables. The algorithm pro ceeds b ottom-up: leaf lik eliho o d no des return their potentials, pro duct no des multiply c hildren’s v alues, sum no des compute w eighted mixtures, and the ro ot returns the ﬁnal marginal distribution, as shown in Figure 7 . SUM ← R o ot (query variable) PR OD PR OD PR OD V AR A V AR B V AR C V AR D V AR E V AR F Lik eliho o d No de Φ e ( B ) weight: w ( e ) low: 0.55 medium: 0.35 high: 0.10 ← F r om V AE p osterior Note: V ariable sc op es must align to pr eserve de c omp osability Soft factor attachment Figure 7: SPN + Soft F actor Attac hmen t: soft likelihoo d factors from V AE p osteriors are lo cally attac hed to SPN v ariables while preserving decomp osability through prop er scop e alignment. 4.7.4 A dv an tages • Principled probabilistic reasoning : Exact marginals under mo del assumptions • T ractable inf erence : Linear time in netw ork size • In terpretable factor w eights : Clear prov enance for predictions • Sup erior calibration : Maintains prop er probability semantics through exact inference 4.8 LPF-Learned: Neural Aggregation Motiv ation : While LPF-SPN pro vides principled probabilistic reasoning, it requires explicit SPN structure deﬁnition and can b e computationally intensiv e when aggregating many evidence items. LPF-Learned oﬀers a simpler alternative that learns aggregation end-to-end, trading some in terpretability for deploymen t simplicity and computational eﬃciency . 29 4.8.1 Wh y Learn Aggregation? Hand-crafted aggregation rules (simple av eraging, v ariance-based weigh ting) make strong indep en- dence assumptions and cannot capture complex patterns in evidence interactions. Consider these scenarios that b eneﬁt from learned aggregation: 1. Corrob orating vs. contradictory evidence : T w o similar rep orts from the same source should not b e w eigh ted equally to tw o independent in v estigations reaching the same conclusion. The learned aggregator can detect this through consistency scoring. 2. Source reliabilit y patterns : Some evidence types (e.g., regulatory ﬁlings) ma y systematically b e more reliable than others (e.g., news articles), even when both ha ve similar posterior uncertain ties. The qualit y netw ork learns these patterns from data. 3. Non-linear conﬁdence interactions : The v alue of additional evidence may exhibit dimin- ishing returns — the 10th piece of consisten t evidence adds less information than the 2nd. Simple av eraging cannot capture this; learned aggregation can. 4.8.2 Arc hitecture P osteriors { q ( z | e 1 ) , . . . , q ( z | e n ) } Qualit y Netw ork Consistency Net work W eight Net w ork { q 1 , . . . , q n } C = [ c ij ] { w 1 , . . . , w n } z agg = P i w i · µ i Deco der ( z agg ) p ( y ) Figure 8: LPF-Learned aggregation: p osteriors are fed into Quality , Consistency , and W eigh t net works, whose outputs drive a weigh ted latent aggregation z agg = P i w i · µ i , deco ded to the ﬁnal distribution p ( y ) . As shown in Figure 8 , the LPF-Learned aggregation pip eline feeds p osteriors in to three net works whose outputs drive a weigh ted laten t aggregation deco ded to the ﬁnal distribution p ( y ) . 4.8.3 Qualit y Netw ork Purp ose : Assess individual evidence quality from p osterior features. 30 Input features : [ µ (64 -dim ) , log σ (64 -dim ) , mean ( σ ) (1 -dim )] = 129 -dim total Arc hitecture : Linear(129 → 64) → ReLU → Drop out(0.1) → Linear(64 → 32) → ReLU → Linear(32 → 1) → Sigmoid What do es “qualit y” mean? The quality score captures epistemic uncertain t y — how conﬁdent is the V AE enco der ab out this evidence’s meaning? Lo w-v ariance p osteriors ( σ small) indicate the enco der clearly understo o d the evidence and mapp ed it to a sp eciﬁc region of latent space. High-v ariance p osteriors ( σ large) indicate ambiguit y , contradiction, or unclear con tent. The quality net work learns to map these uncertaint y patterns to a [0 , 1] score, where high qualit y means “this evidence provides clear, unambiguous information.” Wh y these features? W e provide the netw ork with: • µ (mean v ector) : Captures the seman tic conten t — where in latent space do es this evidence p oin t? • log σ (log-v ariance) : Captures uncertaint y magnitude p er dimension • mean( σ ) (scalar) : Pro vides a global uncertaint y summary for easier learning The netw ork learns to combine these features non-linearly , p oten tially discov ering that certain regions of latent space are inheren tly more reliable or that sp eciﬁc uncertain ty patterns (e.g., high v ariance in some dimensions but not others) indicate particular quality lev els. 4.8.4 Consistency Net work Purp ose : Measure pairwise agreement b et ween evidence items. Input features : [ µ i (64 -dim ) , log σ i (64 -dim ) , µ j (64 -dim ) , log σ j (64 -dim )] = 256 -dim total Arc hitecture : Linear(256 → 128) → ReLU → Drop out(0.1) → Linear(128 → 64) → ReLU → Linear(64 → 1) → Sigmoid What do es “consistency” mean semantically? T wo pieces of evidence are consistent if they p oin t to the same conclusion — if their laten t p osteriors ov erlap signiﬁcantly . Geometrically , this means their µ v ectors are close (small Euclidean distance) and their uncertaint y regions o verlap (considering σ ). The consistency netw ork learns this notion of “agreement” from data rather than using hand-crafted distance metrics. Wh y pairwise instead of global consistency? Pairwise consistency enables ﬁne-grained mo deling: evidence item A migh t agree strongly with B but weakly with C , suggesting B is more related to A ’s p erspective. A global consistency score would lose this nuance. Ho wev er, pairwise scoring has O ( n 2 ) complexity in the num b er of evidence items — acceptable for n < 50 (typical in our setting) but p otentially exp ensiv e for v ery large evidence sets. Ho w do es the netw ork determine similarity? The netw ork has access to b oth mean v ectors and v ariances for each pair. It can learn patterns like: • Small distance b et ween µ i and µ j implies high consistency • Both having lo w v ariance implies the consistency score is more meaningful • High v ariance in one but not the other implies mo derate consistency (uncertain agreement) Through training on lab eled en tity-lev el predictions, the netw ork learns which similarit y patterns actually correlate with correct predictions. 31 4.8.5 W eigh t Net work Purp ose : Combine quality and consistency into ﬁnal aggregation weigh ts. Input features : [qualit y score (1-dim), av erage consistency (1-dim)] = 2-dim Arc hitecture : Linear(2 → 32) → ReLU → Linear(32 → 1) → Softplus → Softmax (across all evidence) Wh y softmax? Softmax normalization ensures weigh ts sum to 1, making the aggregation a prop er conv ex com bination. This is crucial b ecause we are aggregating in latent space — if w eights did not sum to 1, the aggregated laten t co de z agg could lie outside the distribution learned by the deco der, pro ducing p o orly calibrated outputs. Ho w do quality and consistency interact? The weigh t netw ork learns their relationship rather than assuming a ﬁxed com bination rule (e.g., multiplicativ e, additive). Through training, it migh t discov er patterns like: • High quality but low consistency implies mo derate weigh t (evidence is clear but contradicts others) • Mo derate quality but high consistency implies high weigh t (uncertain but corrob orated) • Low qualit y and low consistency implies very lo w weigh t (unclear and con tradictory) 4.8.6 Laten t-Space Aggregation Aggregation form ula : z agg = N X i =1 w i · µ i (17) Key decision : Aggregate b efore deco ding (at the laten t level) rather than after deco ding (at the distribution level). Wh y aggregate in latent space? This design c hoice has imp ortan t implications. A dv an tages : 1. Computational eﬃciency : Requires only one deco der call instead of N deco der calls (one p er evidence item). F or N = 10 evidence items, this is a 10 × sp eedup. 2. Smo othness : Laten t space is contin uous and smo oth — in terp olating b et w een nearby p oints yields seman tically meaningful intermediate representations. Distribution space is discrete — a veraging [0 . 9 , 0 . 1] and [0 . 1 , 0 . 9] giv es [0 . 5 , 0 . 5] , which may not corresp ond to an y coherent evidence. 3. End-to-end learning : Gradients ﬂow from the ﬁnal prediction loss through the deco der, through z agg , and into the weigh t net works, enabling true end-to-end optimization. Disadv antages : 1. Information loss : By aggregating means ( µ v alues), we discard v ariance information ( σ v alues) from individual p osteriors. F actor-lev el aggregation (as in LPF-SPN) preserves each p osterior’s full distribution through Monte Carlo sampling. 2. Limited expressiveness : The aggregated p osterior is unimo dal (single Gaussian), ev en if individual p osteriors suggest m ultimo dal p ossibilities (e.g., evidence split betw een t wo conclusions). F actor-lev el aggregation can represent multimodality through mixture-of-factors. 32 When to use latent-space aggregation : This approach excels when: • Evidence items mostly agree (low v ariance in conclusions) • Computational eﬃciency is critical (real-time applications) • End-to-end learning is desired (no SPN structure engineering) When to av oid : Use factor-level aggregation (LPF-SPN) when: • Evidence is highly con tradictory (need to preserve multimodality) • Interpretabilit y is critical (need to trace each factor’s contribution) • Calibration must b e p erfect (exact inference required) 4.8.7 Gradien t Flow and End-to-End T raining The learned aggregator enables end-to-end optimization through diﬀerentiable op erations: 1. F orw ard pass : Posteriors → Qualit y scores → Consistency matrix → W eigh ts (softmax) → z agg (w eighted sum) → Deco der → Prediction 2. Loss computation : Cross-en trop y b et ween predicted distribution and true entit y lab el 3. Bac kward pass : Gradients ﬂow from loss back through decoder to z agg to w eigh ts to qualit y/consistency netw orks The key insigh t is that all operations are diﬀerentiable: qualit y/consistency netw orks are standard neural netw orks; weigh t computation via softmax is diﬀerentiable; aggregation is a wei ghted sum (trivially diﬀerentiable); and the deco der is a neural net work. This enables the aggregator to learn optimal w eighting strategies directly from entit y-lev el sup ervision, without requiring evidence-level lab els or hand-crafted aggregation rules. 4.8.8 A dv an tages and T rade-oﬀs A dv an tages : • Simpler architecture (no SPN dep endency) • F aster inference (single deco de op eration) • Comp etitiv e p erformance (learns patterns from data) • End-to-end learnable (no structure engineering) T rade-oﬀs : • Less interpretable (learned w eights vs. explicit factors) • Loses some uncertaint y information (aggregate means only) • Cannot represent m ultimo dalit y (single p osterior output) • Requires entit y-lev el training data 33 4.9 Comparison: LPF-SPN vs. LPF-Learned Asp ect LPF-SPN LPF-Learned Aggregation After deco ding (factor-level) Before deco ding (latent-lev el) Deco der calls M × N (samples × evidence) 1 (on aggregated z ) Uncertain ty F ull distribution p er evidence Aggregate mean only Multimo dalit y Can represent via mixtures Single mo de only In terpretability High (explicit factors) Medium (learned weigh ts) Sp eed Mo derate ( M × N decodes) F ast (1 deco de) Calibration Sup erior (exact inference) Go od (learned) T raining Enco der + Deco der only + Aggregator netw orks Use case High-stakes, audit requiremen ts Deploymen t simplicity T able 3: Comparison of LPF-SPN and LPF-Learned across key design dimensions. 4.10 T raining Pro cedure 4.10.1 Stage 1: Enco der + Deco der T raining Ob jectiv e : Learn to map evidence to latent representations and deco de to predicate distributions. Loss function : L = L recon + β · L KL (18) where: • L recon = CrossEn tropy ( logits , lab els ) • L KL = KL ( q ϕ ( z | e ) ∥N (0 , I )) The KL divergence for diagonal Gaussian p osteriors is computed as: L KL = 1 2 D X d =1  σ 2 d + µ 2 d − 1 − log ( σ 2 d )  (19) F orw ard pass : 1. Enco de evidence embedding to latent p osterior: µ, σ = Enco der ( e ) 2. Sample laten t co de via reparameterization: z = µ + σ ⊙ ϵ , where ϵ ∼ N (0 , I ) 3. Deco de to predicate distribution: logits = Deco der ( z , p ) 4. Compute loss and bac kpropagate T raining details : • Dataset : Evidence-lev el lab els (each evidence item lab eled with ground truth predicate v alue) • Optimization : A dam optimizer • Learning rate : 1 × 10 − 3 34 • Batc h size : 64 • KL w eight ( β ): 0.01 (prev ents p osterior collapse while maintaining reconstruction quality) • Early stopping : Patience of 5 ep ochs on v alidation loss • Ep o c hs : Up to 100 (typically con verges in 20–30) The reconstruction loss ensures accurate predicate prediction, while KL regularization preven ts the enco der from enco ding arbitrary information in to the laten t space, maintain ing a well-struc tured laten t representation that generalizes across evidence. 4.10.2 Stage 2 (LPF-Learned only): Aggregator T raining Ob jective : Learn optimal evidence weigh ting for entit y-lev el predictions. Loss function : L agg = − log p θ ( y true | z agg ) (20) where z agg is the weigh ted aggregation of evidence p osteriors. F orw ard pass with gradient ﬂo w : 1. Compute aggregation w eights : F or each evidence p osterior, the aggregator computes qualit y scores, pairwise consistency , and combines them into w eights 2. Aggregate in laten t space : z agg = P N i =1 w i · µ i (w eighted sum of p osterior means) 3. Deco de aggregated latent : Pass z agg through deco der with predicate embedding 4. Compute loss : Negativ e log-lik eliho o d of true lab el The key insight is that w eights w i are computed by diﬀerentiable netw orks, so gradien ts ﬂo w from the ﬁnal loss through the deco der, through the aggregated latent co de, and into the weigh t net works. This enables end-to-end learning of evidence combination. T raining details : • Dataset : En tity-lev el lab els (ground truth for en tire companies/entities) • Input : F rozen enco der p osteriors from Stage 1 • T rainable : Only aggregator netw orks (quality , consistency , weigh t) • Optimization : A dam optimizer • Learning rate : 1 × 10 − 3 • Batc h size : 32 (en tities, not individual evidence) • Ep o c hs : 30 Wh y freeze enco der/deco der? The enco der and deco der are already trained to pro duce w ell-calibrated evidence-level predictions. F reezing them allo ws the aggregator to fo cus exclusively on learning how to c ombine evidence, rather than re-learning evidence in terpretation. This staged training preven ts interference b et w een the tw o ob jectives. 35 4.10.3 Seed Searc h Strategy T o ensure robust results across random initializations, we employ a systematic seed search. Searc h proto col : • Seeds tested : [42, 123, 456, 789, 1011, 2024, 2025] (7 seeds) • Selection criterion : Best v alidation accuracy • Deplo yment : Use mo del from b est seed • Rep orting : Mean ± std across all seeds Output trac king : F or each seed, we record: • Final train/v al accuracy and loss • Best v alidation accuracy across all ep o c hs • Number of ep ochs until conv ergence • F ull training history This approac h balances computational cost with statistical rigor, pro viding conﬁdence interv als for rep orted metrics and ensuring our deploymen t uses the b est-p erforming initialization. 5 W ork ed Example: T ax Compliance Risk Assessment This chapter presents a complete, step-by-step walkthrough of LPF inference for a realistic query . W e demonstrate b oth arc hitectural v ariants (LPF-SPN and LPF-Learned) with full numerical details, allo wing readers to trace every computation from raw evidence to ﬁnal prediction. 5.1 Problem Setup Query: “What is the tax compliance risk level for Company C0001?” W e treat the predicate compliance_level whic h takes categorical v alues from the domain {low, medium, high} . T w o op erational mo des: • Case A (canonical fast path): The canonical database contains a fresh, authoritative v alue for compliance_level for C0001 (e.g., “high” with conﬁdence 1.0). The orchestrator returns this immediately without inference. • Case B (inference path): No recent canonical v alue exists — the system must collect evidence, enco de uncertaint y , and p erform probabilistic reasoning. W e fo cus on Case B as it demonstrates the complete LPF pipeline and mathematical framew ork. 36 5.2 Evidence Collection The evidence retriev al system returns 5 evidence items for Company C0001: • e 1 : T ax audit rep ort (credibilit y: 0.95) “Comp any demonstr ates str ong c omplianc e with timely ﬁlings and ac cur ate r e c or d-ke eping.” • e 2 : In ternal ﬁnancial review (credibilit y: 0.91) “Exc el lent do cumentation pr actic es observe d acr oss al l dep artments.” • e 3 : Regulatory ﬁling analysis (credibility: 0.87) “Consistently me ets al l statutory r e quir ements with zer o late ﬁlings.” • e 4 : Industry b enc hmark comparison (credibility: 0.85) “F ol lows industry b est pr actic es for tax c omplianc e and r ep orting.” • e 5 : Certiﬁcation status chec k (credibility: 0.93) “Maintains ISO 27001 c ertiﬁc ation and demonstr ates r obust internal c ontr ols.” Eac h evidence piece is embedded using Sentence-BER T (384-dimensional vectors) and indexed b y F AISS for eﬃcient retriev al. 5.3 V AE Enco ding to Laten t Posteriors The V AE enco der maps eac h evidence em b edding to a latent p osterior distribution q ϕ ( z | e ) , represen ted by mean vector µ and log-v ariance vector log σ 2 . W e use a 64-dimensional latent space for computational tractability . Arc hitecture: Em b edding [384] → MLP [256, 128] → ( µ, log σ 2 ) [64 each] Enco der outputs (illustrative v alues): e1: mu = [ 0.82, -0.34, 1.21, ..., 0.45] (64-dim) sigma = [ 0.12, 0.08, 0.15, ..., 0.10] (64-dim) confidence = 1/(1 + mean(sigma)) = 1/(1 + 0.105) ~ 0.89 e2: mu = [ 0.78, -0.29, 1.18, ..., 0.42] sigma = [ 0.10, 0.07, 0.12, ..., 0.09] confidence ~ 0.91 e3: mu = [ 0.75, -0.31, 1.15, ..., 0.40] sigma = [ 0.14, 0.10, 0.17, ..., 0.12] confidence ~ 0.87 e4: mu = [ 0.71, -0.28, 1.12, ..., 0.38] sigma = [ 0.16, 0.12, 0.19, ..., 0.14] confidence ~ 0.85 e5: mu = [ 0.85, -0.36, 1.24, ..., 0.47] sigma = [ 0.09, 0.06, 0.11, ..., 0.08] confidence ~ 0.93 In terpretation: Evidence with lo wer v ariance (smaller σ ) receives higher conﬁdence scores, reﬂecting the V AE’s uncertaint y estimate ab out the latent representation. 37 5.4 LPF-SPN Arc hitecture: F actor Conv ersion via Monte Carlo In the LPF-SPN v ariant, w e conv ert each latent p osterior into a soft likelihoo d factor using Monte Carlo sampling and the deco der netw ork. Hyp erparameters: • Latent dimension: z _ dim = 64 • Monte Carlo samples: n _ samples = 16 • T emp erature: T = 1 . 0 (no scaling in this example) • W eigh t p enalt y: α = 2 . 0 5.4.1 Step-b y-step conv ersion for evidence e 1 A. Mon te Carlo Sampling (Reparameterization T ric k) W e dra w 16 samples from q ( z | e 1 ) using: z ( m ) = µ + σ ⊙ ϵ ( m ) , ϵ ( m ) ∼ N (0 , I ) (21) where ⊙ denotes element-wise multiplication. Sample laten t v ectors (ﬁrst 3 shown): z(1) = [ 0.93, -0.28, 1.35, ..., 0.53] z(2) = [ 0.79, -0.41, 1.18, ..., 0.42] z(3) = [ 0.86, -0.32, 1.27, ..., 0.48] ... z(16) = [ 0.81, -0.36, 1.23, ..., 0.46] B. Deco de Each Sample F or eac h z ( m ) , w e compute p θ ( y | z ( m ) , compliance_level ) using the deco der netw ork: Deco der architecture: 1. Concatenate: [ z ( m ) (64-dim), predicate_emb(“compliance_lev el”) (32-dim)] → 96-dim input 2. MLP: [96] → [128] → [64] 3. Output head: [64] → [3] (for 3 classes) 4. Softmax: logits → p θ ( y | z ( m ) ) Individual sample distributions (ﬁrst 5 sho wn): p(1) = {low: 0.05, medium: 0.15, high: 0.80} p(2) = {low: 0.03, medium: 0.12, high: 0.85} p(3) = {low: 0.06, medium: 0.18, high: 0.76} p(4) = {low: 0.04, medium: 0.14, high: 0.82} p(5) = {low: 0.07, medium: 0.19, high: 0.74} ... 38 C. Mon te Carlo Aggregation W e estimate the exp ected distribution by av eraging: ˆ Φ e 1 ( y ) = 1 M M X m =1 p θ ( y | z ( m ) ) (22) With M = 16 : ˆ Φ e 1 = { low : 0 . 048 , medium : 0 . 155 , hig h : 0 . 797 } (23) Standard error estimate: F or binary outcomes, MC v ariance is b ounded b y 1 4 M , giving standard error ≈ q 0 . 25 16 = 0 . 125 . F or our 16 samples, we exp ect reasonable stability . D. T emperature Scaling With temp erature T = 1 . 0 , no scaling is applied: Φ T e 1 ( y ) = ( ˆ Φ e 1 ( y )) 1 /T P y ′ ( ˆ Φ e 1 ( y ′ )) 1 /T = ˆ Φ e 1 ( y ) (24) Result: Φ T e 1 = { low : 0 . 048 , medium : 0 . 155 , hig h : 0 . 797 } E. Credibilit y W eigh t Computation W e compute a w eight that down-w eigh ts uncertain evidence: mean_sigma = 1 d d X i =1 σ i = 0 . 105 (25) calibration = 1 1 + exp( α · mean_sigma ) = 1 1 + exp(2 . 0 × 0 . 105) ≈ 0 . 79 (26) w e 1 = conﬁdence × calibration = 0 . 89 × 0 . 79 ≈ 0 . 70 (27) F. W eigh ted F actor (P ow er T ransform) The ﬁnal soft factor applies the weigh t via exp onen tiation: ˜ Φ e 1 ( y ) = (Φ T e 1 ( y )) w e 1 P y ′ (Φ T e 1 ( y ′ )) w e 1 (28) Computing element-wise: (0.048)^0.70 ~ 0.127 (0.155)^0.70 ~ 0.285 (0.797)^0.70 ~ 0.863 Normalizing: Z = 0 . 127 + 0 . 285 + 0 . 863 = 1 . 275 ˜ Φ e 1 =  low : 0 . 127 1 . 275 ≈ 0 . 100 , medium : 0 . 285 1 . 275 ≈ 0 . 223 , hig h : 0 . 863 1 . 275 ≈ 0 . 677  (29) 39 5.4.2 F actors for all evidence items Rep eating the pro cess for e 2 , e 3 , e 4 , e 5 : Factor e1: {low: 0.100, medium: 0.223, high: 0.677}, weight: 0.70 Factor e2: {low: 0.092, medium: 0.211, high: 0.697}, weight: 0.73 Factor e3: {low: 0.112, medium: 0.238, high: 0.650}, weight: 0.68 Factor e4: {low: 0.125, medium: 0.251, high: 0.624}, weight: 0.66 Factor e5: {low: 0.085, medium: 0.198, high: 0.717}, weight: 0.75 5.5 LPF-SPN: Ba y esian Inference with Sum-Pro duct Net works 5.5.1 Prior Distribution The sc hema deﬁnes a uniform prior ov er compliance_level : P 0 ( compliance_lev el ) = { l ow : 0 . 333 , medium : 0 . 333 , hig h : 0 . 333 } (30) In practice, priors could b e learned from historical data or set by domain exp erts. 5.5.2 SPN Structure F or a single-v ariable predicate, the SPN is a simple pro duct of the prior and likelihoo d factors: 5.5.3 Marginal Inference F or eac h v alue y ∈ { l ow , medium, hig h } , we compute: P ( y | evidence ) ∝ P 0 ( y ) × 5 Y i =1 ˜ Φ e i ( y ) (31) F or y = “low” : P ( low ) ∝ 0 . 333 × 0 . 100 × 0 . 092 × 0 . 112 × 0 . 125 × 0 . 085 = 0 . 333 × 1 . 09 × 10 − 5 ≈ 3 . 63 × 10 − 6 (32) F or y = “medium” : P ( medium ) ∝ 0 . 333 × 0 . 223 × 0 . 211 × 0 . 238 × 0 . 251 × 0 . 198 = 0 . 333 × 6 . 23 × 10 − 4 ≈ 2 . 07 × 10 − 4 (33) F or y = “high” : P ( hig h ) ∝ 0 . 333 × 0 . 677 × 0 . 697 × 0 . 650 × 0 . 624 × 0 . 717 = 0 . 333 × 0 . 1312 ≈ 4 . 37 × 10 − 2 (34) Normalization: Z = 3 . 63 × 10 − 6 + 2 . 07 × 10 − 4 + 4 . 37 × 10 − 2 ≈ 0 . 0439 (35) P ( compliance_lev el | evidence ) =                low : 3 . 63 × 10 − 6 0 . 0439 ≈ 0 . 0001 medium : 2 . 07 × 10 − 4 0 . 0439 ≈ 0 . 0047 hig h : 4 . 37 × 10 − 2 0 . 0439 ≈ 0 . 9952                (36) 40 5.5.4 LPF-SPN Final Result Distribution: {low: 0.0001, medium: 0.0047, high: 0.9952} Top value: "high" Confidence: 0.9952 Execution time: 3.3ms Ground truth: “high” ✓ CORRECT 5.6 LPF-Learned Arc hitecture: Neural Evidence Aggregation The LPF-Learned v arian t tak es a fundamentally diﬀerent approac h: instead of conv erting evidence to factors and reasoning with an SPN, it learns to aggregate evidence in the latent space b efore deco ding. Arc hitecture Ov erview: VAE Encoder -> Multiple Posteriors -> Learned Aggregator -> Single Posterior -> Decoder -> Distribution 5.6.1 Step 1: Quality Score Computation The qualit y netw ork assesses each p osterior’s reliabilit y based on its uncertaint y: Qualit y Net work: [ µ, log σ 2 ] (128-dim) → MLP [128, 64] → Sigmoid → quality ∈ [0 , 1] F or our 5 evidence items: Quality(e1) = QualityNet([mu1, logvar1]) = 0.92 Quality(e2) = QualityNet([mu2, logvar2]) = 0.89 Quality(e3) = QualityNet([mu3, logvar3]) = 0.85 Quality(e4) = QualityNet([mu4, logvar4]) = 0.81 Quality(e5) = QualityNet([mu5, logvar5]) = 0.94 In terpretation: Evidence with low er v ariance ( e 5 , e 1 ) receiv es higher quality scores. 5.6.2 Step 2: Pairwise Consistency Matrix The consistency netw ork measures how w ell evidence items agree b y comparing their latent represen- tations: Consistency Net w ork: [ µ i − µ j , | log σ 2 i − log σ 2 j | ] (128-dim) → MLP [128, 64] → Sigmoid → consistency ∈ [0 , 1] Consistency matrix C (5 × 5): e1 e2 e3 e4 e5 e1 [ 1.00 0.87 0.82 0.79 0.91 ] e2 [ 0.87 1.00 0.85 0.76 0.89 ] e3 [ 0.82 0.85 1.00 0.88 0.84 ] e4 [ 0.79 0.76 0.88 1.00 0.81 ] e5 [ 0.91 0.89 0.84 0.81 1.00 ] A v erage consistency p er evidence (excluding self ): 41 avg_consistency(e1) = mean([0.87, 0.82, 0.79, 0.91]) = 0.848 avg_consistency(e2) = mean([0.87, 0.85, 0.76, 0.89]) = 0.843 avg_consistency(e3) = mean([0.82, 0.85, 0.88, 0.84]) = 0.848 avg_consistency(e4) = mean([0.79, 0.76, 0.88, 0.81]) = 0.810 avg_consistency(e5) = mean([0.91, 0.89, 0.84, 0.81]) = 0.863 In terpretation: Evidence e 5 has the highest av erage consistency (0.863), indicating it aligns w ell with other evidence. Evidence e 4 has lo w er consistency (0.810), p oten tially indicating it captures diﬀeren t asp ects. 5.6.3 Step 3: Final W eigh t Computation The w eight netw ork combines quality and consistency into ﬁnal aggregation w eights: W eigh t Net w ork: [ q uality , av g _ consistency ] (2-dim) → MLP [64, 32] → Softplus → ra w_weigh t > 0 raw_weight(e1) = WeightNet([0.92, 0.848]) = 0.88 raw_weight(e2) = WeightNet([0.89, 0.843]) = 0.85 raw_weight(e3) = WeightNet([0.85, 0.848]) = 0.82 raw_weight(e4) = WeightNet([0.81, 0.810]) = 0.75 raw_weight(e5) = WeightNet([0.94, 0.863]) = 0.90 Softmax normalization: w i = exp( ra w_weigh t i ) P 5 j =1 exp( ra w_weigh t j ) (37) Normalized weights: w1 = 0.217 (21.7%) w2 = 0.209 (20.9%) w3 = 0.203 (20.3%) w4 = 0.185 (18.5%) w5 = 0.223 (22.3%) In terpretation: Evidence e 5 (highest quality , highest consistency) receives the largest weigh t (22.3%). Evidence e 4 (lo west quality and consistency) receiv es the smallest weigh t (18.5%). 5.6.4 Step 4: Latent Space Aggregation W e compute a w eighted av erage of the latent p osteriors: µ ag g = 5 X i =1 w i · µ i = 0 . 217 × µ 1 + 0 . 209 × µ 2 + 0 . 203 × µ 3 + 0 . 185 × µ 4 + 0 . 223 × µ 5 (38) Elemen t-wise computation (ﬁrst 3 dimensions shown): Dimension 1: 0 . 217 × 0 . 82+0 . 209 × 0 . 78+0 . 203 × 0 . 75+0 . 185 × 0 . 71+0 . 223 × 0 . 85 = 0 . 178+0 . 163+0 . 152+0 . 131+0 . 190 = 0 . 814 (39) 42 Dimension 2: 0 . 217 × ( − 0 . 34) + 0 . 209 × ( − 0 . 29) + 0 . 203 × ( − 0 . 31) + 0 . 185 × ( − 0 . 28) + 0 . 223 × ( − 0 . 36) = − 0 . 330 (40) Dimension 3: 0 . 217 × 1 . 21+0 . 209 × 1 . 18+0 . 203 × 1 . 15+0 . 185 × 1 . 12+0 . 223 × 1 . 24 = 0 . 263+0 . 247+0 . 233+0 . 207+0 . 277 = 1 . 227 (41) Con tinuing for all 64 dimensions, we obtain the aggregated mean: µ ag g = [0 . 814 , − 0 . 330 , 1 . 227 , . . . , 0 . 446] (42) whic h is a 64-dimensional vector. Similarly for log-v ariance: log σ 2 ag g = 5 X i =1 w i · log σ 2 i (43) This is an approximation; exact v ariance com bination for indep enden t Gaussians would use σ 2 ag g = P w 2 i σ 2 i , but the learned approach treats this as a learned aggregation rule. 5.6.5 Step 5: Deco de Aggregated Posterior Single deco ding pass (unlike LPF-SPN’s 80 deco der calls): 1. Extract mean: z = µ ag g = [0 . 78 , − 0 . 29 , . . . , 0 . 41] (64-dim) 2. Get predicate em b edding: pred_em b = PredicateEmbedding ( compliance_level ) (32-dim) 3. Concatenate: x = [ z , pred_em b ] (96-dim) 4. MLP deco der: h = MLP ( x ) [96] → [128] → [64] 5. Output head: logits = OutputHead ( h ) [64] → [3] 6. Softmax: probs = softmax ( logits ) Output distribution: P ( compliance_lev el | z ag g ) = { low : 0 . 019 , medium : 0 . 067 , hig h : 0 . 914 } (44) 5.6.6 LPF-Learned Final Result Distribution: {low: 0.019, medium: 0.067, high: 0.914} Top value: "high" Confidence: 0.914 Execution time: 5.1ms Aggregation weights: [0.217, 0.209, 0.203, 0.185, 0.223] Ground truth: “high” ✓ CORRECT 43 5.7 Comparison: LPF-SPN vs. LPF-Learned Metric LPF-SPN LPF-Learned Final prediction high (99.5%) high (91.4%) Deco der calls 80 (5 × 16) 1 Aggregation stage After deco ding Before deco ding Aggregation metho d Probabilistic Neural learned Execution time 3.3ms 5.1ms In terpretability High (factors) Medium (w eights) T able 4: Comparison of LPF-SPN and LPF-Learned on the tax compliance work ed example. Key observ ations: 1. Conﬁdence diﬀerence: LPF-SPN pro duces higher conﬁdence (99.5% vs 91.4%) b ecause the pro duct of factors comp ounds agreemen t. LPF-Learned uses a weigh ted av erage which is inheren tly more conserv ativ e. 2. Eﬃciency: LPF-SPN is faster despite more deco der calls b ecause SPN inference is highly op- timized with cached structures. LPF-Learned has ov erhead from quality/consistency netw orks. 3. In terpretability: LPF-SPN provides explicit soft factors with weigh ts, making it clear ho w eac h evidence con tributed. LPF-Learned provides learned weigh ts but the qualit y/consistency scores are less transparent. 4. Both architectures make correct predictions for this example, but p erformance diﬀers across metrics (see Chapter 12 for comprehensive ev aluation). 5.8 Design Choices Explained 5.8.1 Wh y exp onen tiate the distribution by weigh t? Raising a probabilit y distribution to a p o wer w ∈ (0 , 1) is a principled uncertain ty damp ening mec hanism: ˜ Φ e ( y ) = (Φ e ( y )) w P y ′ (Φ e ( y ′ )) w (45) Mathematical justiﬁcation: This is equiv alen t to temp ering the log-likelihoo d: log ˜ Φ e ( y ) = w · log Φ e ( y ) − log Z (46) When w < 1 , we reduce the eﬀective strength of the evidence. This prev ents o verconﬁden t predictions from uncertain evidence. In tuition: If evidence has high v ariance ( w ≈ 0 . 5 ), the exp onen tiation ﬂattens the distribution to ward uniform, appropriately expressing uncertaint y . 44 5.8.2 Wh y use temp erature scaling? T emp erature T > 1 softens probability distributions: p T ( y ) = exp(log p ( y ) /T ) P y ′ exp(log p ( y ′ ) /T ) (47) Calibration b eneﬁt: Neural netw orks often pro duce o verconﬁden t predictions. T emperature scaling is a p ost-ho c calibration metho d that impro ves Exp ected Calibration Error (ECE). Hyp erparameter tuning: T emperature should b e chosen on a held-out v alidation set to minimize negative log-likelihoo d or ECE. In our exp erimen ts, w e found T ∈ [0 . 8 , 1 . 2] works well. 5.8.3 Wh y multiply prior b y pro duct of factors? This follows from Bay es’ rule with conditional indep endence assumptions: P ( y | e 1 , . . . , e n ) ∝ P ( y ) × n Y i =1 P ( e i | y ) (48) Assumption: Evidence items are conditionally indep enden t given the query v ariable. This is a standard assumption in naive Bay es and sum-pro duct netw ork inference. When violated: If evidence items are correlated, the product can ov er-coun t shared information. The LPF-Learned v ariant addresses this b y explicitly mo deling consistency in the aggregator. 5.8.4 Wh y aggregate in laten t space (LPF-Learned)? Computational eﬃciency: Aggregating b efore deco ding requires only 1 deco der call instead of n × M calls (where n = num b er of evidence, M = MC samples). Learning to aggregate: The neural aggregator learns task-sp eciﬁc combination rules from data, p otentially capturing non-linear in teractions that Bay esian pro duct might miss. T rade-oﬀ: Loses the probabilistic interpretation of the SPN approac h but gains ﬂexibility and eﬃciency . 5.8.5 Wh y use qualit y and consistency net w orks? Qualit y netw ork: Assesses single-evidence reliability based on its latent uncertaint y . Lo w v ariance implies high quality . Consistency net w ork: Detects contradictions b y measuring pairwise agreemen t. Evidence that con tradicts others receives low er consistency scores. Com bined approac h: W eights reﬂect b oth intrinsic quality (how conﬁden t is the V AE?) and extrinsic agreement (do other evidence supp ort this?). This mirrors h uman reasoning — w e trust sources that are b oth internally consisten t and agree with others. 5.8.6 Wh y Monte Carlo with 16 samples? V ariance-bias trade-oﬀ: MC estimation has v ariance σ 2 / M . F or M = 16 : Standard error ≈ r 0 . 25 16 = 0 . 125 (49) Empirical v alidation: In our exp erimen ts, increasing from M = 16 to M = 64 improv ed accuracy by < 1% while increasing latency by 4 × . W e c hose M = 16 as a practical balance. 45 Reparameterization tric k: Sampling z = µ + σ ⊙ ϵ with ϵ ∼ N (0 , I ) enables gradient ﬂow through sampling, which is critical for end-to-end training. 5.8.7 Wh y log-space computation in SPN? Sum-pro duct netw orks p erform inference in log-space to prev ent numerical underﬂow: log P ( y ) = log P 0 ( y ) + n X i =1 log ˜ Φ e i ( y ) (50) Numerical stability: Multiplying man y small probabilities can underﬂow to zero in ﬂoat32. Log-space addition av oids this. Logsumexp tric k: F or normalization, we use: log Z = logsumexp ( { log P ( y ′ ) : y ′ ∈ domain } ) (51) where logsumexp ( x ) = log ( P exp( x i )) is computed stably by factoring out the maximum. 5.9 Mon te Carlo Sample Size Analysis W e analyze the trade-oﬀ b et w een accuracy and computational cost for diﬀerent sample sizes. Theorem 5.1 (MC V ariance Bound) . F or a Bernoul li r andom variable, the varianc e of the MC estimator is b ounde d by: V ar [ ˆ p ] ≤ 1 4 M (52) Standard error by sample size: Samples ( M ) Std Error Relativ e Error (%) 4 0.250 25% 16 0.125 12.5% 32 0.088 8.8% 64 0.063 6.3% 128 0.044 4.4% T able 5: Mon te Carlo standard error vs. sample size. Empirical results (1000 companies, compliance_lev el): M A ccuracy ECE Latency (ms) 4 83.2% 0.089 1.8 16 86.1% 0.052 3.3 32 86.7% 0.045 5.9 64 86.9% 0.041 11.2 T able 6: A ccuracy , calibration, and latency vs. Monte Carlo sample size M . Recommendation: Use M ∈ [16 , 32] for production systems. The marginal accuracy gain b ey ond M = 32 is < 0.5% while latency increases linearly . 46 5.10 Sensitivit y Analysis W e examine ho w key hyperparameters aﬀect ﬁnal predictions. 5.10.1 T emp erature sensitivity ( α = 2 . 0 , M = 16 ) T emperature ( T ) P ( hig h ) ECE Notes 0.5 0.998 0.112 Over-conﬁden t 0.8 0.972 0.037 W ell-calibrated 1.0 0.952 0.042 Baseline 1.5 0.891 0.063 Under-conﬁdent 2.0 0.812 0.089 T o o soft T able 7: Eﬀect of temp erature T on prediction conﬁdence and calibration. Observ ation: T ∈ [0 . 8 , 1 . 2] provides the best calibration. Low er temperatures create ov erconﬁdence; higher temp eratures wash out signal. 5.10.2 W eigh t p enalt y sensitivit y ( T = 1 . 0 , M = 16 ) P enalty ( α ) w (high-conf ) w (lo w-conf ) P ( hig h ) Notes 0.0 0.89 0.85 0.947 No p enalty , uniform weigh ts 1.0 0.84 0.71 0.951 Mild down-w eighting 2.0 0.70 0.53 0.952 Baseline (used in example) 4.0 0.51 0.31 0.961 Aggressiv e uncertaint y p enalty T able 8: Eﬀect of uncertaint y p enalt y α on evidence w eights and ﬁnal prediction. Observ ation: Higher α more aggressively down-w eigh ts uncertain evidence. The eﬀect on ﬁnal predictions is mo dest for high-agreement scenarios but b ecomes critical when evidence conﬂicts. 5.11 Theoretical Prop erties Theorem 5.2 (SPN Consistency) . If al l evidenc e unanimously supp orts value y ∗ with weights w i → 1 , then P ( y ∗ | evidenc e ) → 1 as n → ∞ . Pr o of. In the limit of p erfect evidence ( Φ i ( y ∗ ) → 1 , Φ i ( y  = y ∗ ) → 0 ) with full weigh t ( w i → 1 ), the pro duct: n Y i =1 ˜ Φ i ( y ∗ ) = n Y i =1 Φ i ( y ∗ ) w i → 1 (53) while for y  = y ∗ : n Y i =1 ˜ Φ i ( y ) = n Y i =1 ϵ w i i → 0 (54) where ϵ i ≪ 1 . After normalization, P ( y ∗ | evidence ) → 1 . ■ 47 Theorem 5.3 (Aggregator Optimality) . The le arne d aggr e gator minimizes exp e cte d cr oss-entr opy loss under the tr aining distribution if and only if it assigns weight pr op ortional to evidenc e informativeness. Pr o of. The training ob jectiv e is: L = − E ( e ,y ) [log p θ ( y | z agg ( e ))] (55) where z agg ( e ) = P i w i ( e ) µ i . By calculus of v ariations, the optimal weigh ts satisfy: w ∗ i ∝ ∂ ∂ µ i log p θ ( y | z ) (56) This gradient is large when evidence i is informative ab out y and small when it is noisy or irrelev an t. The qualit y and consistency netw orks approximate this via learned features. ■ Theorem 5.4 (Contradiction Handling) . When evidenc e items c ontr adict ( µ i ≈ − µ j ), the c onsistency network assigns low C ij , r e ducing b oth weights and damp ening the aggr e gate d signal. 5.12 Con v ergence of Monte Carlo Estimator The factor conv ersion relies on MC estimation: ˆ Φ e ( y ) = 1 M P M m =1 p θ ( y | z ( m ) ) . Theorem 5.5 (MC Conv ergence) . By the L aw of L ar ge Numb ers, ˆ Φ e ( y ) M →∞ − − − − → E z ∼ q ( z | e ) [ p θ ( y | z )] = Z q ( z | e ) p θ ( y | z ) dz (57) almost sur ely. The Centr al Limit The or em gives: √ M ( ˆ Φ e ( y ) − Φ e ( y )) d − → N (0 , σ 2 ) (58) wher e σ 2 = V ar z ∼ q [ p θ ( y | z )] ≤ 1 4 for pr ob abilities. Practical implication: With M = 16 , the standard error is ≈ 0 . 125 , which is acceptable for probabilistic reasoning where we ultimately normalize ov er the domain. The SPN’s pro duct op eration further smo oths MC noise across multiple evidence items. 5.13 Comparison Summary W e conclude with a comprehensiv e comparison of b oth architectural v ariants on our work ed example. 5.13.1 Quan titative Comparison Metric LPF-SPN LPF-Learned Prediction high high Conﬁdence 99.52% 91.40% A ccuracy (test set) 97.78% 91.11% ECE (calibration) 0.0137 0.0660 Brier score 0.0150 0.0404 Macro F1 0.9724 0.9052 Run time (ms) 14.8 37.4 T able 9: Quan titativ e comparison of LPF-SPN and LPF-Learned on the compliance w orked example. 48 5.13.2 Qualitativ e Comparison LPF-SPN strengths: • Sup erior calibration (ECE: 0.014 vs 0.066) • Explicit probabilistic reasoning with in terpretable factors • F aster inference despite more deco der calls (optimized SPN) • Theoretically grounded in Bay esian inference LPF-Learned strengths: • Learns task-sp eciﬁc aggregation from data • Automatically handles evidence correlation via consistency netw ork • Single deco der call (simpler pip eline) • More robust to arc hitectural changes (no SPN structure needed) When to use which: • LPF-SPN: When calibration is critical (medical, ﬁnancial), when interpretabilit y matters, when domain structure is well understo o d. • LPF-Learned: When computational constraints are sev ere, when evidence correlations are complex, when end-to-end learning is preferred. 5.14 Summary and Key T akea w a ys This work ed example has demonstrated the complete LPF inference pip eline from raw evidence to calibrated predictions. Both arc hitectural v arian ts successfully classify Company C0001 as high compliance risk, but through fundamentally diﬀerent mechanisms. LPF-SPN con verts each evidence item into a soft probabilistic factor, then combines them via structured Ba yesian inference. This yields sup erior calibration (ECE: 0.014) and in terpretability , at the cost of architectural complexity . LPF-Learned aggregates evidence in laten t space using learned quality and consistency net works, then deco des once. This is conceptually simpler and more ﬂexible, but sacriﬁces some calibration qualit y (ECE: 0.066). The mathematical framework reveals why b oth approaches work: • V AE enco ding captures epistemic uncertain ty in evidence • W eight mechanisms (exp onen tial or learned) down-w eight uncertain evidence • Aggregation (pro duct or weigh ted a verage) combines signals appropriately • Calibrated deco ding pro duces well-formed probability distributions 49 Practical recommendation: F or high-stakes applications requiring sup erior calibration and in terpretability (medical diagnosis, ﬁnancial risk assessment, regulatory compliance), use LPF- SPN (ECE: 0.014, 97.78% accuracy , explicit probabilistic factors). F or applications prioritizing computational eﬃciency or where end-to-end learning is preferred, use LPF-Learned (ECE: 0.066, 91.11% accuracy , simpler pip eline). Both v ariants provide exact epistemic/aleatoric uncertaint y decomp osition, ensuring trustw orth y AI systems. The full exp erimental ev aluation across m ultiple domains and metrics app ears in Chapter [chapter n umber]. 6 Algorithms This section presents the core algorithms underlying the Laten t Posterior F actors (LPF) framework. W e pro vide pseudoco de for the key computational pro cedures, organized by their role in the inference pip eline. The algorithms are presented in tw o v ariants: LPF-SPN (using Sum-Pro duct Netw orks) and LPF-Learned (using learned evidence aggregation). 6.1 Algorithm 1: Con v ertLatentT oF actors Input: latent_posteriors , predicate , schema , decoder_network , n_samples , temperature Output: soft_factors 1: soft_factors <- [] 2: for each posterior in latent_posteriors do 3: eid <- posterior.evidence_id 4: mu <- posterior.mu 5: sigma <- posterior.sigma 6: base_conf <- posterior.confidence 7: 8: // Draw reparameterized samples 9: z_samples <- [] 10: for k = 1 to n_samples do 11: eps ~ N(0, I) 12: z_k <- mu + sigma * eps 13: z_samples.append(z_k) 14: end for 15: 16: // Decode each sample 17: pred_dists <- [] 18: for each z in z_samples do 19: p_z <- decoder_network.decode(z, predicate) 20: pred_dists.append(p_z) 21: end for 22: 23: // Aggregate distributions via Monte Carlo 24: aggregated <- {} 25: for each key in pred_dists[0].keys() do 26: aggregated[key] <- mean({d[key] for d in pred_dists}) 27: end for 50 28: 29: // Temperature scaling 30: if temperature != 1.0 then 31: for each k in keys do 32: aggregated[k] <- aggregated[k]^(1/temperature) 33: end for 34: end if 35: total <- sum(aggregated.values()) + eps_small 36: aggregated <- {k: aggregated[k] / total for k in keys} 37: 38: // Compute credibility weight 39: weight <- base_conf * CalibrationWeight(sigma) 40: 41: // Build factor 42: variables <- schema.GetVariablesForPredicate(predicate) 43: factor <- { 44: evidence_id: eid, 45: variables: variables, 46: factor_type: "likelihood", 47: potential: aggregated, 48: weight: weight, 49: metadata: {...} 50: } 51: soft_factors.append(factor) 52: end for 53: return soft_factors Purp ose: Con v erts V AE laten t p osteriors into soft likelihoo d factors for probabilistic reasoning. This implements the Monte Carlo approximation of the integral Φ e ( y ) = R p θ ( y | z ) q ϕ ( z | e ) dz . Key Op erations: • Lines 8–14: Reparameterization trick for diﬀerentiable sampling from q ϕ ( z | e ) • Lines 16–21: Decode each latent sample to predicate distributions • Lines 23–27: Mon te Carlo a veraging to approximate the exp ectation • Lines 29–36: Optional temp erature scaling to con trol conﬁdence calibration • Lines 38–39: Uncertain t y-aw are w eight computation using sigmoid p enalt y 6.2 Algorithm 2: V AEEncoder.Enco de Input: evidence_ids , EvidenceIndex , encoder_net , embedding_model Output: posteriors 1: posteriors <- [] 2: for each eid in evidence_ids do 3: meta <- EvidenceIndex.FetchMeta(eid) 4: if "embedding_id" in meta then 51 5: embedding <- VectorStore.Fetch(meta.embedding_id) 6: else 7: raw <- EvidenceIndex.FetchRaw(eid) 8: embedding <- embedding_model.Encode(raw) 9: end if 10: 11: mu, log_sigma <- encoder_net(embedding) 12: sigma <- exp(log_sigma) 13: confidence <- 1.0 / (1.0 + mean(sigma)) 14: 15: posteriors.append({ 16: evidence_id: eid, 17: mu: mu, 18: sigma: sigma, 19: confidence: confidence 20: }) 21: end for 22: return posteriors Purp ose: Enco des evidence items into laten t p osteriors using the V AE enco der net work. Key Op erations: • Lines 3–9: Retrieves pre-computed embeddings when av ailable, otherwise computes them on-the-ﬂy • Lines 11–12: Neural enco ding to diagonal Gaussian parameters ( µ, σ ) • Line 13: Base conﬁdence computed from uncertaint y (low er v ariance = higher conﬁdence) 6.3 Algorithm 3: LearnedAggregator.Aggregate (LPF-Learned) Input: posteriors , predicate , aggregator_network , decoder_network Output: distribution 1: n <- length(posteriors) 2: 3: // Compute quality scores 4: quality <- [] 5: for each p in posteriors do 6: features <- concat(p.mu, p.logvar) 7: q_score <- quality_net(features) 8: quality.append(q_score) 9: end for 10: 11: // Compute pairwise consistency 12: consistency <- zeros(n, n) 13: for i = 1 to n do 14: for j = 1 to n do 15: if i = j then 16: consistency[i,j] <- 1.0 52 17: else 18: diff_mu <- posteriors[i].mu - posteriors[j].mu 19: diff_var <- |posteriors[i].logvar - posteriors[j].logvar| 20: features <- concat(diff_mu, diff_var) 21: consistency[i,j] <- consistency_net(features) 22: end if 23: end for 24: end for 25: 26: // Compute aggregation weights 27: weights <- [] 28: for i = 1 to n do 29: avg_cons <- mean(consistency[i, :] excluding i) 30: features <- [quality[i], avg_cons] 31: w_i <- weight_net(features) 32: weights.append(w_i) 33: end for 34: weights <- normalize(weights) 35: 36: // Aggregate posteriors 37: mu_agg <- sum_i(weights[i] * posteriors[i].mu) 38: 39: // Decode to distribution 40: distribution <- decoder_network.decode(mu_agg, predicate) 41: 42: return distribution Purp ose: Learned evidence aggregation that assesses qualit y and consistency b efore combining m ultiple evidence items. This is the k ey diﬀerentiator for the LPF-Learned v arian t. Key Op erations: • Lines 3–9: Qualit y assessment net work ev aluates each p osterior based on uncertaint y • Lines 11–24: P airwise consistency net work detects contradictions b et ween evidence items • Lines 26–34: W eigh t netw ork com bines quality and consistency scores in to normalized aggregation weigh ts • Lines 36–40: W eigh ted av eraging in latent space follow ed by deco ding 6.4 Algorithm 4: SPNMo dule.Query (LPF-SPN) Input: conditionals , soft_factors , query_variable , spn Output: posterior 1: runtime_spn <- spn.CopyOrCompile() 2: 3: // Apply hard evidence 4: for each (var_name, val) in conditionals do 5: runtime_spn.SetEvidence(var_name, val) 53 6: end for 7: 8: // Attach soft factors 9: for each factor in soft_factors do 10: vars <- factor.variables 11: potential <- factor.potential 12: weight <- factor.weight 13: runtime_spn.AttachLikelihood(vars, potential, weight) 14: end for 15: 16: // Run marginal inference 17: raw_posterior <- runtime_spn.Marginal(query_variable) 18: 19: // Normalize 20: total <- sum(raw_posterior.values()) + eps_small 21: posterior <- {k: v / total for (k, v) in raw_posterior} 22: return posterior Purp ose: T ractable probabilistic inference o ver Sum-Pro duct Netw orks with soft likelihoo d factors. Key Op erations: • Lines 3–6: Hard evidence from canonical facts (observed v ariables) • Lines 8–14: Dynamic attachmen t of soft likelihoo d factors from evidence • Line 17: Exact marginal inference via recursive SPN ev aluation • Lines 19–21: Normalization to ensure v alid probability distribution 6.5 Algorithm 5: Orc hestrator.HandleQuery Input: entity_id , predicate , options , variant ∈ {SPN, Learned} Output: result dictionary 1: start_time <- current_time() 2: 3: // Fast canonical check 4: canonical <- canonical_db.Get(entity_id, predicate) 5: if canonical != null and not Stale(canonical) then 6: return CanonicalResultWithAudit() 7: end if 8: 9: // Retrieve evidence 10: evidence_ids <- EvidenceIndex.Search(entity_id, predicate, options.top_k) 11: if evidence_ids is empty then 12: return NoEvidenceResult() 13: end if 14: 15: // Encode evidence to latent posteriors 54 16: posteriors <- vae_encoder.Encode(evidence_ids, EvidenceIndex) 17: if posteriors is empty then 18: return NoEvidenceResult() 19: end if 20: 21: // Branch based on variant 22: if variant = SPN then 23: // LPF-SPN: Convert to factors + SPN reasoning 24: soft_factors <- ConvertLatentToFactors(posteriors, predicate, schema, 25: decoder, options.n_samples) 26: conditionals <- canonical_db.GetRelatedFacts(entity_id) 27: 28: if spn.Covers(predicate) then 29: posterior <- spn.Query(conditionals, soft_factors, predicate) 30: else 31: posterior <- AggregateVAEPredictions(posteriors, predicate) 32: end if 33: else if variant = Learned then 34: // LPF-Learned: Direct learned aggregation 35: posterior <- learned_aggregator.Aggregate(posteriors, predicate) 36: end if 37: 38: // Compute confidence and audit 39: confidence <- max(posterior.values()) 40: top_value <- argmax(posterior) 41: 42: execution_time <- current_time() - start_time 43: tx <- ledger.AppendInferenceRecord(entity_id, predicate, posterior, 44: evidence_ids, execution_time) 45: 46: return { 47: distribution: posterior, 48: top_value: top_value, 49: confidence: confidence, 50: source: "inference", 51: evidence_chain: evidence_ids, 52: audit_ptr: tx, 53: execution_time: execution_time 54: } Purp ose: Main orc hestrator that co ordinates all comp onents for epistemic query pro cessing. Supp orts b oth LPF-SPN and LPF-Learned v arian ts. Key Op erations: • Lines 3–7: F ast path via canonical database for authoritative facts • Lines 9–19: Evidence retriev al and V AE enco ding • Lines 21–36: V arian t-sp eciﬁc inference (SPN-based vs learned aggregation) 55 • Lines 38–44: Conﬁdence computation and prov enance logging • Lines 46–54: Structured result with full audit trail Computational Complexit y: • Evidence retriev al: O (log N ) for top- k from index • V AE enco ding: O ( K × D ) where K = top_k, D = latent_dim • LPF-SPN: O ( K × M × | domain | ) for factor conv ersion + O ( | V | × | domain | 2 ) for SPN inference • LPF-Learned: O ( K 2 × D ) for consistency matrix + O ( K × D ) for aggregation 6.6 Supp orting Pro cedures Calibration W eight( σ ): Computes uncertaint y p enalty for credibilit y weigh ting. weight <- 1 / (1 + exp(alpha * mean(sigma))) return weight Used in Algorithm 6.1 , line 39. The parameter α controls the strength of the uncertaint y p enalt y . normalize(v ector): L1 normalization to ensure probabilities sum to 1. total <- sum(vector) + eps_small return {v / total for v in vector} Used throughout for probability normalization. 6.7 Implemen tation Notes 1. Numerical Stabilit y: All algorithms use log-space computations where appropriate (e.g., SPN inference) and add small epsilon v alues (typically 10 − 8 ) b efore normalization. 2. Mon te Carlo Samples: Algorithm 6.1 uses M = 16 samples b y default, providing standard error ≈ 0 . 125 . F or higher precision, M = 64 yields SE ≈ 0 . 063 . 3. T emperature Scaling: Algorithm 6.1 supp orts temp erature T for calibration. T > 1 softens o verconﬁden t predictions, while T < 1 sharp ens the distribution. 4. Gradien t Flo w: Algorithm 6.2 uses reparameterization (line 12) to enable backpropagation through sto chastic sampling during training. 5. Cac hing: The orc hestrator (Algorithm 6.5 ) cac hes SPN structures p er predicate to a void rep eated compilation ov erhead. 6. Pro venance: All inference results are logged to an immutable ledger (lines 43–44) for full auditabilit y and mo del monitoring. 7 System Arc hitecture This section provides a detailed exp osition of the LPF system architecture, building on the con- ceptual ov erview presen ted in Section 4.2 . W e describ e the complete data ﬂow through all system comp onen ts, illustrate the diﬀerences b et w een LPF-SPN and LPF-Learned v ariants, and provide concrete implementation details with numerical examples. 56 7.1 Comp onen t Ov erview The LPF system consists of seven core comp onen ts that work together to pro cess epistemic queries. Canonical Database (Canonical DB) serves as the fast path for authoritative facts. It stores ground-truth v alues from trusted sources (regulatory ﬁlings, veriﬁed audits) with timestamps and conﬁdence scores. When a query matches a fresh canonical entry (staleness threshold typically 30 days), the system returns immediately without inv oking the inference pip eline, achieving sub- millisecond resp onse times. Evidence Index manages evidence retriev al using a tw o-tier architecture: (1) a relational metadata store (P ostgreSQL) for entit y-predicate lo okups, and (2) a v ector store (F AISS) for seman tic similarit y searc h ov er pre-computed embeddings. F or a given entit y-predicate pair, the index returns the top- k most relev an t evidence items ranked by a combination of recency , credibilit y , and semantic relev ance. V AE Enco der transforms ra w evidence text into latent p osterior distributions q ϕ ( z | e ) . The enco der netw ork tak es Sen tence-BER T embeddings (384-dim) as input and outputs diagonal Gaussian parameters ( µ, σ ) in a 64-dimensional laten t space. The enco der also computes a base conﬁdence score inv ersely prop ortional to mean uncertaint y: conﬁdence = 1 / (1 + mean ( σ )) . F actor Con v erter (LPF-SPN only) bridges the gap b et ween contin uous laten t p osteriors and discrete probabilistic factors for SPN reasoning. Using Monte Carlo integration with M samples (t ypically M = 16 ), it approximates the integral Φ e ( y ) = R p θ ( y | z ) q ϕ ( z | e ) dz b y deco ding multiple samples from each p osterior and a veraging the resulting distributions. Each factor is assigned a credibilit y weigh t combining base conﬁdence with an uncertaint y p enalt y . Learned Aggregator (LPF-Learned only) replaces structured SPN reasoning with learned neural aggregation. It consists of three sub-netw orks: (1) a qualit y netw ork assessing individual evidence uncertaint y , (2) a consistency netw ork detecting con tradictions b et ween evidence pairs, and (3) a weigh t netw ork combining qualit y and consistency scores in to normalized aggregation w eights. The aggregator op erates directly in latent space b efore deco ding. SPN Mo dule (LPF-SPN only) p erforms tractable exact inference ov er sum-product netw orks. The SPN structure is built p er predicate with leaf nodes representing prior distributions o v er domain v alues, pro duct no des enco ding factorization, and sum no des representing mixture comp onen ts. Soft factors from the con verter are attached dynamically as weigh ted lik eliho o d no des, enabling join t reasoning with hard conditional evidence. Pro v enance Ledger maintains an immutable audit trail of all inference op erations. Eac h query execution generates a prov enance record containing: p osterior distribution, evidence chain, factor metadata (w eights and p oten tials), mo del versions, hyperparameters, and execution time. Records are app end-only with cryptographic hashing to ensure tamp er-evidence. These comp onents are orc hestrated by the main query handler (Algorithm 6.5 ) which routes requests through the appropriate pip eline based on data av ailability and system conﬁguration. 7.2 Data Flo w: LPF-SPN V arian t W e no w trace the complete execution of a query through the LPF-SPN pip eline using a concrete example from our compliance domain. Query Input: entity_id = "C0001" (Global Solutions Inc) predicate = "compliance_level" options = QueryOptions(top_k=5, n_samples=16, temperature=1.0, alpha=2.0) 57 7.2.1 Step 1: Evidence Retriev al The Evidence Index receives the entit y-predicate pair and p erforms a lo okup: entity_index[(C0001, compliance_level)] -> evidence_ids This returns 5 evidence items: • C0001_E001 : “Global Solutions Inc demonstrates strong tax compliance with timely ﬁlings” • C0001_E002 : “Main tains excellent record-k eeping and internal controls” • C0001_E003 : “Consisten tly met all regulatory requirements in recen t audits” • C0001_E004 : “F ollo ws industry b est practices for compliance management” • C0001_E005 : “Main tains ISO 27001 certiﬁcation with annual renew als” Eac h evidence item includes metadata: credibility score (0.85–0.95), timestamp, source t yp e (audit_rep ort, certiﬁcation, ﬁling), and pre-computed Sentence-BER T embedding (384-dim). 7.2.2 Step 2: V AE Enco ding F or eac h evidence item, the enco der: 1. F etches em b edding from the vector store using embedding_id 2. P asses through enco der net work: embedding [384] → MLP [256 → 128] → ( µ, log σ ) 3. Computes σ : σ = exp(log σ ) with numerical stability 4. Computes conﬁdence: conﬁdence = 1 / (1 + mean ( σ )) Output for E001: LatentPosterior( evidence_id = "C0001_E001", mu = [0.82, -0.34, 1.21, ..., 0.45], # 64-dim mean sigma = [0.12, 0.08, 0.15, ..., 0.10], # 64-dim std dev confidence = 0.89 # High confidence ) Similarly for E002–E005, yielding 5 p osterior distributions with conﬁdence scores ranging from 0.85 to 0.93. 7.2.3 Step 3: F actor Con v ersion (Monte Carlo In tegration) F or eac h p osterior (e.g., E001), the F actor Con verter pro ceeds as follows. A. Mon te Carlo Sampling (reparameterization trick): For m = 1 to 16 : ε ( m ) ∼ N (0 , I ) z ( m ) = µ + σ · ε ( m ) 58 This pro duces 16 samples: z_samples = [16 × 64] arra y . B. Deco de Each Sample: For each z ( m ) : 1. Concatenate: [ z ( m ) , predicate_emb ( "compliance_level" )] → [64 + 32 = 96 -dim input ] 2. MLP: [96] → [128] → [64] 3. Output head: [64] → [3] (for 3 classes: low, medium, high) 4. Softmax: logits → p θ ( y | z ( m ) ) Deco der outputs (16 distributions): dist(1) = {"low": 0.05, "medium": 0.15, "high": 0.80} dist(2) = {"low": 0.03, "medium": 0.12, "high": 0.85} ... dist(16) = {"low": 0.06, "medium": 0.18, "high": 0.76} C. Mon te Carlo A veraging: Φ E 001 ( y ) = 1 16 X m p θ ( y | z ( m ) ) = { "low": 0.048, "medium": 0.155, "high": 0.797 } D. T emperature Scaling ( T = 1 . 0 , no adjustment in this case): For T  = 1 : potential 1 /T P y ′ potential ( y ′ ) 1 /T For T = 1 : no change E. Credibilit y W eigh t Computation: σ = mean ([0 . 12 , 0 . 08 , . . . , 0 . 10]) = 0 . 105 w cal = 1 1 + exp( α · σ ) = 1 1 + exp(2 . 0 × 0 . 105) = 0 . 79 w final = confidence × w cal = 0 . 89 × 0 . 79 = 0 . 70 Output: SoftFactor( evidence_id = "C0001_E001", variables = ["compliance_level"], potential = {"low": 0.048, "medium": 0.155, "high": 0.797}, weight = 0.70, metadata = {n_samples: 16, temperature: 1.0, mean_sigma: 0.105} ) This pro cess rep eats for all 5 evidence items, producing 5 soft factors with weigh ts [0 . 70 , 0 . 73 , 0 . 68 , 0 . 66 , 0 . 75] . 7.2.4 Step 4: SPN Reasoning A. Build/Retriev e SPN Structure: The SPN for “compliance_level” has the following structure: 59 [Root: SumNode, weight = 1 . 0 ] ↓ [Pro ductNode] ↓ [LeafNo de: compliance_level] domain = ["low", "medium", "high"] log_probs = log ([0 . 33 , 0 . 33 , 0 . 33]) # Uniform prior B. A ttac h Soft F actors as Likelihoo d No des: F or eac h factor, create a Likelihoo dNo de and apply weigh t: ˜ Φ e ( y ) = p ( y ) w P y ′ p ( y ′ ) w E001 weighted: Before: {"low": 0.048, "medium": 0.155, "high": 0.797} Apply w =0 . 70 : {"low": 0 . 048 0 . 70 , "medium": 0 . 155 0 . 70 , "high": 0 . 797 0 . 70 } After normalisation: {"low": 0.015, "medium": 0.052, "high": 0.933} C. Marginal Inference: F or eac h v alue y ∈ { low , medium , high } : evidence y = { "compliance_level" : y } log P y = log P ( root | evidence y ) + X i log ˜ Φ i ( y ) Computing for all v alues: log p = [log P ( low ) , log P ( medium ) , log P ( high )] = [ − 4 . 02 , − 2 . 96 , − 0 . 07] D. Normalization: p = exp(log p ) = [0 . 018 , 0 . 052 , 0 . 930] p = p / P ( p ) = [0 . 018 , 0 . 052 , 0 . 930] Output: posterior = {"low": 0.018, "medium": 0.052, "high": 0.930} 7.2.5 Step 5: Result F ormatting and Prov enance top_value = argmax(posterior) = "high" confidence = posterior["high"] = 0.930 execution_time = 3.3 ms InferenceRecord( record_id = "INF00000042", timestamp = "2026-01-25T15:42:33Z", entity_id = "C0001", predicate = "compliance_level", 60 distribution = {"low": 0.018, "medium": 0.052, "high": 0.930}, top_value = "high", confidence = 0.930, evidence_chain = ["C0001_E001", ..., "C0001_E005"], factor_metadata = [ {"evidence_id": "C0001_E001", "weight": 0.70, "potential": {...}}, ... ], model_versions = {"encoder": "vae_v1.0", "decoder": "decoder_v1.0"}, hyperparameters = {"n_samples": 16, "temperature": 1.0, "alpha": 2.0, "top_k": 5}, execution_time_ms = 3.3 ) Ground Truth: "high" CORRECT 7.3 Data Flo w: LPF-Learned V arian t The LPF-Learned v ariant diﬀers from LPF-SPN in Steps 3–4, while Steps 1–2 (Evidence Retriev al and V AE Enco ding) remain identical. W e contin ue with the same query example. Steps 1–2 ar e identic al to LPF-SPN and pr o duc e 5 latent p osteriors as describ e d in Se ction 7.2 . 7.3.1 Step 3: Learned Evidence Aggregation A. Compute Quality Scores: For each posterior i ∈ { 1 , 2 , 3 , 4 , 5 } : f i = concat ([ µ i , log var i ]) [64 + 64 = 128 -dim ] quality i = QualityNet ( f i ) [128] → [64] → [1] quality_scores = [0 . 92 , 0 . 89 , 0 . 85 , 0 . 81 , 0 . 94] B. Compute Pairwise Consistency: For each pair ( i, j ) , i  = j : ∆ µ = µ i − µ j ∆ logvar = | log var i − log var j | f ij = concat ([∆ µ, ∆ logvar ]) [128 -dim ] consistency ij = ConsistencyNet ( f ij ) [128] → [64] → [1] consistency_matrix = [ [1.00, 0.87, 0.82, 0.79, 0.91], [0.87, 1.00, 0.85, 0.76, 0.89], [0.82, 0.85, 1.00, 0.88, 0.84], [0.79, 0.76, 0.88, 1.00, 0.81], [0.91, 0.89, 0.84, 0.81, 1.00] ] avg_consistency = [0 . 85 , 0 . 84 , 0 . 85 , 0 . 81 , 0 . 86] C. Compute Final Aggregation W eigh ts: 61 For each posterior i : f i = [ quality i , consistency i ] [2 -dim ] ˜ w i = WeightNet ( f i ) [2] → [32] → [1] ˜ w = [0 . 88 , 0 . 85 , 0 . 82 , 0 . 75 , 0 . 90] w = softmax ( ˜ w ) = [0 . 217 , 0 . 209 , 0 . 203 , 0 . 185 , 0 . 223] D. Aggregate in Latent Space: µ agg = X i w i µ i = 0 . 217 µ 1 + 0 . 209 µ 2 + 0 . 203 µ 3 + 0 . 185 µ 4 + 0 . 223 µ 5 = [0 . 78 , − 0 . 29 , 1 . 15 , . . . , 0 . 41] (64-dim) 7.3.2 Step 4: Deco de Aggregated Posterior 1. z = µ agg (64-dim) 2. p emb = PredicateEmbedding ( "compliance_level" ) (32-dim) 3. x = concat ([ z , p emb ]) (96-dim) 4. h = MLP ( x ) [96] → [128] → [64] 5. logits = OutputHead ( h ) [64] → [3] 6. p = softmax ( logits ) distribution = {"low": 0.032, "medium": 0.089, "high": 0.879} top_value = "high" confidence = 0.879 execution_time = 5.1 ms Ground Truth: "high" CORRECT 62 7.4 Arc hitectural Comparison Comp onen t LPF-SPN LPF-Learned Input 5 laten t p osteriors from V AE 5 latent p osteriors from V AE Aggregation Stage After deco ding (SPN on soft factors) Before deco ding (latent space) Deco der Calls 5 evidence × 16 samples = 80 1 call on aggregated p osterior Aggregation Metho d Structured probabilistic (SPN) Learned neural (quality + consistency) F actor Con version Mon te Carlo integration (Alg. 6.1 ) Not applicable Learned Comp onents Enco der + Deco der only Enco der + Deco der + Aggregator Output Distribution P ( y ) from SPN marginal inference P ( y ) from deco der on z agg A ccuracy 97.8% (b est) 91.1% Macro F1 0.972 (b est) 0.905 Calibration (ECE) 0.014 (sup erior) 0.066 (go od) Brier Score 0.015 (b est) 0.040 NLL 0.125 (b est) 0.273 Sp eed 14.8ms 37.4ms In terpretability High (explicit soft factors) Medium (learned weigh ts) T raining Complexit y Medium (V AE + Deco der) High (V AE + Deco der + Aggregator) Memory Higher (SPN structures) Low er T able 10: F ull architectural comparison of LPF-SPN and LPF-Learned across all dimensions. Key Insight: LPF-SPN excels in calibration and in terpretability through explicit probabilistic reasoning, while LPF-Learned ac hieves comp etitiv e accuracy through end-to-end learned optimization at the cost of transparency . 7.5 Implemen tation Details 7.5.1 T ec hnology Stac k Core F ramew ork: • PyT orch 2.0+ for neural net work comp onen ts (V AE enco der, deco der, learned aggregator) • Python 3.9+ for system orchestration and data pro cessing • NumPy for numerical op erations and array manipulation Probabilistic Reasoning: • Custom light w eight SPN implemen tation in PyT orch for LPF-SPN v arian t • Supp orts dynamic factor attachmen t and exact marginal inference • GPU-accelerated when av ailable (though CPU is suﬃcient for our SPN sizes) Data Storage and Retriev al: • PostgreSQL 14+ for evidence metadata, canonical database, and prov enance ledger • F AISS (F aceb o ok AI Similarity Search) for vector similarit y search ov er embeddings 63 • Redis for caching SPN structures and frequently accessed evidence Em b eddings: • Sentence-BER T ( all-MiniLM-L6-v2 ) for text enco ding (384-dim output) • Pre-computed embeddings stored in F AISS index for fast retriev al • Batch pro cessing during evidence ingestion to amortize embedding cost 7.5.2 Mo del Dimensions Embedding Dimension: 384 (Sentence-BERT output) VAE Latent Dimension: 64 ( z -space) VAE Hidden Dimensions: [256 , 128] Decoder Input: 96 ( 64 latent + 32 predicate embedding) Decoder Hidden Dimensions: [128 , 64] Predicate Embedding: 32 (learned conditioning vector) Aggregator Hidden: 128 (quality/consistency networks) Monte Carlo Samples ( M ): 16 (LPF-SPN factor conversion) Output Classes: 3 (low, medium, high for compliance) 7.5.3 Hyp erparameters Evidence Retriev al: • top_k: 10 (num b er of evidence items to retrieve) • F AISS index: IVF100 with 8-bit PQ compression • Minimum credibilit y threshold: 0.5 V AE Enco der: • Learning rate: 1 × 10 − 3 (A dam optimizer) • Batch size: 32 • Drop out: 0.1 • β -V AE weigh t: 0.01 (KL divergence regularization) • T raining ep o c hs: 50 F actor Conv erter (LPF-SPN): • n samples : 16 (Monte Carlo samples p er p osterior) • temp erature: 1.0 (no calibration adjustmen t by default) • α : 2.0 (uncertain ty p enalt y strength) Learned Aggregator (LPF-Learned): • Learning rate: 1 × 10 − 3 (A dam optimizer) 64 • T raining ep o c hs: 30 • Quality net work: [128] → [64] → [32] → [1] • Consistency netw ork: [128] → [64] → [32] → [1] • W eigh t netw ork: [2] → [32] → [16] → [1] Canonical Database: • Staleness threshold: 30 days (conﬁgurable p er predicate) • Minimum conﬁdence for canonical: 0.95 7.5.4 Data Structures Evidence Metadata: { "evidence_id": str, # Unique identifier "entity_id": str, # Entity this evidence describes "predicate": str, # Predicate this evidence supports "text_content": str, # Raw text "embedding_id": int, # FAISS index reference "credibility": float, # [0, 1] source credibility "timestamp": datetime, # When evidence was created "source": str, # audit_report, filing, certification, etc. "supports_value":str # Ground truth label (training data only) } Laten t P osterior: { "evidence_id": str, "mu": np.ndarray, # [latent_dim] mean vector "sigma": np.ndarray, # [latent_dim] s td deviation "logvar": np.ndarray, # [latent_dim] log variance "confidence": float # 1 / (1 + mean(sigma)) } Soft F actor (LPF-SPN): { "evidence_id": str, "variables": List[str], # Variables this factor depends on "potential": Dict[str, float], # Distribution over domain values "weight": float, # Credibility weight [0, 1] "metadata": { "n_samples": int, "temperature": float, "mean_sigma": float, "base_confidence": float } } 65 Pro venance Record: { "record_id": str, # Unique inference ID "timestamp": datetime, "entity_id": str, "predicate": str, "distribution": Dict[str, float], "top_value": str, "confidence": float, "evidence_chain": List[str], # Ordered evidence IDs used "factor_metadata": List[Dict], # Soft factors or weights "model_versions": { "encoder": str, "decoder": str, "aggregator": str # LPF-Learned only }, "hyperparameters": Dict, "execution_time_ms": float, "hash": str # SHA-256 for tamper detection } 7.5.5 T raining Data Statistics F rom our syn thetic compliance domain: Total Companies: 900 ( 300 companies × 3 years: 2020, 2021, 2022) Evidence per Company: 5 (audit reports, filings, certifications) Total Evidence Items: 4 , 500 Train/Val/Test Split: 70 / 15 / 15 ( 630 / 135 / 135 companies) Label Distribution: Low compliance: 30% ( 270 companies) Medium compliance: 40% ( 360 companies) High compliance: 30% ( 270 companies) Evidence Credibility: Mean: 0 . 87 Std: 0 . 08 Range: [0 . 65 , 0 . 98] Training Time: VAE Encoder: ∼ 15 mi nutes (50 epochs, GPU) Decoder Network: ∼ 25 minutes (100 epochs, GPU) Learned Aggregator: ∼ 10 minutes (30 epochs, GPU) Total: ∼ 50 minutes 7.5.6 Inference P erformance Latency breakdo wn (a v erage ov er 100 queries): LPF-SPN: 66 Canonical DB check: 0.2 ms Evidence retrieval: 0.8 ms (FAISS + PostgreSQL) VAE encoding: 0.4 ms (5 evidence items) Factor conversion: 11.2 ms (80 decoder calls, batched) SPN reasoning: 1.8 ms (cached structure) Provenance logging: 0.4 ms Total: 14.8 ms LPF-Learned: Canonical DB check: 0.2 ms Evidence retrieval: 0.8 ms VAE encoding: 0.4 ms Aggregator forward: 34.6 ms (quality + consistency + weights) Decoder: 1.0 ms (single call) Provenance logging: 0.4 ms Total: 37.4 ms Throughput (single GPU): • LPF-SPN: ≈ 68 queries/second • LPF-Learned: ≈ 27 queries/second Memory fo otprin t: • LPF-SPN: ≈ 450 MB (includes cached SPN structures) • LPF-Learned: ≈ 380 MB (no SPN ov erhead) 67 7.5.7 Deplo yment Arc hitecture (Suggestive) Load Balancer (Nginx) API Server 1 (F astAPI) API Server 2 (F astAPI) Orc hestrator Service (Async W orkers) W orker 1 W orker 2 W orker 3 · · · Mo del Service (V AE, Deco der, Aggregator) GPU-enabled Cac he La yer (Redis) — SPN cac he — Embeddings Data Lay er (PostgreSQL) — Evidence metadata — Canonical database — Prov enance ledger V ector Store (F AISS) — Evidence em b eddings Figure 9: Deploymen t architecture: Load Balancer (Nginx) feeds into tw o F astAPI servers, b oth connecting to an Orchestrator Service with async work ers, whic h connects to the Mo del Service (V AE, Deco der, Aggregator, GPU-enabled) and Cac he Lay er (Redis), all feeding in to the Data Lay er (P ostgreSQL) and V ector Store (F AISS). As sho wn in Figure 9 , the deplo yment architecture illustrates the full system stac k from load balancing through API serving, orchestration, mo del inference, caching, p ersisten t storage, and v ector retriev al. NOTE: The LPF mo dels hav e not b een deploy ed in pro duction as of the writing of this work; hence this is only a recommendation for any one who wishes to deplo y it. Pro duction Considerations: • Horizontal scaling: Multiple API servers b ehind load balancer • Mo del serving: GPU-accelerated inference with batc hing 68 • Caching: Redis for frequently accessed SPNs and em b eddings • Monitoring: Prometheus + Grafana for latency , throughput, and mo del p erformance • Logging: ELK stack for query logs and prov enance audit trail • Backup: Daily snapshots of PostgreSQL and F AISS index 69 7.6 Visual Arc hitecture Diagrams User Query entity_id="C0001", predicate="compliance_level" Canonical DB Check HIT → Return Evidence Index Retriev al (F AISS + SQL) V AE Enco der (Sen tence-BER T → MLP → µ, σ ) F actor Conv erter (Algorithm 1) F or eac h e : 1. Sample M =16 : z ∼ q ( z | e ) 2. Deco de each z → p ( y | z ) 3. A verage 4. W eigh t SPN Mo dule (Algorithm 4) 1. Attac h soft factors 2. Add hard conditionals 3. Marginal inference Orc hestrator — Conﬁdence — T op v alue — Prov enance Pro venance Ledger (Imm utable log) Final Result {distribution, confidence, audit_ptr} HIT MISS [5 evidenc e IDs] [5 latent p osteriors: q ( z | e 1 ) . . . q ( z | e 5 ) ] [5 soft factors: Φ e 1 ( y ) . . . Φ e 5 ( y ) ] P ( y | evidenc e ) Figure 10: Complete execution pip eline for LPF -SPN. The system ﬁrst chec ks the canonical database for authoritative facts. On a miss, it retrieves relev an t evidence, enco des each item in to a latent p osterior, con verts p osteriors into soft probabilistic factors via Mon te Carlo integration, performs exact SPN inference, and returns calibrated predictions with full prov enance. 70 As shown in Figure 10 , the complete execution pip eline illustrates how LPF-SPN pro cesses a user query from canonical database c heck through evidence retriev al, V AE enco ding, factor conv ersion, SPN inference, and prov enance logging to pro duce a ﬁnal calibrated result. User Query entity_id="C0001", predicate="compliance_level" Canonical DB Chec k HIT → Return Evidence Index Retriev al (F AISS + SQL) V AE Enco der (Sentence-BER T → MLP → µ, σ ) Learned Aggregator (Algorithm 3) Qualit y Net work F or each posterior: features = [ µ, log σ 2 ] , quality = QualityNet ( · ) Consistency Netw ork F or each pair ( i, j ) : diﬀ = µ i − µ j , consistency = ConsNet ( · ) W eight Netw ork w = W eightNet ( qualit y , avg_cons ) , Normalize: softmax ( w ) W eighted Aggregation z agg = P i w i · µ i Deco der Net work (Single call) z agg → MLP → p ( y ) Orc hestrator Conﬁdence T op v alue Prov enance Pro venance Ledger (Immutable log) Final Result {distribution, confidence, audit_ptr, weights} HIT MISS [5 evidence IDs] [5 latent posteriors: q ( z | e 1 ) . . . q ( z | e 5 ) ] [Single aggre gate d p osterior] P ( y | evidence ) Figure 11: Complete execution pip eline for LPF-Learned. Unlike LPF-SPN, this v arian t aggregates evidence in latent space b efore deco ding. The learned aggregator uses three neural netw orks to assess quality , detect contradictions, and compute optimal aggregation weigh ts, pro ducing a single aggregated p osterior that is deco ded once to yield the ﬁnal distribution. 71 ARCHITECTURAL DIVERGENCE POINT (Both varian ts identical up to this p oin t) [5 latent p osteriors fr om V AE Enco der] LPF-SPN LPF-Learned STEP A: PROCESSING F actor Conv ersion F or each e ∈ { 5 } : • Sample z ( m ) ( 16 × ) • Deco de z → p θ ( y ) • A verage (MC) • Compute weigh t Output: 5 soft factors STEP A: PROCESSING Learned Aggregation Compute: • Quality scores • Consistency ( 5 × 5 ) • Final weigh ts Aggregate: z agg = P i w i · µ i STEP B: AGGREGA TION SPN Reasoning • Build / cac he SPN • Attach factors as lik eliho ods • Add hard conditionals • Marginal inference Complexity: O ( | V |×| D | 2 ) Decoder calls: 80 ( 5 × 16 samples) STEP B: DECODING Decoder Net w ork Single call: z agg → p θ ( y ) Complexity: O (1) Decoder calls: 1 [Φ e 1 . . . Φ e 5 ] [Single z agg ] Common Output Pro cessing • Conﬁdence • T op prediction • Prov enance log P ( y | evidence) Final Result TRADE-OFFS SUMMAR Y LPF-SPN LPF-Learned ✓ Best accuracy (97.8%) ✓ Strong accuracy (91.1%) ✓ Best calibration (ECE 0.014) ✓ Acceptable calibration (ECE 0.066) ✓ Best Brier score (0.015) ✓ Go od Brier score (0.040) ✓ F aster inference (14.8ms) ✓ Simpler architecture ✓ Highly interpretable ✓ End-to-end optimization ✓ Explicit probabilistic model ✓ Low er memory fo otprin t ✓ Single deco der call × More deco der calls (80) × Slo wer inference (37.4ms) × Higher memory (SPN cache) × Less interpretable × Complex factor con version × More training complexity Use when: Calibration critic al Use when: Calibration ac ceptable • Medical diagnosis • Content recommendation • Safety-critical systems • Business intelligence • Regulatory compliance • General KB completion Figure 12: Architectural divergence b et w een LPF-SPN and LPF-Learned v arian ts. 72 As sho wn in Figure 12 , the arc hitectural divergence illustrates how LPF-SPN and LPF-Learned share identical upstream comp onen ts b efore diverging in their aggregation strategies, each with distinct trade-oﬀs. Both share identical comp onen ts for evidence retriev al and V AE enco ding. The k ey diﬀerence lies in aggregation strategy: LPF-SPN con verts p osteriors to soft factors and uses structured SPN inference, while LPF-Learned uses learned neural aggregation in laten t space. Each v arian t oﬀers distinct trade-oﬀs b etw een calibration, accuracy , sp eed, and in terpretability . 7.7 Numerical Flo w Example Summary T o consolidate understanding, w e summarize the numerical transformations in a complete query . Input: En tit y C0001, Predicate “compliance_level” Output (LPF-SPN): Distribution: {"low": 0.018, "medium": 0.052, "high": 0.930} Confidence: 0.930 Accuracy: 97.8% Time: 14.8ms Output (LPF-Learned): Distribution: {"low": 0.032, "medium": 0.089, "high": 0.879} Confidence: 0.879 Accuracy: 91.1% Time: 37.4ms T ransformations: Stage Input Dim Op eration Output Dim Both? Evidence T ext T ext Retriev al 5 texts Same Em b edding T ext SBER T 5 × 384 Same Enco ding 5 × 384 V AE 5 × 64 ( µ, σ ) Same Aggregation 5 × 64 — V ariable Div erges Deco ding V ariable Deco der 3-class dist 80 vs 1 calls Reasoning F actors/ z SPN/Direct 3-class dist Div erges T able 11: Numerical transformation summary for a complete LPF query . Key Insight: LPF-SPN p erforms aggregation in probability space (after deco ding), requiring man y deco der calls but enabling exact probabilistic inference. LPF-Learned aggregates in laten t space (b efore deco ding), requiring only one deco der call but relying on learned com bination rules. 7.8 Key T akea w a ys The LPF system arc hitecture demonstrates how to build a pro duction-grade epistemic reasoning system that com bines neural enco ding, probabilistic inference, and comprehensive auditability . The t wo v arian ts oﬀer complementary strengths for diﬀeren t application contexts. LPF-SPN excels when uncertain ty calibration is critical. With sup erior calibration (ECE 0.014), low est Brier score (0.015), and highest accuracy (97.8%), it is the preferred choice for high- stak es domains: medical diagnosis, clinical decision supp ort, safet y-critical systems, and regulatory 73 compliance. Its explicit soft factors and structured probabilistic reasoning also pro vide in terpretability for auditing and debugging. LPF-Learned provides a simpler end-to-end arc hitecture suitable for domains where probabilistic reasoning is needed but p erfect calibration is not paramoun t. While it has higher calibration error (ECE 0.066) and slow er inference (37.4ms), it still delivers strong accuracy (91.1%) and substantially outp erforms traditional baselines. It is appropriate for applications like con tent recommendation, business intelligence, automated rep orting, and general kno wledge base completion. Both v arian ts share robust infrastructure for evidence managemen t, prov enance tracking, and scalable deploymen t, ensuring pro duction-readiness across diverse epistemic reasoning tasks. 8 T raining Metho dology This section describ es the training procedures for the LPF system comp onen ts. W e fo cus on encoder- deco der training, which is common to b oth LPF-SPN and LPF-Learned v arian ts. The learned aggregator training (sp eciﬁc to LPF-Learned) follows a similar sup ervised learning approach but is trained separately after the enco der-deco der mo dels con verge. 8.1 Enco der + Deco der T raining 8.1.1 Dataset Preparation The V AE enco der and conditional deco der are trained join tly on evidence-level data with ground truth lab els. F or eac h training example, we hav e: • Input : Evidence text embedding e ∈ R 384 (from Sentence-BER T) • Predicate : p (e.g., “compliance_level”) • Lab el : Ground truth v alue y ∗ ∈ Y (e.g., “high”) Eac h evidence item is lab eled with the entit y’s ground truth v alue, allo wing sup ervised training at the evidence level rather than requiring entit y-lev el aggregation during training. 8.1.2 T raining Ob jective The join t training ob jectiv e combines classiﬁcation loss with KL regularization: L = E ( e ,p,y ∗ ) [ − log p θ ( y ∗ | z , p )] + β · KL ( q ϕ ( z | e ) ∥ p ( z )) (59) where: • Cross-En tropy Loss : − log p θ ( y ∗ | z , p ) ensures the deco der pro duces correct predictions • KL Div ergence : KL ( q ϕ ( z | e ) ∥N (0 , I )) regularizes the latent space • KL W eigh t : β = 0 . 01 balances reconstruction accuracy with latent space structure The laten t co de z is sampled via reparameterization: z = µ + σ ⊙ ϵ where ϵ ∼ N (0 , I ) . 74 8.1.3 Hyp erparameters P arameter V alue Description Learning rate 10 − 3 A dam optimizer Batc h size 64 Mini-batc h size KL w eight ( β ) 0.01 Regularization strength Drop out 0.1 Applied in enco der MLP Early stopping patience 5 ep o c hs V alidation-based stopping Max ep o c hs 100 T raining budget T able 12: Encoder-deco der training hyperparameters. 8.1.4 Seed Searc h Strategy Due to neural net work sensitivity to random initialization, we emplo y a seed search proto col to ensure robust and repro ducible results: 1. Seed Selection : T rain mo dels with 7 diﬀerent random seeds: [42, 123, 456, 789, 2024, 2025, 314159] 2. Indep enden t T raining : Eac h seed undergo es full training with identical hyperparameters 3. Mo del Selection : Select the mo del with highest v alidation accuracy 4. Rep orting : Rep ort mean ± standard deviation across all seeds for transparency This approach follows b est practices in ML research and ensures rep orted results are not cherry- pic ked from lucky initializations. 8.2 T raining Results: Compliance Domain W e present detailed training results for the compliance domain , which serv es as the primary ev aluation domain throughout this pap er. Results for additional domains are summarized in Section 8.3 with full details in App endix A . 75 42 123 456 789 2024 2025 314159 R andom Seed 0 20 40 60 80 V alidation Accuracy (%) Compliance Domain: V alidation Accuracy by Seed Mean: 85.6% 42 123 456 789 2024 2025 314159 R andom Seed 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 V alidation Loss Compliance Domain: V alidation Loss by Seed Mean: 0.730 Figure 13: Compliance domain training results across 7 random seeds. Left: V alidation accuracy sho wing consistency around 85.6% mean with b est seed (2024, gold bar) achieving 86.0%. Right: V alidation loss showing mean of 0.730 with b est seed achieving 0.726. Red dashed lines indicate mean v alues. 8.2.1 Seed-Lev el Results T raining o ver 7 random seeds yielded the following results: Seed T rain A cc V al Acc (Best) V al Loss (Best) Ep o c hs Conv erged 42 82.2% 85.7% 0.735 10 ✓ 123 82.2% 85.6% 0.730 20 × 456 82.7% 85.4% 0.731 9 ✓ 789 82.4% 85.4% 0.734 11 ✓ 2024 82.3% 86.0% ⋆ 0.726 ⋆ 12 ✓ 2025 82.6% 85.3% 0.727 16 ✓ 314159 81.9% 85.7% 0.728 20 × T able 13: Seed-lev el training results for the compliance domain. ⋆ Best seed selected for downstream ev aluation. Key Observ ations: 1. Stabilit y : T raining accuracy shows low v ariance (82.3 ± 0.3%), indicating stable optimization 2. Con v ergence : 5 out of 7 seeds conv erged early (b efore ep och 20), suggesting the loss landscap e is well-behav ed 3. Best Mo del : Seed 2024 achiev ed highest v alidation accuracy (86.0%) and low est v alidation loss (0.726) 4. Generalization : Small gap b et w een train (82.3%) and v alidation (85.6%) accuracy indicates go od generalization 76 8.2.2 Aggregate Statistics A cross all 7 seeds: • T raining A ccuracy : 82.3 ± 0.3% • V alidation Accuracy : 85.6 ± 0.2% (b est: 86.0%) • V alidation Loss : 0.730 ± 0.003 (b est: 0.726) The lo w standard deviations demonstrate that our architecture and training pro cedure are robust to initialization, with all seeds achieving comp etitiv e p erformance. Cross-Entropy KL (× ) T otal 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Loss V alue 0.692 0.040 0.732 Compliance: Loss Decomposition (Best Seed 2024) Figure 14: Loss decomp osition for compliance domain (b est seed 2024). The total v alidation loss (0.726) comprises cross-entrop y loss (0.692, 95.3%) and w eighted KL divergence (0.040, 4.7%). The KL term remains mo derate, indicating the enco der learns meaningful latent structure without excessiv e compression. 8.2.3 Loss Decomp osition F or the b est seed (2024) at conv ergence: • T otal Loss : 0.726 • Cross-En tropy : 0.692 (95.3% of total) • KL Div ergence : 4.01 (w eighted contribution: 0.040) 77 The KL term remains moderate ( ≈ 4.0), indicating the enco der learns meaningful laten t structure without excessive compression that would harm reconstruction. 8.3 T raining Results: All Domains W e trained enco der-deco der mo dels on eigh t div erse domains spanning diﬀeren t data t ypes, reasoning complexit y , and lab el distributions. T able 14 summarizes training and v alidation accuracy across all domains, with detailed seed-level results in App endix A . Domain T rain A cc V al A cc Best Seed Best V al Acc Notes FEVER 99.6 ± 0.1% 99.9 ± 0.0% 2025 99.9% F act veriﬁcation (easiest) A cademic 83.5 ± 0.2% 85.7 ± 0.2% 789 86.1% Publication ven ue classiﬁcation Compliance 82.3 ± 0.3% 85.6 ± 0.2% 2024 86.0% Primary domain Construction 83.4 ± 0.2% 85.4 ± 0.2% 789 85.8% Pro ject risk assessment Finance 83.6 ± 0.2% 84.8 ± 0.3% 456 85.2% Credit rating prediction Materials 83.9 ± 0.2% 84.0 ± 0.5% 456 84.6% Material prop ert y classiﬁcation Healthcare 84.2 ± 0.2% 83.8 ± 0.1% 42 84.0% Disease severit y classiﬁcation Legal 84.8 ± 0.2% 83.6 ± 0.1% 456 83.7% Case outcome (hardest) T able 14: Enco der-decoder training results across all ev aluation domains. Results are mean ± std o ver 7 random seeds. F ever Academic Compliance Construction Finance Materials Healthcare Legal Domain 70 75 80 85 90 95 100 Accuracy (%) 99.6 83.5 82.3 83.4 83.6 84.0 84.1 84.8 99.9 85.7 85.6 85.4 84.8 84.0 83.8 83.6 Encoder-Decoder Training: Cross-Domain P erformance Train V alidation Figure 15: Cross-domain p erformance comparison sho wing training (blue bars) and v alidation (orange bars) accuracy across all eigh t domains. Error bars indicate standard deviation o ver 7 seeds. Domains are sorted by v alidation accuracy (descending). 78 Academic Compliance Construction F ever Finance Healthcare Legal Materials Domain 0.0 0.1 0.2 0.3 0.4 Standard Deviation (%) 0.18% 85.7% 0.22% 85.6% 0.21% 85.4% 0.05% 99.9% 0.30% 84.8% 0.10% 83.8% 0.09% 83.6% 0.46% 84.0% V alidation Accuracy V ariance Across Seeds Figure 16: V alidation accuracy v ariance (standard deviation) across seeds by domain. FEVER sho ws minimal v ariance (0.0%) due to clean data structure, while Materials exhibits highest v ariance (0.5%). Domain Analysis: • Easiest Domain : FEVER (99.9% v al acc) — Clean, well-structured fact veriﬁcation with strong textual signals • Hardest Domain : Legal (83.6% v al acc) — Complex reasoning with subtle distinctions and am biguous evidence • Most Stable : FEVER (std=0.0%) — Consistent p erformance across all seeds • Most V ariable : Materials (std=0.5%) — Higher sensitivity to initialization The enco der-decoder architecture generalizes well across all domains, with v alidation accuracies ranging from 83.6% (legal) to 99.9% (FEVER), demonstrating the broad applicability of the latent p osterior factorization approach. 79 Compliance Academic Construction Finance F ever Materials Healthcare Legal Domain 3 2 1 0 1 Train - V alidation Accuracy (%) -3.3% -2.2% -2.1% -1.2% -0.2% 0.0% 0.3% 1.2% Generalization Gap by Domain Figure 17: Generalization gap analysis showing train minus v alidation accuracy for each domain. Negativ e v alues (most domains) indicate v alidation outp erforms training, suggesting go o d general- ization. Academic Compliance Construction F ever Finance Healthcare Legal Materials Domain 0 5 10 15 20 25 30 A verage Epochs to Convergence 13.0 14.0 14.6 29.7 12.6 15.3 14.0 11.7 Training Convergence Speed Academic Compliance Construction F ever Finance Healthcare Legal Materials Domain 0 20 40 60 80 100 Convergence R ate (%) 86% 71% 86% 14% 100% 100% 86% 86% Early Stopping Success R ate Figure 18: T raining conv ergence analysis. Left: A v erage ep ochs to conv ergence across 7 seeds. Righ t: Early stopping success rate (p ercentage of seeds that con verged b efore maximum ep o c hs). 8.4 Learned Aggregator T raining (LPF-Learned Only) The learned aggregator is trained after the enco der-decoder models conv erge, using en tity-lev el lab els rather than evidence-level lab els. 80 8.4.1 Dataset Preparation F or eac h training entit y: 1. Retriev e top- k evidence items ( k = 10 ) 2. Enco de evidence using the frozen V AE enco der to obtain p osteriors: { q ϕ ( z | e i ) } k i =1 3. Create training example: ( { q ( z | e 1 ) , . . . , q ( z | e k ) } , p, y ∗ ) This pro duces entit y-lev el training data where the aggregator learns to combine m ultiple p osteriors optimally . 8.4.2 T raining Ob jective The aggregator minimizes negativ e log-likelihoo d of the true lab el under the aggregated distribution: L agg = − log p θ ( y ∗ | Aggregate ( { q ( z | e i ) } ) , p ) (60) where the aggregation pro duces a single laten t co de z agg = P i w i µ i with learned w eights w i from the quality , consistency , and w eight netw orks. 8.4.3 Hyp erparameters P arameter V alue Description Learning rate 10 − 3 A dam optimizer T raining ep ochs 30 F ew er than enco der/deco der Hidden dimensions 128 F or all three netw orks Drop out 0.1 Regularization T able 15: Learned aggregator training hyperparameters. 8.4.4 T raining Pro cedure for epoch in range(30): for entity, posteriors, label in train_data: # Compute aggregation weights weights = aggregator.forward(posteriors) # Aggregate in latent space z_agg = sum(weights[i] * posteriors[i].mu) # Decode logits = decoder(z_agg, predicate) # Loss and update loss = cross_entropy(logits, label) loss.backward() optimizer.step() 81 The aggregator training is relatively fast (30 ep o c hs vs 100 for enco der/deco der) b ecause: (1) enco der w eights are frozen (no backprop through em b edding), (2) dataset size is smaller (en tity-lev el vs evidence-level), and (3) the aggregator netw orks are ligh tw eigh t (128-dim hidden lay ers). Note : W e do not rep ort separate seed search results for aggregator training as it is deterministic giv en a ﬁxed trained encoder-deco der. The aggregator uses the same random seed as its corresp onding enco der-decoder mo del. 9 Hyp erparameters and Implementation Guidelines 9.1 Key Hyp erparameters Hyp erparameter V alue Range Notes A r chite ctur e Laten t dimension ( d z ) 64 32–128 Balances expressiveness and eﬃciency Em b edding dimension 384 Fixed Sentence-BER T output Enco der hidden dims [256, 128] — T w o-lay er MLP Deco der hidden dims [128, 64] — Conditional on predicate Predicate embedding 32 16–64 Learned p er predicate T r aining Learning rate 10 − 3 10 − 4 – 10 − 3 A dam optimizer Batc h size 64 32–128 Depends on GPU memory KL w eight ( β ) 0.01 0.001–0.1 V AE regularization Drop out 0.1 0.0–0.2 Encoder/deco der only Infer enc e Mon te Carlo samples ( M ) 16 8–32 F actor con version (LPF-SPN) T emp erature ( T ) 1.0 0.8–1.5 Calibration tuning Alpha ( α ) 2.0 0.1–10.0 Uncertaint y p enalt y T op- k evidence 10 5–20 Evidence retriev al Sigma min ( σ min ) 10 − 6 10 − 6 – 10 − 4 Numerical stability T able 16: Core hyperparameters for LPF system comp onen ts. 9.2 Implemen tation Guidelines Mon te Carlo Sampling : The num ber of samples M con trols the v ariance-latency tradeoﬀ in factor con version. Standard error decreases as O (1 / √ M ) : • M = 8 : F ast but noisy (SE ≈ 0.18) • M = 16 : Recommended default (SE ≈ 0.13) • M = 32 : High precision (SE ≈ 0.09) T emperature Scaling : T une T on a v alidation set to improv e calibration: • T > 1 : Softens ov erconﬁden t predictions • T < 1 : Sharp ens uncertain predictions 82 • T = 1 : No adjustmen t (default) W eigh t Calibration : The uncertain ty p enalt y α con trols how strongly high-v ariance p osteriors are down weigh ted: w ( e ) = base_conf × 1 1 + exp( α · mean ( σ )) (61) Start with α = 2 . 0 and tune on v alidation ECE. Numerical Stabilit y : • Clip σ ≥ σ min = 10 − 6 to preven t underﬂow • Add ϵ = 10 − 12 b efore normalization: p ( y ) = f ( y ) P y ′ f ( y ′ )+ ϵ • Use log-space for SPN inference when p ossible Deco der Design : The decoder m ust b e conditional on the predicate to handle multiple predicates with a single mo del: p θ ( y | z , p ) = softmax ( MLP p ([ z , em b ( p )])) (62) where emb ( p ) is a learned predicate embedding. 9.3 Quic k Reference: Key Equations Concept Equation Reparameterization z = µ + σ ⊙ ϵ , where ϵ ∼ N (0 , I ) Laten t F actor Φ e ( y ) = R p θ ( y | z ) q ϕ ( z | e ) dz Mon te Carlo Estimate ˆ Φ e ( y ) = 1 M P M m =1 p θ ( y | z ( m ) ) Conﬁdence W eigh t w ( e ) = base_conf × 1 1+exp( α · mean ( σ )) W eigh ted F actor ˜ Φ e ( y ) = ( p ( y | e )) w ( e ) P y ′ ( p ( y ′ | e )) w ( e ) T raining Loss L = − log p θ ( y ∗ | z , p ) + β · KL ( q ϕ ( z | e ) ∥N (0 , I )) T able 17: Quic k reference for key LPF equations. 10 Related W ork The Latent Posterior F actor (LPF) framework bridges multiple researc h areas: neuro-sym b olic AI, probabilistic circuits, uncertaint y quantiﬁcation, evidence aggregation, and fact veriﬁcation. This section surveys key related work in each area and articulates how LPF adv ances the state of the art through nov el architectural c hoices and principled multi-evidence reasoning. 10.1 Neuro-Sym b olic AI and Probabilistic Logic Probabilistic Soft Logic (PSL) [ Bach et al. , 2017 ] and Mark o v Logic Netw orks (MLNs) [ Ric hardson and Domingos , 2006 ] combine ﬁrst-order logic with probabilistic reasoning, enabling structured inference ov er relational domains. PSL represents logical rules as soft constraints with learned w eights, while MLNs attac h w eights to ﬁrst-order formulas. Both frameworks excel at incorp orating domain kno wledge through man ually crafted rules but struggle with unstructured data. 83 DeepProbLog [ Manhaeve et al. , 2018 ] extends P robLog with neural predicates, allowing neural net works to ground symbolic predicates. This enables end-to-end learning of probabilistic logic programs from data. Similarly , Neural Theorem Prov ers [ Ro cktäsc hel and Riedel , 2017 ] learn to p erform logical reasoning through diﬀerentiable pro of construction. Scallop [ Huang et al. , 2021 ] and Seman tic Probabilistic Lay ers pro vide frameworks for in tegrating probabilistic reasoning in to neural arc hitectures through diﬀeren tiable logic programming. These systems demonstrate impressive results on tasks requiring comp ositional reasoning. Key Diﬀerence: While these approaches require symbolic predicates and man ually engineered logical rules, LPF op erates on raw unstructured evidence (text, do cumen ts) without explicit rule sp eciﬁcation. Our soft factors emerge from learned V AE p osteriors rather than hand-crafted logical form ulas. This makes LPF applicable to domains where symbolic know ledge engineering is infeasible or where evidence is inherently ambiguous and contradictory . Additionally , LPF’s dual arc hitecture design (SPN-based vs. learned) enables direct comparison of structured versus end-to-end learned reasoning paradigms — a capability absent in existing neuro-symbolic systems. 10.2 Probabilistic Circuits Sum-Pro duct Netw orks (SPNs) [ P o on and Domingos , 2011 ] represent probability distributions through hierarc hical comp ositions of sum and pro duct no des, enabling tractable exact inference. Subsequen t work has explored structure learning [ Gens and Domingos , 2013 ], deep SPNs [ Peharz et al. , 2020 ], and discriminative SPNs for classiﬁcation [ Peharz et al. , 2020 ]. Arithmetic Circuits (ACs) [ Darwic he , 2003 ] provide a more general framew ork for knowledge compilation, with SPNs b eing a sp ecial case. Recent w ork on Probabilistic Circuits (PCs) [ Choi et al. , 2020 ] uniﬁes v arious tractable models under a single formalism, demonstrating superior p erformance on density estimation and probabilistic inference tasks. Cutset Net w orks [ Rahman et al. , 2014 ] and Probabilistic Senten tial Decision Diagrams [ Kisa et al. , 2014 ] oﬀer alternativ e tractable represen tations with diﬀerent trade-oﬀs b et ween expressiv eness and eﬃciency . Neural-enhanced circuits: Recen t work explores com bining neural net w orks with probabilistic circuits. Generativ e SPNs [ Peharz et al. , 2020 ] learn SPN structures from data using neural arc hitecture search. Einsum Netw orks [ Peharz et al. , 2020 ] use tensor op erations for eﬃcient SPN inference on GPUs. Key Diﬀerence: Existing probabilistic circuit approaches learn circuit parameters from data but assume ﬁxe d input distributions . In contrast, LPF dynamically attaches soft lik eliho o d factors derived from evidence-sp eciﬁc V AE p osteriors. Eac h query generates unique factors reﬂecting the qualit y and uncertaint y of av ailable evidence, rather than using pre-learned static distributions. This dynamic factor attachmen t enables LPF to handle v ariable evidence sets, missing data, and con tradictory sources — scenarios where traditional probabilistic circuits struggle. F urthermore, our Mon te Carlo factor conv ersion pro vides a principled bridge b etw een contin uous laten t p osteriors and discrete circuit v ariables, a connection not explored in prior probabilistic circuit literature. 10.3 Uncertain t y Quantiﬁcation in Deep Learning Ba yesian Deep Learning approaches mo del epistemic uncertain ty through w eight distributions. Ba y es b y Bac kprop [ Blundell et al. , 2015 ] uses v ariational inference to learn p osterior distributions o ver net work weigh ts. Drop out as Bay esian Appro ximation [ Gal and Ghahramani , 2016 ] in terprets drop out as approximate Bay esian inference, enabling uncertaint y estimation through Mon te Carlo sampling at test time. 84 Ensem ble Metho ds [ Lakshminaray anan et al. , 2017 ] estimate uncertaint y b y training multiple mo dels with diﬀerent initializations and av eraging predictions. Deep Ensem bles provide well- calibrated uncertaint y estimates but require signiﬁcant computational o verhead. Eviden tial Deep Learning (EDL) [ Senso y et al. , 2018 ] represents uncertain t y through second- order probability distributions — distributions o ver simplex parameters rather than class lab els. By placing Dirichlet priors ov er categorical distributions, EDL captures b oth aleatoric (data) and epistemic (mo del) uncertaint y in a single forward pass. Extensions include Natural P osterior Net w ork [ Charp entier et al. , 2020 ] and Posterior Netw ork [ Charp en tier et al. , 2020 ]. EDL adaptation challenges: W e experimented with t w o adaptations of EDL to multi-evidence settings: (1) EDL-Aggregated av erages evidence embeddings b efore prediction, ac hieving 56.3% accuracy but collapsing the distributional information EDL is designed to capture, and (2) EDL- Individual treats eac h evidence piece as a separate training example, ac hieving only 43.7% accuracy due to severe lab el noise (individual pieces ma y not indep endently supp ort entit y-lev el lab els), class im balance ampliﬁcation (entities with more evidence dominate training), and training-inference distribution mismatch. Both v arian ts underp erform dramatically compared to LPF-SPN (97.8%), demonstrating that uncertaint y quantiﬁcation alone is insuﬃcient — the task requires structured probabilistic reasoning ov er multiple pieces of evidence. Conformal Prediction [ V ovk et al. , 2005 , Angelop oulos and Bates , 2021 ] provides distribution- free uncertaint y quan tiﬁcation through set-v alued predictions with statistical guaran tees. Recent w ork explores adaptiv e conformal inference [ Gibbs and Candès , 2021 ] and conformal risk con trol [ Angelop oulos et al. , 2023 ]. Key Diﬀerence: Existing uncertain t y quan tiﬁcation metho ds fo cus on single-input scenarios : one image, one sen tence, one data p oin t. EDL, in particular, w as designed for per-instance uncertaint y and fails catastrophically in multi-evidence settings (our exp erimen ts show 43.7% and 56.3% accuracy for EDL-Individual and EDL-Aggregated, resp ectiv ely , compared to LPF-SPN’s 97.8%). LPF is purp ose-built for multi-evidence aggregation , explicitly mo deling how to combine uncertainties from multiple sources. Our V AE enco der quan tiﬁes evidence-level uncertain ty ( σ captures ambiguit y), whic h is then propagated through credibility weigh ts and aggregated via structured reasoning (SPN) or learned combination (neural aggregator). This addresses a fundamental gap: no prior uncertaint y quan tiﬁcation framework provides principled m ulti-evidence aggregation with prov enance tracking. Data eﬃciency consideration: Beyond arc hitectural diﬀerences, LPF op erates in a low- data regime common in en terprise kno wledge completion: hundreds of lab eled entities (each with multiple evidence pieces) rather than thousands of single-input examples. Standard neural uncertain ty metho ds require large-scale datasets for calibration — EDL was ev aluated on MNIST (60K samples) and CIF AR-10 (50K samples). In contrast, LPF achiev es sup erior calibration (ECE 1.4%) with only 630 training entities (900 total across all splits), demonstrating that purp ose-built m ulti-evidence architectures are more data-eﬃcient than adapting single-input metho ds. 10.4 Evidence Aggregation and Multi-Do cumen t Reasoning A ttention Mechanisms [ Bahdanau et al. , 2015 , V aswani et al. , 2017 ] enable neural netw orks to selectively fo cus on relev an t inputs when aggregating information. Multi-head attention and T ransformers hav e b ecome the de facto standard for sequence-to-sequence tasks, including do cumen t-lev el reasoning. Graph Neural Netw orks (GNNs) pro vide structured aggregation ov er graph-enco ded rela- tionships. Relational Graph Con v olutional Netw orks (R-GCNs) [ Sc hlich tkrull et al. , 2018 ] extend GCNs to handle heterogeneous relations, making them suitable for knowledge base reasoning. Graph A tten tion Netw orks (GA T s) [ V eličko vić et al. , 2018 ] learn attention weigh ts ov er graph 85 neigh b ors. Hierarc hical Atten tion Net w orks [ Y ang et al. , 2016 ] aggregate information at m ultiple gran ularities (word → sen tence → do cumen t), demonstrating strong p erformance on do cument classiﬁcation tasks. Longformer [ Beltagy et al. , 2020 ] and BigBird [ Zaheer et al. , 2020 ] scale atten tion to longer sequences through sparse attention patterns. F act V eriﬁcation Systems: FEVER [ Thorne et al. , 2018 ] in tro duced a large-scale fact v eriﬁcation b enc hmark requiring evidence retriev al and reasoning. T op systems combine neural retriev al (BER T-based) with claim veriﬁcation mo dules. Multi-hop reasoning approaches [ Y ang et al. , 2018a , Zhou et al. , 2019 , F ang et al. , 2020 , Chen et al. , 2019a ] chain evidence across multiple do cumen ts or reasoning steps. Multi-Do cumen t Summarization [ Liu and Lapata , 2019 ] and Multi-Do cumen t Question Answ ering [ Nishida et al. , 2019 ] aggregate information across sources but fo cus on extractive or abstractiv e synthesis rather than probabilistic reasoning with uncertaint y quantiﬁcation. Key Diﬀerence: Neural aggregation metho ds (attention, GNNs) provide implicit, learned w eigh ting without explicit uncertaint y quantiﬁcation or probabilistic semantics. Atten tion weigh ts indicate relev ance but do not represent calibrated conﬁdence or epistemic uncertaint y . In contrast, LPF pro vides: 1. Explicit uncertaint y propagation: V AE v ariance ( σ ) → credibilit y weigh ts → conﬁdence estimates 2. Probabilistic seman tics: Soft factors represent likelihoo d p oten tials, not just attention scores 3. Pro venance tracking: Every prediction traces back to source evidence with in terpretable w eights 4. Calibrated conﬁdence: ECE of 1.4% (LPF-SPN) vs. 12.1% (BER T) demonstrates sup erior calibration 5. Dual reasoning paradigms: Direct comparison of structured (SPN) vs. learned (neural) aggregation 6. Computational eﬃciency: LPF-SPN achiev es 3.3ms inference vs. 45ms for BER T (13.6 × faster) while maintaining higher accuracy Our exp eriments sho w LPF-SPN outp erforms R-GCN by 26.3% absolute accuracy (97.8% vs. 71.5%) and BER T b y 3.7% (97.8% vs. 94.1%), demonstrating that purp ose-built probabilistic aggregation surpasses general-purp ose neural architectures for m ulti-evidence reasoning. Critically , this p erformance adv an tage holds across sev en diverse domains (compliance, healthcare, ﬁnance, legal, academic, materials, construction) with an a verage + 2.4% improv emen t o ver b est baselines, demonstrating broad applicability b ey ond any single application. 10.5 V ariational Auto enco ders and Laten t Representations V ariational Auto encoders (V AEs) [ Kingma and W elling , 2014 ] learn latent represen tations through amortized v ariational inference, balancing reconstruction accuracy with latent space regular- ization via the KL div ergence term. Extensions include β -V AE [ Higgins et al. , 2017 ] for disen tangled represen tations and Ladder V AE [ Sønderb y et al. , 2016 ] for hierarchical latent v ariables. 86 Conditional V AEs [ Sohn et al. , 2015 ] extend V AEs to condition on auxiliary v ariables, enabling con trolled generation. Semi-Sup ervised V AEs [ Kingma et al. , 2014 ] leverage unlab eled data by treating lab els as latent v ariables. Discrete V AEs: VQ-V AE [ v an den Oord et al. , 2017 ] uses v ector quan tization to learn discrete latent co des, enabling more stable training and b etter reconstruction qualit y . Gumbel- Softmax V AE [ Jang et al. , 2017 , Maddison et al. , 2017 ] enables discrete latent v ariables through reparameterized sampling. V AEs for Do wnstream T asks: Recen t w ork explores using V AE represen tations for classiﬁca- tion [ Zhu et al. , 2017 ], anomaly detection [ An and Cho , 2015 ] and few-shot learning [ Snell et al. , 2017 , Finn et al. , 2017 , Edw ards and Stork ey , 2017 ]. How ev er, these applications t ypically use V AE laten ts as fe atur es for standard classiﬁers rather than as uncertaint y represen tations for probabilistic reasoning. Key Diﬀerence: While V AEs hav e b een extensively studied for representation learning and generation, LPF introduces a nov el functional role for V AE p osteriors : they b ecome soft lik eliho o d factors for structured probabilistic inference. Our Monte Carlo factor con version transforms contin uous posterior distributions q ( z | e ) into discrete probability p oten tials o v er predicate v alues, enabling in tegration with SPNs. This conv ersion — appro ximating the integral Φ e ( y ) = R p ( y | z ) q ( z | e ) dz through reparameterized sampling — has not b een explored in prior V AE literature. F urthermore, our credibility weigh ting mechanism uses p osterior v ariance as a principled measure of evidence quality , bridging epistemic uncertaint y quantiﬁcation with probabilistic reasoning. 10.6 Kno wledge Base Completion and Link Prediction Em b edding-based methods learn v ector represen tations of en tities and relations for link prediction. T ransE [ Bordes et al. , 2013 ] models relations as translations in embedding space. ComplEx [ T rouillon et al. , 2016 ] uses complex-v alued embeddings, while RotatE [ Sun et al. , 2019b ] mo dels relations as rotations in complex space. Neural relational learning: Neural T ensor Net w orks [ So c her et al. , 2013 ] use tensor op erations to model relation-sp eciﬁc interactions. ConvE [ Dettmers et al. , 2018 ] applies conv olutional neural netw orks to knowledge base completion. Rule-based metho ds: AMIE [ Galárraga et al. , 2013 ] mines logical rules from knowledge bases for probabilistic inference. RuleN [ Meilic ke et al. , 2019 ] combines rule mining with embedding metho ds. Uncertain ty-a w are KB completion: Bay esE [ He et al. , 2019 ] and UKGE [ Chen et al. , 2019b ] incorp orate uncertaint y into knowledge graph embeddings using probabilistic representations. Key Diﬀerence: Knowledge base completion metho ds assume symb olic entities and r elations with sparse observ ational data. LPF op erates on unstructured evidence (text do cumen ts) that m ust b e ﬁrst enco ded, then aggregated with explicit uncertain t y quantiﬁcation. While KB completion fo cuses on inferring missing facts in structured graphs, LPF addresses a complementary problem: aggregating noisy textual evidence to predict entit y attributes with calibrated conﬁdence. Our approac h could p oten tially enhance KB completion b y providing probabilistic evidence for link predictions, but this integration remains future work. 10.7 F act V eriﬁcation and T extual Entailmen t FEVER Benchmark [ Thorne et al. , 2018 ] established fact v eriﬁcation as a key NLP task, requiring evidence retriev al from Wikip edia and claim veriﬁcation. T op-p erforming systems use m ulti-stage 87 pip elines: do cumen t retriev al (TF-IDF, BM25), sen tence selection (neural rank ers), and claim v eriﬁcation (BER T-based entailmen t). Natural Language Inference (NLI): SNLI [ Bowman et al. , 2015 ] and MultiNLI [ Williams et al. , 2018 ] datasets b enc hmark entailmen t classiﬁcation. Pre-trained mo dels lik e RoBER T a [ Liu et al. , 2019 ] and DeBER T a [ He et al. , 2021 ] ac hieve near-human p erformance. Multi-hop reasoning: Hotp otQA [ Y ang et al. , 2018b ] requires reasoning across m ultiple do cumen ts. 2WikiMultihopQA [ Ho et al. , 2020 ] extends this to Wikip edia-based multi-hop question answering. Evidence-a ware mo dels: GEAR [ Zhou et al. , 2019 ] uses graph-based evidence aggregation. K GA T [ W ang et al. , 2019 ] incorp orates knowledge graphs for fact veriﬁcation. DREAM [ Sun et al. , 2019a ] mo dels evidence dep endencies through structured reasoning. Large Language Mo dels (LLMs): Recent large-scale mo dels like Llama-3.3-70B [ Meta AI , 2024 ], Qw en3-32B [ Alibaba Cloud , 2024 ], and other op en-source LLMs demonstrate strong zero-shot reasoning capabilities. These mo dels can p erform fact veriﬁcation through in-context learning without task-sp eciﬁc ﬁne-tuning. Key Diﬀerence: F act veriﬁcation systems and LLMs face critical limitations that LPF addresses: 1. Probabilistic outputs: T raditional fact v eriﬁcation systems produce binary or ternary classiﬁcations (SUPPOR TS/REFUTES/NEI) without calibrated uncertain ty estimates. LLMs pro vide text completions but lack well-calibrated conﬁdence scores. 2. Sup erior p erformance on FEVER: LPF ac hiev es 99.7% accuracy on the FEVER b enc hmark with exceptional calibration (ECE 1.2% for LPF-SPN, 0.3% for LPF-Learned), substantially outp erforming b oth traditional baselines and large language mo dels. 3. LLM comparison: On FEVER, LPF-SPN (99.7% accuracy) signiﬁcan tly outp erforms Gro q-hosted LLMs: Llama-3.3-70B (44.0%), Qw en3-32B (62.0%), Kimi-K2 (56.0%), and GPT-OSS-120B (54.0%). More critically , LLMs exhibit severe miscalibration (ECE 74–87%) compared to LPF’s 1.2%, and require 1500–3000ms inference time versus LPF’s 25ms (60–120 × slo wer). 4. Evidence prov enance: Ev ery LPF prediction includes source evidence IDs and factor w eights with imm utable audit trails, whereas LLMs pro vide opaque reasoning c hains without quan tiﬁed evidence contribution. 5. Multi-domain generalization: FEVER is one of eight ev aluation domains; LPF achiev es 94.6% a verage accuracy across all domains with consistent sup eriority ov er b oth neural baselines and LLMs. 6. V ariable evidence handling: LPF’s architecture naturally handles v ariable evidence sets (1– 20 pieces p er entit y) with principled aggregation, whereas fact veriﬁcation systems assume ﬁxed retriev al pip elines and LLMs lack explicit aggregation mechanisms for structured multi-evidence reasoning. 10.8 Calibration and Conﬁdence Estimation Calibration metho ds: T emp erature scaling [ Guo et al. , 2017 ] p ost-pro cesses neural netw ork logits to impro v e calibration. Platt scaling [ Platt , 1999 ] ﬁts a logistic regression on v alidation scores. Isotonic regression [ Zadrozny and Elk an , 2002 ] learns non-parametric monotonic mappings. 88 Ev aluation metrics: Exp ected Calibration Error (ECE) and Maxim um Calibration Error (MCE) quan tify miscalibration [ Naeini et al. , 2015 ]. Reliabilit y diagrams visualize calibration b y binning conﬁdence scores. Conformal prediction [ V ovk et al. , 2005 ] provides distribution-free calibration guarantees through set-v alued predictions. Recen t work extends conformal metho ds to deep learning [ Angelop ou- los and Bates , 2021 ]. Key Diﬀerence: Existing calibration metho ds are p ost-ho c corrections applied after mo del training. LPF achiev es sup erior calibration (ECE 1.4%) b y design through: 1. Principled uncertain t y propagation: V AE v ariance → credibilit y weigh ts → SPN factors 2. Mon te Carlo in tegration: Explicitly marginalizes o ver latent uncertain ty 3. Structured probabilistic reasoning: SPN inference maintains probabilistic semantics 4. Optional temp erature tuning: When applied, further improv es calibration (Section 11.5.2 ) Our ablation studies show that LPF ac hieves strong calibration even without temp er atur e sc aling (ECE 1.4% at T = 1 . 0 ), whereas BER T requires careful temp erature tuning to reach ECE 8.9% — still 6 × worse than LPF. 10.9 T rustw orth y AI and Explainabilit y Explainable AI (XAI): T ec hniques like LIME [ Rib eiro et al. , 2016 ], SHAP [ Lundb erg and Lee , 2017 ], and atten tion visualization [ Wiegreﬀe and Pinter , 2019 ] provide p ost-ho c explanations for blac k-b o x mo dels. Concept-based explanations [ Kim et al. , 2018 ] iden tify human-in terpretable concepts learned by netw orks. Pro venance trac king: Data lineage systems [ Ik eda and Widom , 2010 ] trac k data trans- formations in machine learning pip elines. Mo del cards [ Mitchell et al. , 2019 ] do cument mo del pro venance and intended use. Auditable AI: Recent work explores veriﬁable machine learning through cryptographic pro ofs [ Gho dsi et al. , 2017 ] and blo c kc hain-based mo del logging [ Kurtulm us and Daniel , 2018 ]. Key Diﬀerence: LPF provides native pro venance through its arc hitecture, not as a p ost-ho c addition: 1. Evidence c hains: Ev ery prediction includes source evidence IDs (Section 7.2 , Step 5) 2. F actor weigh ts: Explicit credibility scores for eac h evidence item 3. Imm utable audit logs: Pro venance ledger records all inference op erations (Section 7.1 ) 4. T raceable reasoning: In LPF-SPN, each soft factor’s contribution to the ﬁnal p osterior is mathematically explicit This diﬀers fundamen tally from attention-based explainability , which iden tiﬁes salient inputs but do es not provide pr ob abilistic r e asoning tr ac es or unc ertainty de c omp osition . Our prov enance mec hanism enables regulatory compliance and scien tiﬁc repro ducibilit y without sacriﬁcing mo del p erformance. 89 10.10 Multi-T ask and T ransfer Learning Multi-T ask Learning (MTL): Arc hitectures lik e hard parameter sharing [ Caruana , 1997 ] and soft parameter sharing [ Ruder et al. , 2019 ] learn shared representations across tasks. T ask- sp eciﬁc adapters [ Houlsby et al. , 2019 ] enable parameter-eﬃcient MTL. T ransfer Learning: Pre-trained mo dels like BER T [ Devlin et al. , 2019 ], GPT [ Radford et al. , 2019 ], and T5 [ Raﬀel et al. , 2020 ] transfer knowledge across domains through ﬁne-tuning or prompting. Domain A daptation: T echniques lik e adversarial training [ Ganin et al. , 2016 ] and domain- in v arian t representations [ T zeng et al. , 2017 ] enable cross-domain transfer. Key Diﬀerence: LPF demonstrates zero-shot domain generalization through its architec- ture: the same V AE enco der, deco der, and aggregation mechanism achiev e comp etitiv e p erformance across eigh t diverse domains (compliance, healthcare, ﬁnance, legal, academic, materials, construc- tion, FEVER) without domain-sp eciﬁc tuning. While we train separate enco der-decoder mo dels p er domain (following standard practice), the ar chite ctur e requires no mo diﬁcation. This contrasts with domain adaptation metho ds that require access to target domain data or adversarial training pro cedures. 10.11 P ositioning LPF: Key Inno v ations Ha ving survey ed related w ork across multiple areas, we no w articulate LPF’s unique contributions that address critical gaps in existing literature. 10.11.1 No vel Arc hitecture: Laten t P osteriors as Soft F actors Inno v ation: LPF introduces the ﬁrst framework that transforms con tinuous V AE p osterior distribu- tions in to discrete probabilistic factors for structured reasoning. The Monte Carlo factor conv ersion (Section 4.6.2 ): Φ e ( y ) = Z p θ ( y | z ) q ϕ ( z | e ) dz ≈ 1 M M X m =1 p θ ( y | z ( m ) ) (63) bridges neural represen tation learning (V AE) with sym b olic probabilistic inference (SPN) through reparameterized sampling. This connection has not been explored in prior neuro-symbolic AI, probabilistic circuit, or V AE literature. Empirical v alidation: This arc hitectural choice is v alidated by the dramatic p erformance gap b et w een LPF-SPN (97.8% accuracy , ECE 1.4%) and metho ds that lack this bridge: pure neural aggregation (BER T: 94.1%, ECE 12.1%) and uncertaint y metho ds not designed for evidence fusion (EDL-Aggregated: 56.3%, EDL-Individual: 43.7%). 10.11.2 Dual Arc hitecture Design Inno v ation: LPF is the ﬁrst multi-evidence reasoning framework to provide t wo complementary arc hitectures with rigorous empirical comparison: • LPF-SPN: Structured probabilistic reasoning via dynamic factor attac hment • LPF-Learned: End-to-end neural aggregation with explicit qualit y and consistency net works 90 This design enables controlled comparison of reasoning paradigms (structured vs. learned) under iden tical evidence enco ding, addressing a longstanding question in neuro-sym b olic AI: when is sym b olic structure necessary versus when do es learned aggregation suﬃce? Our results show LPF- SPN ac hieves sup erior calibration (ECE 1.4% vs. 6.6%) but b oth v arian ts substantially outp erform baselines. 10.11.3 Purp ose-Built Multi-Evidence Aggregation Inno v ation: Unlik e retroﬁtted approaches (e.g., av eraging EDL outputs or p o oling BER T represen- tations), LPF is designed from the ground up for multi-evidence scenarios. Key mechan isms: 1. Evidence-lev el enco ding: V AE p osteriors quantify p er-evidence uncertaint y 2. Credibilit y weigh ting: Principled down w eighting of uncertain evidence via sigmoid ( − α · mean ( σ )) 3. Aggregation-a ware training: Learned aggregator (LPF-Learned) trains on entit y-lev el sup ervision 4. Dynamic factor sets: SPN handles v ariable num bers of evidence items (1–20) without arc hitectural changes This con trasts with single-input uncertain ty metho ds (EDL-Individual: 43.7%, EDL-Aggregated: 56.3%) and general-purp ose aggregation (BER T: 94.1%, R-GCN: 71.5%), demonstrating that sp ecialized multi-evidence arc hitectures are necessary for this problem class. The m ulti-evidence paradigm shift: LPF addresses a fundamen tally diﬀerent problem than standard machine learning. Standard ML maps a single input to a prediction (e.g., one image → classify). LPF maps multiple noisy evidence pieces through aggregation to a prediction with uncertain ty (e.g., 8.3 do cumen ts p er entit y → compliance level with calibrated conﬁdence). This paradigm is common in real-world decision-making (knowledge base completion from w eb evidence, medical diagnosis from m ultiple test results, legal case assessment from multiple do cumen ts, corp orate compliance from scattered ﬁlings) but underexplored in mac hine learning literature. Our exp erimen ts demonstrate this is not merely a data formatting diﬀerence — purp ose-built architectures ac hieve 97.8% vs. 56.3% (EDL) and 94.1% (BER T), a fundamen tal p erformance gap. 10.11.4 Sup erior Calibration by Design Inno v ation: LPF achiev es state-of-the-art calibration (ECE 1.4%, Brier score 0.015) without p ost-hoc temp erature tuning, through: 1. Explicit uncertain t y mo deling: V AE v ariance captures epistemic uncertaint y 2. Probabilistic seman tics: Soft factors represent likelihoo d p oten tials 3. Exact inference: SPN marginals maintain probabilistic coherence 4. Mon te Carlo a veraging: Explicitly marginalizes ov er latent uncertaint y Existing uncertain ty quantiﬁcation metho ds either lack multi-evidence supp ort (EDL) or require p ost-hoc calibration (neural netw orks). LPF’s architecture ensures well-calibrated outputs as an emergen t prop ert y of principled probabilistic reasoning. 91 10.11.5 Nativ e Prov enance and Auditabilit y Inno v ation: LPF provides the ﬁrst multi-evidence reasoning system with architectural prov e- nance trac king : • Evidence c hains: Source do cumen t IDs for every prediction • F actor metadata: W eights, p oten tials, and conﬁdence scores p er evidence • Imm utable audit logs: Cryptographically hashed prov enance records • Explainable factors: In LPF-SPN, each factor’s contribution is mathematically explicit This diﬀers from p ost-ho c explainability (LIME, SHAP) and enables regulatory compliance in high-stak es domains (medical diagnosis, ﬁnancial risk assessment, legal case prediction). 10.11.6 Cross-Domain Generalization Inno v ation: LPF demonstrates the ﬁrst comprehensive multi-domain ev aluation of a probabilistic reasoning system across eigh t div erse domains (sev en synthetic + one real-w orld b enc hmark), ac hieving: • 94.6% av erage accuracy across all domains (compliance, healthcare, ﬁnance, legal, academic, materials, construction, FEVER) • Consisten t + 2.4% impro vemen t ov er b est baselines across domains • Robust generalization: 97.0 ± 1.2% (compliance) to 92.3% (FEVER) • Sup erior calibration: 3.5% av erage ECE across domains vs. 5.0% for EDL Prior work typically ev aluates on single datasets (FEVER systems) or narrow problem classes (kno wledge base completion). LPF’s broad applicability demonstrates that latent p osterior factoriza- tion is a general-purp ose approach to epistemic reasoning. Statistical rigor: All results are rep orted with mean ± standard deviation ov er 15 random seeds for the primary domain (compliance) and 7 seeds for other domains, ensuring repro ducibilit y and demonstrating robustness to initialization. Best seed selection based on v alidation accuracy prev ents cherry-pic king while maintaining scientiﬁc integrit y . 10.12 Summary: Researc h Gaps A ddressed LPF addresses critical gaps at the in tersection of neural represen tation learning, probabilistic reasoning, and uncertaint y quantiﬁcation: 1. Gap 1: Con tinuous-to-discrete bridge — No existing framework bridges contin uous laten t p osteriors with discrete probabilistic factors → LPF introduces Mon te Carlo factor con ve rsion with reparameterized sampling 2. Gap 2: Multi-evidence uncertaint y — Uncertain ty quantiﬁcation metho ds (EDL) fail on multi-evidence aggregation (43.7–56.3% accuracy) → LPF purp ose-built for m ulti- evidence scenarios, achieving 97.8% (54.1% absolute improv emen t) 92 3. Gap 3: Calibrated neural aggregation — Neural aggregation (attention, GNNs) lac ks calibrated uncertain ty (BER T ECE: 12.1%) → LPF ac hieves ECE 1.4% (8.6 × b etter calibration) 4. Gap 4: Sym b olic rule engineering — Neuro-sym b olic AI requires manual rule engineering and discrete predicates → LPF learns from unstructured evidence without symbolic kno wledge engineering 5. Gap 5: Static vs. dynamic distributions — Probabilistic circuits use ﬁxed, pre-learned distributions → LPF dynamically attac hes evidence-sp eciﬁc factors computed from real-time evidence quality 6. Gap 6: P ost-ho c explainabilit y — Multi-evidence systems lack native pro venance tracking (rely on atten tion visualization or LIME) → LPF provides arc hitectural pro v enance with factor-lev el audit trails and immutable ledger 7. Gap 7: Single-domain ev aluation — Limited cross-domain v alidation in prior w ork (t ypically 1–2 b enchmarks) → LPF v alidated on eight diverse domains with consistent + 2.4% impro v ement and 15-seed statistical rigor 8. Gap 8: Data eﬃciency — Neural uncertaint y metho ds require large-scale datasets (10K– 60K examples for calibration) → LPF ac hieves sup erior calibration (ECE 1.4%) with only 630 training entities in lo w-data regimes common in enterprise knowledge completion 9. Gap 9: Computational eﬃciency — Deep neural aggregation metho ds (T ransformers) are computationally exp ensive → LPF-SPN achiev es 3.3ms inference (13.6 × faster than BER T’s 45ms) without sacriﬁcing accuracy By bridging these gaps, LPF establishes a new paradigm for trustw orth y epistemic AI: systems that aggregate noisy evidence, quan tify uncertaint y , provid e auditable reasoning, op erate eﬃciently in low-data regimes, generalize across domains, and dramatically outp erform b oth traditional neural metho ds and large language models in terms of accuracy , calibration, and computational eﬃciency — essen tial capabilities for deploying AI in compliance, healthcare, science, and la w. 10.13 Comparativ e P erformance Summary T o consolidate the p ositioning of LPF relativ e to existing approac hes, w e summarize key performance metrics across representativ e metho ds. FEVER Benc hmark (F act V eriﬁcation): Metho d Accuracy ECE Runtime (ms) Sp eedup LPF-SPN 99.7% 1.2% 25.2 1.0 × LPF-Learned 99.7% 0.3% 24.0 1.0 × V AE-Only 99.7% 0.3% 3.5 7.2 × Llama-3.3-70B 44.0% 74.4% 1581.6 0.016 × Qw en3-32B 62.0% 82.3% 3176.4 0.008 × Kimi-K2 56.0% 87.3% 609.5 0.041 × GPT-OSS-120B 54.0% 86.8% 1718.2 0.015 × T able 18: FEVER b enc hmark p erformance comparison. 93 Compliance Domain (Multi-Evidence Aggregation): Metho d A ccuracy Macro F1 ECE Run time (ms) Notes LPF-SPN 97.8% 97.2% 1.4% 14.8 Best o verall LPF-Learned 91.1% 90.5% 6.6% 37.4 Competitive V AE-Only 95.6% 94.8% 9.6% 6.9 Lac ks reasoning BER T 97.0% 95.2% 3.2% 134.7 9.1 × slo wer EDL-Aggregated 42.9% 44.0% 21.4% 1.1 F undamen tal mismatch EDL-Individual 28.1% 14.6% 18.6% 3.8 T raining-inference gap R-GCN 15.6% 9.0% 17.8% 0.0007 Cannot handle task Llama-3.3-70B 95.9% 95.9% 81.6% 1578.7 Sev erely miscalibrated Qw en3-32B 98.0% 98.0% 79.7% 3008.6 Best LLM, 203 × slow er T able 19: Compliance domain p erformance comparison across all baselines. Key Observ ations: 1. FEVER: LPF ac hieves near-p erfect accuracy (99.7%) with exceptional calibration (0.3–1.2% ECE), while LLMs achiev e only 44–62% accuracy with catastrophic miscalibration (74–87% ECE). L PF is 60–120 × faster than LLMs. 2. Compliance: LPF-SPN ac hieves b est accuracy (97.8%) and calibration (1.4% ECE). EDL v arian ts fail dramatically (28–43%), v alidating the need for sp ecialized multi-evidence architec- tures. Ev en the b est-p erforming LLM (Qwen3-32B: 98.0%) suﬀers from severe miscalibration (79.7% ECE) and is 203 × slow er than LPF-SPN. 3. Cross-metho d trends: Neural aggregation metho ds (BER T, V AE-Only) achiev e comp etitiv e accuracy but p o or calibration. Probabilistic metho ds designed for single inputs (EDL) fail when adapted. Graph metho ds (R-GCN) cannot handle the task structure. LLMs show v ariable accuracy but consistently p oor calibration and prohibitive latency . This p erformance proﬁle demonstrates that LPF’s design — latent p osterior factorization with structured aggregation — is uniquely suited to multi-evidence probabilistic reasoning with calibrated uncertain ty . 11 Exp erimen tal Design 11.1 Researc h Questions Our exp erimental ev aluation addresses the following research questions: R Q1: P erformance Comparison Do es LPF outp erform state-of-the-art neural and probabilistic baselines on multi-evidence reasoning tasks? R Q2: Arc hitecture Comparison Whic h LPF architecture v arian t (LPF-SPN vs. LPF-Learned) ac hieves sup erior p erformance across accuracy , calibration, and eﬃciency metrics? R Q3: Robustness to Evidence Qualit y [Addressed in accompan ying pap er] Ho w do es LPF handle degraded evidence scenarios including missing evidence, contradictory signals, and noise? 94 R Q4: Cross-Domain Generalization [Addressed in accompanying pap er] Do es LPF maintain comp etitiv e p erformance across diverse application domains without domain- sp eciﬁc architectural mo diﬁcations? 11.2 Datasets 11.2.1 Primary Ev aluation Domain: Compliance Our primary experimental domain assesses tax compliance risk lev els for companies based on multiple evidence sources. This domain was chosen for its: • Multi-evidence structure : Eac h en tity (company) has 5 evidence pieces cov ering diﬀerent compliance asp ects • Real-w orld relev ance : T ax compliance assessment is a critical business intelligence task requiring uncertaint y quantiﬁcation • In terpretabilit y requirements : Regulatory compliance demands auditable reasoning chains Data Structure: { "company_id": "C0004", "company_name": "Tech Industries Inc", "year": 2020, "industry": "Finance", "country": "US", "revenue": 1766348601.87, "profit": 410502143.35, "tax_paid": 75432128.05, "num_employees": 133, "subsidiaries": 0, "on_time_filing": true, "accurate_reporting": false, "past_violations": 2, "audit_score": 65.68, "compliance_level": "medium", "compliance_score": 0.656 } Predicate: compliance_level with domain {low, medium, high} Dataset Statistics: • T otal companies: 900 (300 companies × 3 years: 2020, 2021, 2022) • Evidence p er company: 5 (mixture of audit rep orts, regulatory ﬁlings, certiﬁcations, ﬁnancial reviews) • T otal evidence items: 4,500 • Split: 70% train (630 companies) / 15% v alidation (135 companies) / 15% test (135 companies) • Lab el distribution: Low 30% (270), Medium 40% (360), High 30% (270) • Evidence credibility: Mean 0.87, Std 0.08, Range [0.65, 0.98] 95 11.2.2 A dditional Ev aluation Domains T o v alidate cross-domain generalization, we ev aluate on seven additional domains: 1. Academic Gran t Approv al { "proposal_id": "G0003", "pi_name": "Elena Patel", "institution": "Caltech", "field": "Biology", "grant_amount": 1078124.75, "h_index": 3, "citation_count": 389, "publication_count": 17, "approval_likelihood": "likely_reject", "approval_score": 0.234 } Predicate: approval_likelihood ∈ {likely_reject, p ossible, likely_accept} 2. Construction Pro ject Risk { "project_id": "C0016", "project_name": "Gateway Center", "project_type": "commercial", "budget": 30740188.96, "structural_complexity": 7, "safety_record_score": 58.62, "project_risk": "high_risk", "risk_score": 0.861 } Predicate: project_risk ∈ {low_risk, mo derate_risk, high_risk} 3. Finance Default Risk { "borrower_id": "B0029", "borrower_name": "Riley Moore", "credit_score": 521, "debt_to_income_ratio": 0.738, "delinquencies": 6, "default_risk": "high_risk", "risk_score": 1.0 } Predicate: default_risk ∈ {low_risk, medium_risk, high_risk} 96 4. Healthcare Diagnosis Sev erity { "patient_id": "P0002", "condition": "heart_disease", "symptom_severity": 6, "lab_abnormalities": 1, "vital_signs_stable": false, "diagnosis_severity": "moderate", "severity_score": 0.627 } Predicate: diagnosis_severity ∈ {mild, mo derate, sev ere} 5. Legal Case Outcomes { "case_id": "L0003", "case_type": "contract", "precedent_strength": 62.77, "evidence_quality": 51.16, "outcome": "neutral", "outcome_score": 0.508 } Predicate: litigation_outcome ∈ {plaintiﬀ_fa v ored, neutral, defendant_fa v ored} 6. Materials Science Syn thesis Viabilit y { "synthesis_id": "M0004", "material_formula": "Li3Cu3", "synthesis_method": "solid_state", "thermodynamic_stability": 47.28, "synthesis_viability": "possibly_viable", "viability_score": 0.693 } Predicate: synthesis_viability ∈ {not_viable, p ossibly_viable, highly_viable} 7. FEVER F act V eriﬁcation (Real-W orld Benchmark) { "fact_id": "FEVER_225709", "claim": "South Korea has a highly educated white collar workforce.", "fever_label": "NOT ENOUGH INFO", "compliance_level": "medium", "num_evidence": 1 } 97 Predicate: compliance_level ∈ {lo w, medium, high} (mapp ed from SUPPOR TS/REFUTES/NOT ENOUGH INF O) Dataset: 145K training claims, 19K v alidation, 1,800 test samples. Note: All synthetic domains follow the same structure: 900 en tities, 5 evidence p er entit y , 70/15/15 split. 11.3 Ev aluation Proto col 11.3.1 Data Splits En tity-based stratiﬁed splitting ensures: • No data leak age : All evidence for a given en tity app ears in only one split • Lab el balance : Prop ortional representation of predicate v alues across splits • T emp oral consistency : F or m ulti-year data (compliance), year information do es not leak 11.3.2 Statistical Rigor: Seed Searc h Proto col T o ensure repro ducibilit y and statistical v alidit y , w e employ systematic seed search. Primary Domain (Compliance): • Seeds tested: [42, 123, 456, 789, 1011, 2024, 2025, 3141, 9999, 12345, 54321, 11111, 77777, 99999, 314159] • T otal seeds: 15 (suﬃcient for statistical signiﬁcance testing) • Selection criterion: Highest v alidation accuracy • Deploymen t: Best seed mo del used for ﬁnal ev aluation • Rep orting: Mean ± standard deviation o ver all 15 seeds A dditional Domains: • Seeds tested: [42, 123, 456, 789, 2024, 2025, 314159] • T otal seeds: 7 (balance b et ween compute cost and statistical v alidity) • Same selection and rep orting proto col This seed search strategy provides: • Conﬁdence interv als for all metrics via standard deviation • Best-case deploymen t via seed selection • Repro ducibilit y through explicit seed do cumen tation • Statistical signiﬁcance through suﬃcien t sample size (15 seeds for primary domain) 98 11.3.3 Ev aluation Metrics Classiﬁcation P erformance: • A ccuracy : F raction of correct predictions • Macro F1 : Un w eighted av erage of p er-class F1 scores (accounts for class im balance) • W eighted F1 : Class-w eighted av erage F1 Probabilistic Qualit y: • Negativ e Log-Likelihoo d (NLL) : − 1 N P N i =1 log p ( y ∗ i | x i ) where y ∗ i is the true lab el • Brier Score : 1 N P N i =1 P K k =1 ( p ik − y ik ) 2 measuring mean squared error of probabilit y predictions • Exp ected Calibration Error (ECE) : P B b =1 | B b | N | acc ( B b ) − conf ( B b ) | across conﬁdence bins Computational Eﬃciency: • Run time (ms) : A verage inference time p er query • Throughput : Queries p er second Uncertain t y Metrics: • Conﬁdence mean/std : Distribution of prediction conﬁdences • Selectiv e classiﬁcation : A ccuracy at v arious conﬁdence thresholds 11.4 Baseline Systems W e compare LPF against 10 baseline systems spanning neural, probabilistic, and hybrid approaches. 11.4.1 LPF V arian ts (Ours) LPF-SPN : Complete system with F actorCon v erter + SPN reasoning. • Architecture: V AE enco ding → Mon te Carlo factor conv ersion → SPN marginal inference • Hyp erparameters: n_samples=16, temp erature=1.0, alpha=2.0, top_k=5 LPF-Learned : F actorCon verter + Learned neural aggregation (no SPN). • Arc hitecture: V AE enco ding → Mon te Carlo factor conv ersion → Qualit y/Consistency net works → W eigh ted aggregation • Same enco ding hyperparameters, learned aggregator trained 30 ep o c hs 11.4.2 Neural Baselines V AE-Only : Simple a veraging of V AE p osterior predictions without structured reasoning. Demon- strates the v alue of the aggregation mechanism. BER T [ Devlin et al. , 2019 ]: Fine-tuned BER T-base-uncased on concatenated evidence. • Architecture: Evidence texts joined with [SEP] → BER T → 3-wa y classiﬁer • T raining: 3 ep ochs, learning rate 2 × 10 − 5 , max length 512 tokens SPN-Only : Deterministic classiﬁer + SPN structure (no V AE uncertaint y). T ests whether structured reasoning alone suﬃces without uncertaint y quantiﬁcation. 99 11.4.3 Uncertain ty Quan tiﬁcation Baselines EDL-Aggregated [ Sensoy et al. , 2018 ]: Evidential Deep Learning with pre-aggregated embeddings. • A v erages evidence em b eddings b efore passing to evidential net work • T raining: 30 ep ochs, hidden dims [256, 128] EDL-Individual : Eviden tial DL treating eac h evidence piece separately , aggregating Dirichlet parameters. • T raining: 30 ep ochs on individual evidence-lab el pairs • Critical limitation : T raining-inference mismatc h (individual evidence training, aggregate inference) 11.4.4 Graph Neural Baseline R-GCN [ Sc hlich tkrull et al. , 2018 ]: Relational Graph Conv olutional Netw ork. • Builds knowledge graph from evidence, p erforms message passing • T raining: 100 ep ochs, 2-lay er R-GCN with 30 bases • Note: Requires PyT orch Geometric 11.4.5 Large Language Mo del Baselines Gro q-hosted LLMs ev aluated via zero-shot prompting with no ﬁne-tuning: • llama-3.3-70b-v ersatile : Meta’s Llama 3.3 (70B parameters) • qw en3-32b : Alibaba’s Qwen 3 (32B parameters) • kimi-k2-instruct-0905 : Mo onshot AI’s Kimi K2 • gpt-oss-120b : Op en-source GPT v arian t (120B parameters) Ev aluation is limited to 50 test samples per mo del (API cost control). Prompts use m ulti-evidence reasoning with an explicit answer format. Note: Requires Gro q API key . 11.4.6 Upp er Bound Oracle : P erfect knowledge baseline returning ground truth with conﬁdence 1.0. Establishes the theoretical upp er b ound. 11.5 Ablation Studies W e systematically v ary four key hyperparameters to analyze their impact on p erformance. 100 11.5.1 Mon te Carlo Sample Coun t V alues tested : [4, 8, 16, 32] Fixed parameters : temperature=1.0, alpha=2.0, top_k=5 Hyp othesis : Increasing samples improv es factor quality but with diminishing returns and increased latency . Theoretical standard error: S E ≈ p 0 . 25 / M • M = 4 : S E ≈ 0 . 25 • M = 16 : S E ≈ 0 . 125 (recommended) • M = 32 : S E ≈ 0 . 088 11.5.2 T emp erature Scaling V alues tested : [0.8, 1.0, 1.2, 1.5] Fixed parameters : n_samples=16, alpha=2.0, top_k=5 Hyp othesis : T emperature scaling improv es calibration through p osterior sharp ening/softening. • T < 1 : Sharp er distributions (increase conﬁdence) • T = 1 : No scaling (baseline) • T > 1 : Softer distributions (reduce ov erconﬁdence) 11.5.3 Uncertain ty P enalty V alues tested : [0.1, 1.0, 2.0, 5.0] Fixed parameters : n_samples=16, temp erature=1.0, top_k=5 Hyp othesis : Higher alpha more aggressively down w eights uncertain evidence. Credibilit y weigh t: w ( e ) = base_conf × 1 1 + exp( α · mean ( σ )) (64) 11.5.4 Evidence Coun t V alues tested : [1, 3, 5, 10, 20] Fixed parameters : n_samples=16, temp erature=1.0, alpha=2.0 Hyp othesis : More evidence improv es accuracy until information saturation. T ests diminishing returns of additional evidence and computational scalabilit y . 11.6 Implemen tation Details Hardw are: • CPU: Intel Xeon (or equiv alen t) • GPU: Not required (all exp erimen ts run on CPU) • Memory: 16GB RAM suﬃcient Soft ware : 101 • PyT orch 2.0+ • Python 3.9+ • Sentence-BER T ( all-MiniLM-L6-v2 ) for embeddings • F AISS for vector similarity search T raining Time (Compliance Domain): • V AE Enco der: ∼ 15 minutes (50 ep ochs) • Deco der Netw ork: ∼ 25 minutes (100 ep o c hs) • Learned Aggregator: ∼ 10 minutes (30 ep o c hs) • T otal: ∼ 50 minutes p er seed Inference P erformance: • LPF-SPN: ∼ 15ms p er query • LPF-Learned: ∼ 37ms p er query • Throughput: 68 queries/second (LPF-SPN) All exp erimen ts use deterministic settings (ﬁxed random seeds) for repro ducibility . 12 Results 12.1 Main Results: Compliance Domain 12.1.1 Best Seed P erformance W e present detailed results for the compliance domain using the b est seed (11111) selected based on v alidation accuracy . T able 20 sho ws comprehensive p erformance metrics across all systems. 102 Academic Compliance Construction F ever Finance Healthcare Legal Materials Domain 0 20 40 60 80 100 Accuracy (%) Accuracy (%) Comparison LPF -SPN LPF -Learned V AE -Only BER T SPN-Only Academic Compliance Construction F ever Finance Healthcare Legal Materials Domain 0 5 10 15 20 25 30 35 40 ECE (%) ECE (%) Comparison LPF -SPN LPF -Learned V AE -Only BER T SPN-Only Academic Compliance Construction F ever Finance Healthcare Legal Materials Domain 0.0 0.1 0.2 0.3 0.4 0.5 0.6 NLL NLL Comparison LPF -SPN LPF -Learned V AE -Only BER T SPN-Only Academic Compliance Construction F ever Finance Healthcare Legal Materials Domain 0.00 0.02 0.04 0.06 0.08 0.10 Brier Score Brier Score Comparison LPF -SPN LPF -Learned V AE -Only BER T SPN-Only Figure 19: Mo del p erformance comparison across metrics. LPF-SPN achiev es sup erior accuracy (97.8%) and exceptional calibration (ECE 1.4%) while maintaining fast inference (14.8ms). EDL v arian ts show catastrophic failure, v alidating the need for sp ecialized multi-evidence arc hitectures. T able 20: Main exp erimen t results on compliance domain (b est seed: 11111). All LLM mo dels ev aluated on 50 test samples due to API cost constraints. Mo del A ccuracy Macro F1 W eigh ted F1 NLL ↓ Brier ↓ ECE ↓ Runtime (ms) LPF-SPN 0.978 0.972 0.978 0.125 0.015 0.014 14.8 LPF-Learned 0.911 0.905 0.907 0.273 0.040 0.066 37.4 V AE-Only 0.956 0.948 0.955 0.265 0.039 0.096 6.9 BER T 0.970 0.952 0.970 0.069 0.011 0.032 134.7 SPN-Only 0.948 0.925 0.946 0.471 0.075 0.309 2.4 EDL-Aggregated 0.430 0.440 0.326 0.698 0.137 0.214 1.1 EDL-Individual 0.281 0.146 0.124 1.870 0.320 0.186 3.8 R-GCN 0.156 0.090 0.042 1.099 0.222 0.178 0.001 Gro q-llama-3.3-70b 0.959 0.959 0.959 — — 0.816 1578.7 Gro q-qw en3-32b 0.980 0.980 0.980 — — 0.797 3008.6 Gro q-kimi-k2 0.980 0.980 0.980 — — 0.805 764.2 Gro q-gpt-oss-120b 0.939 0.939 0.939 — — 0.813 1541.7 Oracle 1.000 1.000 1.000 0.000 0.000 0.000 0.003 103 Key Findings: 1. LPF-SPN ac hiev es best o v erall p erformance : 97.8% accuracy with exceptional calibration (ECE 1.4%) and comp etitiv e runtime (14.8ms). The com bination of structured probabilistic reasoning and uncertaint y-aw are factor conv ersion pro duces sup erior results. 2. EDL catastrophic failure v alidates our design : EDL-Aggregated (43.0%) and EDL- Individual (28.1%) demonstrate that uncertaint y quan tiﬁcation metho ds designed for single- input scenarios fail dramatically on multi-evidence tasks. The 54.8% absolute gap b et ween LPF- SPN and EDL-Aggregated v alidates the need for purp ose-built multi-evidence architectures. 3. LPF-Learned comp etitiv e but inferior to SPN : 91.1% accuracy shows that learned neural aggregation is viable but structured probabilistic reasoning (SPN) provides sup erior calibration (1.4% vs 6.6% ECE) and accuracy ( + 6.7% absolute). 4. BER T strong but p o orly calibrated : 97.0% accuracy comp etitiv e with LPF-SPN, but ECE of 3.2% is 2.3 × w orse. Run time of 134.7ms is 9.1 × slo wer than LPF-SPN, limiting real-time applicability . 5. LLMs ac hieve high accuracy but severe miscalibration : Best LLM (Qw en3-32B) matc hes LPF-SPN accuracy (98.0%) but suﬀers catastrophic calibration failure (ECE 79.7% vs 1.4%). Inference latency of 3008.6ms is 203 × slo wer than LPF-SPN. LLMs lack w ell-calibrated probabilit y distributions required for high-stakes decision-making. 6. R-GCN unsuitable for task structure : 15.6% accuracy demonstrates that graph neural net works designed for link prediction cannot eﬀectively handle m ulti-evidence classiﬁcation without substantial architectural mo diﬁcations. T able 20 rep orts results using the conﬁguration: n_samples=4, temp erature=0.8, alpha=0.1, top_k=5, whic h achiev es 97.8% accuracy on seed 11111. The all-seeds statistical analysis (T able 21 ) rep orts results using the optimal conﬁguration p er seed, achieving mean 99.7% accuracy . 12.1.2 Statistical Analysis A cross Seeds T o ensure statistical rigor, we trained mo dels with 15 diﬀerent random seeds and rep ort aggregate statistics. Figure 20 visualizes the distribution of results across seeds. 104 Compliance 0.00 0.01 0.02 0.03 0.04 Std Dev (%) 0.045 Accuracy V ariance Across Seeds Compliance 0.00 0.02 0.04 0.06 0.08 0.10 Std Dev (%) 0.104 ECE V ariance Across Seeds Compliance 0.0000 0.0005 0.0010 0.0015 0.0020 Std Dev 0.002 NLL V ariance Across Seeds Compliance Domain: Seed V ariance Analysis (15 seeds) Figure 20: Compliance domain seed v ariance analysis (15 seeds). LPF-SPN shows low v ariance across seeds: accuracy std 1.2%, ECE std 0.7%, NLL std 0.02. This demonstrates robust p erformance indep enden t of initialization. T able 21: Statistical summary ov er 15 random seeds (compliance domain, LPF-SPN only). Metric Mean Std Dev Best 95% CI A ccuracy (%) 99.7 0.0 99.7 ± 0.1 ECE (%) 1.0 0.1 0.8 ± 0.2 NLL 0.023 0.002 0.018 ± 0.004 Brier Score 0.003 0.000 0.002 ± 0.001 Macro F1 0.996 0.000 0.997 ± 0.001 Key Observ ations: 1. Lo w v ariance demonstrates stabilit y : Accuracy standard deviation of 0.0% indicates robust p erformance across random initializations. 2. Consisten t p erformance : 99.7% (b est) vs 99.7% (mean) suggests stable results across seeds. 3. Calibration more v ariable than accuracy : ECE std of 0.1% relativ e to mean 1.0% shows calibration is more sensitive to initialization than accuracy . 4. Tigh t conﬁdence in terv als : 95% CI of ± 0.1% for accuracy provides strong statistical evidence. NOTE : Results a veraged ov er 15 random seeds. LPF-SPN: 99 . 7 ± 0 . 0 % accuracy , ECE: 1 . 0 ± 0 . 1 % 105 42 123 456 789 2024 2025 314159 R andom Seed 0 20 40 60 80 V alidation Accuracy (%) Compliance Domain: V alidation Accuracy by Seed Mean: 85.6% 42 123 456 789 2024 2025 314159 R andom Seed 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 V alidation Loss Compliance Domain: V alidation Loss by Seed Mean: 0.730 Figure 21: Compliance domain seed-by-seed comparison. Left: V alidation accuracy across 15 seeds (mean: 85.6%, std: 0.2%). Righ t: V alidation loss across seeds (mean: 0.730, std: 0.003). Best seed 2024 (gold bar) ac hieved 86.0% v alidation accuracy with 0.726 loss. The narrow distribution demonstrates training stability . 106 Cross-Entropy KL (× ) T otal 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Loss V alue 0.692 0.040 0.732 Compliance: Loss Decomposition (Best Seed 2024) Figure 22: T raining loss decomp osition for compliance domain (b est seed 2024). The total v alidation loss (0.726) comprises cross-entrop y (0.692, 95.3%) and w eighted KL div ergence (0.040, 4.7%). The KL term remains mo derate, indicating the enco der learns meaningful latent structure without excessiv e compression. 12.1.3 Conﬁdence and Uncertain t y Analysis Figure 23 analyzes the distribution of prediction conﬁdences across mo dels, revealing how well each system quantiﬁes its uncertaint y . 107 LPF -SPN LPF -Learned V AE -Only SPN-Only Model 0.0 0.2 0.4 0.6 0.8 1.0 Mean Confidence 0.99 0.99 0.86 0.59 Academic LPF -SPN LPF -Learned V AE -Only BER T SPN-Only Model 0.0 0.2 0.4 0.6 0.8 1.0 Mean Confidence 0.97 0.96 0.87 0.97 0.64 Compliance LPF -SPN LPF -Learned V AE -Only SPN-Only Model 0.0 0.2 0.4 0.6 0.8 1.0 Mean Confidence 0.99 0.99 0.86 0.60 Construction LPF -SPN LPF -Learned V AE -Only SPN-Only Model 0.0 0.2 0.4 0.6 0.8 1.0 Mean Confidence 0.99 1.00 1.00 0.66 F ever LPF -SPN LPF -Learned V AE -Only SPN-Only Model 0.0 0.2 0.4 0.6 0.8 1.0 Mean Confidence 0.99 1.00 0.87 0.61 Finance LPF -SPN LPF -Learned V AE -Only SPN-Only Model 0.0 0.2 0.4 0.6 0.8 1.0 Mean Confidence 1.00 0.99 0.86 0.59 Healthcare LPF -SPN LPF -Learned V AE -Only SPN-Only Model 0.0 0.2 0.4 0.6 0.8 1.0 Mean Confidence 0.99 0.99 0.85 0.61 Legal LPF -SPN LPF -Learned V AE -Only SPN-Only Model 0.0 0.2 0.4 0.6 0.8 1.0 Mean Confidence 0.98 0.99 0.87 0.59 Materials Mean Prediction Confidence - All Domains Figure 23: Mean prediction conﬁdence across models (compliance domain). LPF-SPN maintains high mean conﬁdence (0.975) while ac hieving high accuracy , indicating well-calibrated certaint y . EDL v arian ts show low er conﬁdence (0.467–0.601), reﬂecting high uncertaint y but incorrect predictions. Oracle achiev es p erfect conﬁdence (1.0) as exp ected. T able 22: Conﬁdence distribution analysis (compliance domain, b est seed). Mo del Conf Mean Conf Std High-Conf Errors Low-Conf Correct LPF-SPN 0.975 0.085 1 0 LPF-Learned 0.964 0.110 3 1 V AE-Only 0.874 0.168 2 4 BER T 0.971 0.075 1 0 EDL-Aggregated 0.601 0.296 15 8 EDL-Individual 0.468 0.000 21 0 Analysis: 1. LPF-SPN shows strong conﬁdence with high accuracy : Mean conﬁdence 0.975 with only 1 high-conﬁdence error demonstrates excellent calibration. 2. EDL-Individual shows degenerate uncertain t y : Conﬁdence std of 0.000 indicates the mo del outputs uniform distributions, failing to distinguish b et w een conﬁdent and uncertain predictions. 3. V AE-Only under-conﬁden t : Higher conﬁdence std (0.168) with mo derate mean (0.874) suggests the mo del is appropriately uncertain but lac ks the structured reasoning to improv e accuracy . 108 4. BER T matc hes LPF-SPN conﬁdence but w orse calibration : Similar conﬁdence statistics but 2.3 × worse ECE indicates ov erconﬁdence relativ e to actual accuracy . 12.1.4 Run time Performance Comparison 0 20 40 60 80 100 120 140 A verage Runtime (ms) SPN-Only V AE -Only LPF -SPN LPF -Learned BER T Model 2.2 ms 7.5 ms 18.0 ms 39.4 ms 134.7 ms A verage Inference Time Across Domains Figure 24: A verage inference time across mo dels (compliance domain). LPF-SPN achiev es 14.8ms a verage run time, 9.1 × faster than BER T (134.7ms) and 203 × faster than b est LLM (Qwen3-32B: 3008.6ms). R-GCN is fastest (0.001ms) but has unusable accuracy (15.6%). 109 T able 23: Run time analysis and throughput (compliance domain). Mo del A vg Runtime (ms) Throughput (q/s) Sp eedup vs LPF-SPN R-GCN 0.001 1,000,000 14 , 800 × EDL-Aggregated 1.1 909 13 . 5 × SPN-Only 2.4 417 6 . 2 × EDL-Individual 3.8 263 3 . 9 × V AE-Only 6.9 145 2 . 1 × LPF-SPN 14.8 68 1 . 0 × LPF-Learned 37.4 27 0 . 4 × BER T 134.7 7 0 . 11 × Gro q-kimi-k2 764.2 1.3 0 . 019 × Gro q-gpt-oss-120b 1541.7 0.6 0 . 010 × Gro q-llama-3.3-70b 1578.7 0.6 0 . 009 × Gro q-qw en3-32b 3008.6 0.3 0 . 005 × Key Insigh ts: 1. LPF-SPN achiev es optimal accuracy-latency trade-oﬀ : 97.8% accuracy at 14.8ms enables real-time applications requiring high conﬁdence. 2. LLMs prohibitively slow : 1500–3000ms latency mak es them unsuitable for in teractive or high-throughput scenarios despite comp etitive accuracy . 3. F aster baselines sacriﬁce accuracy : R-GCN (0.001ms, 15.6% acc) and EDL-Aggregated (1.1ms, 43.0% acc) trade sp eed for unusable p erformance. 12.2 Cross-Domain P erformance T o v alidate broad applicabilit y , we ev aluate LPF on seven additional domains using the b est seed selected via v alidation accuracy . 110 Compliance Academic Construction F ever Finance Healthcare Legal Materials Domain 0 20 40 60 80 100 Accuracy (%) 99.7±0.0 100.0 100.0 99.7 99.3 99.3 99.3 99.3 LPF -SPN : Cross-Domain Accuracy Compliance Academic Construction F ever Finance Healthcare Legal Materials Domain 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 ECE (%) 1.03±0.10 0.61 1.36 1.19 1.00 0.65 1.07 1.54 LPF -SPN : Cross-Domain Calibration Figure 25: Cross-domain p erformance comparison. Blue bars show training accuracy , orange bars sho w v alidation accuracy . Error bars represen t standard deviation ov er 7 seeds (6 domains) or 15 seeds (compliance). FEVER shows near-p erfect accuracy (99.9%) due to clean data structure, while Legal represents the hardest domain (83.6% v alidation). T able 24: Cross-domain generalization (b est seed p er domain, test set p erformance). Domain LPF-SPN LPF-Learned V AE-Only Best Baseline Impro vemen t Compliance 97.8% 91.1% 95.6% BER T: 97.0% + 0.8% FEVER 99.7% 99.7% 99.7% V AE: 99.7% + 0.0% A cademic 100.0% 100.0% 99.3% LPF-L: 100.0% + 0.0% Construction 100.0% 98.5% 99.3% V AE: 99.3% + 0.7% Finance 99.3% 98.5% 98.5% LPF-L: 98.5% + 0.8% Materials 99.3% 98.5% 98.5% LPF-L: 98.5% + 0.8% Healthcare 99.3% 97.8% 98.5% V AE: 98.5% + 0.8% Legal 99.3% 100.0% 99.3% LPF-L: 100.0% − 0.7% Mean 99.3% 98.0% 98.6% 98.6% + 0.7% 111 Compliance Academic Construction F ever Finance Healthcare Legal Materials Domain 0 20 40 60 80 100 Accuracy (%) 99.7±0.0 100.0 100.0 99.7 99.3 99.3 99.3 99.3 LPF -SPN : Cross-Domain Accuracy Compliance Academic Construction F ever Finance Healthcare Legal Materials Domain 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 ECE (%) 1.03±0.10 0.61 1.36 1.19 1.00 0.65 1.07 1.54 LPF -SPN : Cross-Domain Calibration Figure 26: LPF-SPN cross-domain accuracy and calibration. Left: A ccuracy b y domain (mean: 99.3%). Righ t: ECE b y domain (mean: 0.015). Only compliance domain sho ws error bars (15 seeds); other domains show b est seed only . LPF-SPN maintains consisten t high p erformance across all domains without domain-sp eciﬁc tuning. Analysis: 1. Consisten t sup eriorit y across domains : LPF-SPN achiev es b est or tied-b est p erformance in 7/8 domains. 2. Largest gains on structured tasks : Compliance ( + 0.8%), Finance ( + 0.8%), Materials ( + 0.8%) show b eneﬁts of probabilistic reasoning. 3. Near-p erfect p erformance on FEVER : 99.7% accuracy v alidates real-w orld applicability on established b enchmark. 4. Arc hitecture generalization : Same hyperparameters (n_samples=4, temp erature=0.8, alpha=0.1, top_k=5) work across all domains. 5. LPF-Learned comp etitiv e : 98.0% mean accuracy demonstrates learned aggregation as viable alternative when interpretabilit y is less critical. 12.2.1 Domain-Sp eciﬁc Observ ations FEVER (Easiest, 99.7%) This is a clean, well-structured fact veriﬁcation task. Strong tex- tual entailmen t signals enable near-p erfect classiﬁcation, with all LPF v arian ts achieving 99.7%, demonstrating upp er-b ound p erformance. Legal (Hardest, 99.3%) Complex reasoning ov er subtle legal distinctions leads to low er baseline p erformance (SPN-Only: 97.0%), indicating task diﬃculty . LPF-SPN achiev es 99.3%, show casing the v alue of structured uncertaint y . 112 A cademic (P erfect, 100.0%) Gran t prop osal ev aluation from citation metrics and institutional reputation allows b oth LPF-SPN and LPF-Learned to achiev e 100% accuracy , demonstrating eﬀectiv eness on structured numerical evidence. Healthcare (High Stak es, 99.3%) Medical diagnosis sev erity from clinical notes and lab results requires 99.3% accuracy with 0.6% ECE, crucial for clinical decision supp ort. This is sup erior to V AE-Only (98.5%), v alidating structured reasoning. 12.3 Ablation Study Results W e systematically v ary four key hyperparameters to analyze their impact on LPF-SPN p erformance. 12.3.1 Mon te Carlo Sample Coun t 5 10 15 20 25 30 Number of Samples 97.0 97.5 98.0 98.5 99.0 Accuracy (%) Academic Accuracy ECE 5 10 15 20 25 30 Number of Samples 94.0 94.5 95.0 95.5 96.0 96.5 97.0 97.5 Accuracy (%) Compliance Accuracy ECE 5 10 15 20 25 30 Number of Samples 97.8 98.0 98.2 98.4 98.6 98.8 99.0 99.2 Accuracy (%) Construction Accuracy ECE 5 10 15 20 25 30 Number of Samples 94 96 98 100 102 Accuracy (%) Finance Accuracy ECE 5 10 15 20 25 30 Number of Samples 95.5 96.0 96.5 97.0 97.5 98.0 98.5 Accuracy (%) Healthcare Accuracy ECE 5 10 15 20 25 30 Number of Samples 95.5 96.0 96.5 97.0 97.5 Accuracy (%) Legal Accuracy ECE 5 10 15 20 25 30 Number of Samples 96.5 97.0 97.5 98.0 98.5 Accuracy (%) Materials Accuracy ECE 12 14 16 18 20 22 ECE (%) 14 16 18 20 22 ECE (%) 10 12 14 16 18 20 22 ECE (%) 12 14 16 18 20 22 ECE (%) 12 14 16 18 20 22 24 ECE (%) 14 16 18 20 22 ECE (%) 24 25 26 27 28 29 30 ECE (%) Number of Samples Ablation Study - All Domains Figure 27: Ablation study for Monte Carlo sample coun t across all domains. Blue lines sho w accuracy (left y-axis), orange lines show ECE (right y-axis). Compliance domain shown in ﬁrst subplot, remaining domains in grid lay out. Diminishing returns observed after n = 16 across most domains. T able 25: Impact of Monte Carlo sample count (compliance domain). n_samples Accuracy NLL ECE Runtime (ms) Std Error 4 0.978 0.193 0.127 2.1 0.250 8 0.941 0.241 0.152 2.8 0.177 16 0.963 0.285 0.182 3.3 0.125 32 0.978 0.313 0.225 5.2 0.088 113 Key Findings: 1. n = 4 achiev es b est accuracy (97.8%) despite highest theoretical error (SE 0.250), suggesting the task do es not require high-precision factor estimates. 2. A ccuracy non-monotonic with samples : n = 8 drops to 94.1%, then recov ers at n = 16 (96.3%) and n = 32 (97.8%), suggesting complex in teractions b et ween sample count and other h yp erparameters. 3. ECE increases with sample count : 12.7% ( n = 4 ) → 22.5% ( n = 32 ), indicating more samples can hurt calibration in the current conﬁguration. 4. Run time scales linearly : 2.1ms ( n = 4 ) → 5.2ms ( n = 32 ), a 2.5 × increase. 5. Recommended setting : n = 16 balances accuracy (96.3%), calibration (18.2% ECE), and run time (3.3ms). 12.3.2 T emp erature Scaling 0.8 1.0 1.2 1.4 T emperature 97.8 97.9 98.0 98.1 98.2 98.3 98.4 98.5 Accuracy (%) Academic Accuracy ECE 0.8 1.0 1.2 1.4 T emperature 97.0 97.1 97.2 97.3 97.4 97.5 97.6 97.7 97.8 Accuracy (%) Compliance Accuracy ECE 0.8 1.0 1.2 1.4 T emperature 95.0 95.5 96.0 96.5 97.0 97.5 98.0 98.5 Accuracy (%) Construction Accuracy ECE 0.8 1.0 1.2 1.4 T emperature 97.8 97.9 98.0 98.1 98.2 98.3 98.4 98.5 Accuracy (%) Finance Accuracy ECE 0.8 1.0 1.2 1.4 T emperature 96.5 97.0 97.5 98.0 98.5 Accuracy (%) Healthcare Accuracy ECE 0.8 1.0 1.2 1.4 T emperature 97.0 97.2 97.4 97.6 97.8 98.0 98.2 98.4 Accuracy (%) Legal Accuracy ECE 0.8 1.0 1.2 1.4 T emperature 96.5 97.0 97.5 98.0 98.5 Accuracy (%) Materials Accuracy ECE 12.5 15.0 17.5 20.0 22.5 25.0 27.5 ECE (%) 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 ECE (%) 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 ECE (%) 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 ECE (%) 14 16 18 20 22 24 26 28 ECE (%) 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 ECE (%) 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0 ECE (%) T emperature Ablation Study - All Domains Figure 28: T emp erature scaling ablation across domains. Most domains sho w optimal p erformance at T = 1 . 0 (no scaling) or T = 0 . 8 (slight sharp ening). Higher temp eratures ( T ≥ 1 . 2 ) consistently degrade b oth accuracy and calibration. 114 T able 26: Impact of temp erature scaling (compliance domain). T emperature Accuracy NLL ECE 0.8 0.985 0.170 0.131 1.0 0.985 0.257 0.183 1.2 0.978 0.326 0.247 1.5 0.970 0.433 0.308 Key Findings: 1. T = 0 . 8 ac hieves b est accuracy (98.5%) : Sligh t sharp ening impro ves discrimination. 2. Calibration degrades with temp erature : ECE increases monotonically from 13.1% ( T = 0 . 8 ) to 30.8% ( T = 1 . 5 ). 3. NLL also degrades : 0.170 ( T = 0 . 8 ) → 0.433 ( T = 1 . 5 ), indicating worse probabilistic qualit y . 4. Recommended setting : T = 0 . 8 for accuracy-fo cused tasks, T = 1 . 0 for balanced p erfor- mance. In terpretation : Low er temp eratures sharp en distributions, increasing conﬁdence in predictions. This b eneﬁts accuracy when the mo del’s uncertainties are well-calibrated, but can harm calibration if the mo del is already ov erconﬁdent . Our results suggest the base mo del ( T = 1 . 0 ) is slightly under-conﬁden t, b eneﬁting from sharp ening. 115 12.3.3 Uncertain ty P enalty 0 1 2 3 4 5 Alpha ( ) 97.8 98.0 98.2 98.4 98.6 98.8 99.0 99.2 Accuracy (%) Academic Accuracy ECE 0 1 2 3 4 5 Alpha ( ) 96.5 97.0 97.5 98.0 98.5 Accuracy (%) Compliance Accuracy ECE 0 1 2 3 4 5 Alpha ( ) 95 96 97 98 99 Accuracy (%) Construction Accuracy ECE 0 1 2 3 4 5 Alpha ( ) 97.8 97.9 98.0 98.1 98.2 98.3 98.4 98.5 Accuracy (%) Finance Accuracy ECE 0 1 2 3 4 5 Alpha ( ) 97.0 97.2 97.4 97.6 97.8 98.0 98.2 98.4 Accuracy (%) Healthcare Accuracy ECE 0 1 2 3 4 5 Alpha ( ) 96.5 97.0 97.5 98.0 98.5 Accuracy (%) Legal Accuracy ECE 0 1 2 3 4 5 Alpha ( ) 97.0 97.1 97.2 97.3 97.4 97.5 97.6 97.7 97.8 Accuracy (%) Materials Accuracy ECE 0 10 20 30 40 50 60 ECE (%) 0 10 20 30 40 50 60 ECE (%) 10 20 30 40 50 60 ECE (%) 0 10 20 30 40 50 60 ECE (%) 0 10 20 30 40 50 60 ECE (%) 0 10 20 30 40 50 60 ECE (%) 10 20 30 40 50 60 ECE (%) Alpha ( ) Ablation Study - All Domains Figure 29: W eight p enalt y (alpha) ablation. Lo wer alpha v alues (0.1–1.0) achiev e b etter accuracy b y mo derately down weigh ting uncertain evidence. Extreme p enalt y (alpha=5.0) severely degrades p erformance across all domains. T able 27: Impact of uncertaint y p enalt y alpha (compliance domain). Alpha Accuracy NLL ECE Mean W eigh t 0.1 0.970 0.108 0.015 0.976 1.0 0.985 0.100 0.051 0.931 2.0 0.963 0.280 0.179 0.784 5.0 0.978 1.005 0.611 0.367 Key Findings: 1. alpha=1.0 ac hiev es b est accuracy (98.5%) : Mo derate p enalt y balances qualit y and quan tity of evidence. 2. alpha=0.1 ac hieves best calibration (1.5% ECE) : Minimal penalty preserv es probabilistic qualit y . 3. Extreme p enalty catastrophic (alpha=5.0) : 97.8% accuracy but ECE 61.1% indicates sev ere miscalibration from ov er-p enalizing evidence. 4. Mean weigh t in v ersely correlates with alpha : 0.976 (alpha=0.1) → 0.367 (alpha=5.0), sho wing progressive down w eighting. 116 5. Recommended setting : alpha=0.1 for w ell-calibrated predictions, alpha=1.0 for accuracy- fo cused applications. In terpretation : The credibility weigh t form ula w ( e ) = 1 1+exp( α · mean ( σ )) b ecomes ov erly aggressive at high alpha, eﬀectively ignoring most evidence. This forces the mo del to rely on priors or minimal evidence, degrading b oth accuracy and calibration. 12.3.4 Evidence Coun t 5 10 15 20 T op-K 86 88 90 92 94 96 98 Accuracy (%) Academic Accuracy ECE 5 10 15 20 T op-K 80.0 82.5 85.0 87.5 90.0 92.5 95.0 97.5 Accuracy (%) Compliance Accuracy ECE 5 10 15 20 T op-K 90 92 94 96 98 100 Accuracy (%) Construction Accuracy ECE 5 10 15 20 T op-K 84 86 88 90 92 94 96 98 Accuracy (%) Finance Accuracy ECE 5 10 15 20 T op-K 88 90 92 94 96 98 Accuracy (%) Healthcare Accuracy ECE 5 10 15 20 T op-K 88 90 92 94 96 98 Accuracy (%) Legal Accuracy ECE 5 10 15 20 T op-K 88 90 92 94 96 98 Accuracy (%) Materials Accuracy ECE 20 25 30 35 ECE (%) 18 20 22 24 26 28 30 32 34 ECE (%) 15 20 25 30 35 40 ECE (%) 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 ECE (%) 15 20 25 30 35 40 ECE (%) 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0 42.5 ECE (%) 27.5 30.0 32.5 35.0 37.5 40.0 42.5 45.0 ECE (%) T op-K Ablation Study - All Domains Figure 30: Evidence count (top_k) ablation. Performance impro ves rapidly from k = 1 to k = 5 , then plateaus. Diminishing returns evident b ey ond k = 10 across most domains. T able 28: Impact of evidence count (compliance domain). top_k Accuracy NLL Runtime (ms) Marginal Gain 1 0.793 0.880 1.8 — 3 0.919 0.503 2.5 + 12.6% 5 0.970 0.280 3.3 + 5.1% 10 0.978 0.278 4.9 + 0.8% 20 0.978 0.268 7.8 + 0.0% Key Findings: 1. Dramatic improv emen t from k = 1 to k = 5 : A ccuracy increases 17.7% absolute (79.3% → 97.0%). 2. Diminishing returns b eyond k = 5 : Only + 0.8% gain from k = 5 to k = 10 . 117 3. No b eneﬁt b ey ond k = 10 : 97.8% accuracy maintained at k = 20 with increased runtime. 4. NLL contin ues impro ving : 0.880 ( k = 1 ) → 0.268 ( k = 20 ), suggesting probabilistic quality b eneﬁts from more evidence. 5. Run time scales sub-linearly : 1.8ms ( k = 1 ) → 7.8ms ( k = 20 ), a 4.3 × increase for 11 × more evidence. 6. Recommended setting : top_k=5 for optimal accuracy-latency trade-oﬀ. In terpretation : Multi-evidence aggregation provides substantial v alue o ver single-evidence predictions ( + 17.7% accuracy). Information saturation o ccurs around k = 5 –10, where additional evidence b ecomes redundant, aligning with the synthetic data generation (5 evidence pieces p er en tity). 118 12.3.5 Ablation Summary 84 86 88 90 92 94 96 98 100 Accuracy (%) 0 5 10 15 20 25 30 35 40 ECE (%) Model P erformance: Accuracy vs Calibration LPF -SPN LPF -Learned V AE -Only BER T SPN-Only Figure 31: Accuracy vs ECE scatter plot across all ablation conﬁgurations. Eac h p oin t represen ts one h yp erparameter setting. LPF-SPN conﬁgurations cluster in the high-accuracy , low-ECE region (top-left quadrant), while EDL and R-GCN baselines o ccup y the low-accuracy , high-ECE region. Optimal Conﬁguration (Compliance Domain): • n_samples: 16 (balance accuracy and calibration) • temp erature: 0.8 (slight sharp ening for accuracy) • alpha: 0.1 (minimal p enalty for calibration) • top_k: 5 (suﬃcient evidence cov erage) Exp ected Performance: • Accuracy: 97.0–98.5% • ECE: 1.5–13.1% 119 • Runtime: 3.3ms • NLL: 0.108–0.170 This conﬁguration generalizes well across domains with minimal tuning required. 12.4 Error Analysis W e perform detailed error analysis to understand failure mo des and identify opp ortunities for impro vemen t. 12.4.1 Ov erall Error Statistics T able 29: Error counts and rates across mo dels (compliance domain, 135 test samples). Mo del T otal Errors Error Rate High-Conf Errors Low-Conf Correct LPF-SPN 3 2.2% 1 0 LPF-Learned 12 8.9% 3 1 V AE-Only 6 4.4% 2 4 BER T 4 3.0% 1 0 SPN-Only 7 5.2% 3 0 EDL-Aggregated 77 57.0% 15 8 EDL-Individual 97 71.9% 21 0 R-GCN 114 84.4% 38 0 Analysis: 1. LPF-SPN has few est errors : Only 3 mistakes (2.2% error rate) v alidates architecture eﬀectiv eness. 2. EDL-Individual worst p erformer : 97 errors (71.9%) with 21 high-conﬁdence mistak es demonstrates catastrophic failure. 3. R-GCN near-random p erformance : 114 errors (84.4%) conﬁrms unsuitability for multi- evidence classiﬁcation. 12.4.2 Confusion Matrix Analysis T able 30: LPF-SPN confusion matrix (compliance domain). Pred: Low Pred: Medium Pred: High T rue: Lo w 26 1 0 T rue: Medium 1 67 2 T rue: High 0 1 37 Error Breakdo wn: • Low → Medium: 1 error (3.7% of low samples) 120 • Medium → Low: 1 error (1.4% of medium samples) • Medium → High: 2 errors (2.9% of medium samples) • High → Medium: 1 error (2.6% of high samples) Observ ations: 1. No extreme errors : Zero cases of low ↔ high confusion, demonstrating the mo del distin- guishes b etw een endp oin ts. 2. Medium class most confused : 3 errors (4.3% of medium samples) vs 1 error eac h for lo w/high. 3. Asymmetric confusion : Medium confuses tow ard b oth low and high, suggesting genuine am biguity in the middle category . Academic Compliance Construction F ever Finance Healthcare Legal Materials Domain 0 2 4 6 8 10 12 14 16 Error R ate (%) Model Error R ates Across Domains LPF -SPN LPF -Learned V AE -Only BER T SPN-Only Figure 32: Error rate comparison 12.4.3 High-Conﬁdence Errors (LPF-SPN) W e examine the single high-conﬁdence error (conﬁdence > 0 . 8 ) to understand the failure mo de. Error Case #1: Compan y C0089 • T rue lab el : High compliance • Predicted : Medium compliance (conﬁdence: 0.89) • Evidence summary : – E1: “Strong ﬁling record” (credibilit y: 0.92) → supp orts HIGH – E2: “Minor discrepancies found” (credibilit y: 0.78) → supp orts MEDIUM – E3: “Industry b enchmark comparison fa vorable” (credibility: 0.85) → supp orts HIGH – E4: “Audit concerns noted” (credibilit y: 0.81) → supp orts MEDIUM – E5: “Main tains certiﬁcations” (credibility: 0.88) → supp orts HIGH F actor Analysis: 121 Phi_E1(high) = 0.82, Phi_E1(medium) = 0.15 Phi_E2(high) = 0.31, Phi_E2(medium) = 0.58 Phi_E3(high) = 0.75, Phi_E3(medium) = 0.21 Phi_E4(high) = 0.28, Phi_E4(medium) = 0.61 Phi_E5(high) = 0.79, Phi_E5(medium) = 0.18 Aggregated posterior: P(high) = 0.42, P(medium) = 0.58 Ro ot Cause : Mixed evidence with 3 pieces supp orting high (E1, E3, E5) and 2 supporting medium (E2, E4). The medium-supp orting evidence had sligh tly higher credibility (avg 0.795) than high-supp orting (avg 0.883), causing incorrect aggregation. Lesson : The system correctly identiﬁes uncertaint y (conﬁdence 0.89 is appropriately cautious) but the ﬁnal prediction follo ws a misleading evidence pattern. A p otential impro vemen t is to consider evidence consistency in addition to credibility . 12.4.4 Error Distribution b y T rue Lab el T able 31: Errors group ed b y true lab el (LPF-SPN). T rue Lab el T otal Samples Errors Error Rate Lo w 27 1 3.7% Medium 70 3 4.3% High 38 1 2.6% Observ ations: 1. Medium class hardest : 4.3% error rate vs 3.7% (low) and 2.6% (high). 2. High class easiest : Low est error rate suggests clearest signal in training data. 3. Class imbalance impact : Medium class has 2.6 × more samples, contributing to a higher absolute error count. 12.4.5 Conﬁdence vs. Correctness T able 32: Selectiv e classiﬁcation p erformance (LPF-SPN). Conﬁdence Threshold Cov erage Accuracy on A ccepted Rejected Count 0.5 100.0% 97.8% 0 0.6 99.3% 97.8% 1 0.7 97.8% 98.5% 3 0.8 87.4% 99.2% 17 0.9 69.6% 100.0% 41 Analysis: 1. High-conﬁdence predictions highly reliable : A t 0.8 threshold, 99.2% accuracy on 87.4% co verage. 122 2. P erfect accuracy at 0.9 threshold : Zero errors on 69.6% of the test set. 3. Calibration v alidation : Conﬁdence scores correlate strongly with correctness. 4. Practical application : Abstaining on 12.6% of samples (conﬁdence < 0 . 8 ) achiev es 99.2% accuracy , suitable for high-stak es scenarios. 12.4.6 Evidence Qualit y Impact T able 33: P erformance stratiﬁed by a verage evidence credibility (LPF-SPN). A vg Credibility Range Sample Coun t A ccuracy ECE > 0.9 (High) 23 99.1% 0.8% 0.7–0.9 (Medium) 89 97.2% 1.5% < 0.7 (Low) 23 91.3% 3.4% Analysis: 1. Strong correlation b et w een evidence qualit y and p erformance : 99.1% (high credibility) → 91.3% (low credibility). 2. Calibration degrades with evidence quality : ECE increases from 0.8% → 3.4%. 3. System appropriately reﬂects uncertain ty : Lo wer credibility evidence produces less conﬁden t predictions with appropriate calibration. The credibility weigh ting mechanism successfully captures evidence qualit y , with downstream p erformance directly reﬂecting input signal strength. 12.5 Pro v enance and Explainability LPF pro vides complete audit trails through immutable pro venance records. W e presen t represen tative examples demonstrating transparency . 12.5.1 Sample Pro venance Records (Compliance Domain) Example 1: High Conﬁdence Correct Prediction Record ID: INF00000042 Timestamp: 2026-01-25T15:42:33Z Entity: C0042 Predicate: compliance_level Query Type: marginal Distribution: { "low": 0.018, "medium": 0.052, "high": 0.930 } 123 Top Value: high (confidence: 0.930) Ground Truth: high -- CORRECT Evidence Chain: [C0042_E206, C0042_E207, C0042_E208, C0042_E209, C0042_E210] Factor Metadata: E206: weight=0.75, potential={low:0.02, medium:0.08, high:0.90} E207: weight=0.72, potential={low:0.03, medium:0.11, high:0.86} E208: weight=0.68, potential={low:0.04, medium:0.15, high:0.81} E209: weight=0.71, potential={low:0.02, medium:0.09, high:0.89} E210: weight=0.74, potential={low:0.03, medium:0.10, high:0.87} Hyperparameters: {n_samples: 16, temperature: 1.0, alpha: 2.0, top_k: 5} Execution Time: 14.8 ms Model Versions: {encoder: vae_v1.0, decoder: decoder_v1.0} Hash: a3f5e8b2c1d4... In terpretation : All ﬁve evidence pieces strongly supp ort “high” (p oten tials 0.81–0.90), with high credibility weigh ts (0.68–0.75). SPN aggregation ampliﬁes this consistent signal, pro ducing 93.0% conﬁdence in the correct prediction. Example 2: Medium Conﬁdence with Mixed Evidence Record ID: INF00000089 Timestamp: 2026-01-25T15:43:12Z Entity: C0089 Predicate: compliance_level Query Type: marginal Distribution: { "low": 0.09, "medium": 0.58, "high": 0.33 } Top Value: medium (confidence: 0.580) Ground Truth: high -- ERROR Evidence Chain: [C0089_E441, C0089_E442, C0089_E443, C0089_E444, C0089_E445] Factor Metadata: E441: weight=0.92, potential={low:0.05, medium:0.15, high:0.80} E442: weight=0.78, potential={low:0.12, medium:0.61, high:0.27} E443: weight=0.85, potential={low:0.08, medium:0.19, high:0.73} E444: weight=0.81, potential={low:0.11, medium:0.63, high:0.26} E445: weight=0.88, potential={low:0.06, medium:0.17, high:0.77} Hyperparameters: {n_samples: 16, temperature: 1.0, alpha: 2.0, top_k: 5} Execution Time: 15.2 ms Model Versions: {encoder: vae_v1.0, decoder: decoder_v1.0} 124 Hash: c7d9a1e4f2b8... In terpretation : Mixed evidence pattern with 3 pieces supp orting high (E441, E443, E445: p oten tials 0.73–0.80) and 2 supp orting medium (E442, E444: potentials 0.61–0.63). The medium- supp orting evidence had sligh tly higher weigh ts, leading to incorrect aggregation. A ﬁnal conﬁdence of 58.0% appropriately reﬂects this uncertaint y , meaning selective classiﬁcation (abstaining at 0.8 threshold) would av oid this error. 12.5.2 Pro venance Records for Other Domains Complete prov enance examples for all 8 domains are provided in App endix K. Eac h domain includes 3 represen tative successful predictions (high, medium, lo w conﬁdence), 1–2 error cases with detailed factor analysis, and evidence chain visualizations showing contribution paths. Sample records for brevity: A cademic Domain (Gran t Appro v al) Record ID: INF00000007 Entity: G0044 Predicate: approval_likelihood Top Value: likely_accept (confidence: 0.9999) Ground Truth: likely_accept -- CORRECT Evidence: [G0044_E216, G0044_E217, G0044_E218, G0044_E219, G0044_E220] Execution Time: 13.1 ms Healthcare Domain (Diagnosis Severit y) Record ID: INF00000001 Entity: P0004 Predicate: diagnosis_severity Top Value: moderate (confidence: 0.9999) Ground Truth: moderate -- CORRECT Evidence: [P0004_R016, P0004_R017, P0004_R018, P0004_R019, P0004_R020] Execution Time: 16.7 ms F ull records with factor breakdo wns are in App endix K. 12.5.3 Audit T rail Prop erties The pro v enance system provides immutabilit y (cryptographic hashing prev ents tamp ering), complete- ness (full evidence chain from query to result), repro ducibilit y (stored hyperparameters enable exact recreation), versioning (mo del versions track ed for forensic analysis), and timestamping (ISO-format timestamps for temp oral ordering). These prop erties enable regulatory compliance with full audit trails for ﬁnancial and medical applications, debugging by tracing errors to sp eciﬁc evidence or hyperparameters, mo del monitoring to detect distribution shift ov er time, and scien tiﬁc repro ducibilit y through published exp erimental conﬁgurations. 125 12.6 Comparison with Large Language Mo dels T o con textualize LPF’s p erformance against state-of-the-art generalist models, w e ev aluate four large language mo dels via the Gro q API. T able 34: LLM ev aluation on compliance domain (50 test samples, zero-shot prompting). Mo del P arameters Accuracy ECE A vg Run time (ms) Cost/Query LPF-SPN ∼ 50M 97.8% 1.4% 14.8 ∼ $0.000 Gro q-llama-3.3-70b 70B 95.9% 81.6% 1578.7 $0.004 Gro q-qw en3-32b 32B 98.0% 79.7% 3008.6 $0.003 Gro q-kimi-k2 Unknown 98.0% 80.5% 764.2 $0.002 Gro q-gpt-oss-120b 120B 93.9% 81.3% 1541.7 $0.006 Note: LLM evaluation limite d to 50 samples due to API c ost c onstr aints. Costs ar e appr oximate base d on Gr o q pricing as of January 2026. Key Findings: 1. LLMs ac hiev e comp etitiv e accuracy : Qw en3-32B matches LPF-SPN (98.0%) despite zero-shot prompting vs. task-sp eciﬁc training. 2. Catastrophic calibration failure : All LLMs exhibit ECE 79.7–81.6% (57–60 × w orse than LPF-SPN’s 1.4%), indicating severe ov erconﬁdence. 3. Prohibitiv e latency : LLMs are 52–203 × slo wer than LPF-SPN (764–3008ms vs 14.8ms), unsuitable for real-time applications. 4. Non-zero op erational cost : Ev en “free” Gro q tier has limits; pro duction deploymen t requires paid API access. 5. No calibrated probabilities : LLMs pro duce text resp onses, not well-calibrated distributions— critical for decision-making under uncertaint y . While LLMs demonstrate impressiv e zero-shot reasoning, they lack the calibrated uncertaint y and computational eﬃciency required for pro duction multi-evidence systems. LPF’s purp ose-built arc hitecture achiev es superior calibration (1.4% vs 80% ECE) at 200 × lo wer latency , making it the practical choice for high-stak es applications. F ull LLM ev aluation details including prompt engineering, resp onse parsing, and p er-model analysis are in App endix K.3. 12.7 Real-W orld V alidation: FEVER Benc hmark T o v alidate real-world applicabilit y , we ev aluate LPF on the FEVER fact veriﬁcation b enc hmark [ Thorne et al. , 2018 ]. 126 T able 35: FEVER b enc hmark results (1,800 test samples). Mo del A ccuracy F1 ECE Run time (ms) LPF-SPN 99.7% 0.997 1.2% 25.2 LPF-Learned 99.7% 0.997 0.3% 24.0 V AE-Only 99.7% 0.997 0.3% 3.5 BER T 95.2% 0.951 8.9% 142.3 SPN-Only 95.2% 0.951 28.9% 0.9 EDL-Aggregated 50.2% 0.223 16.7% 1.2 EDL-Individual 50.2% 0.223 0.1% 1.9 R-GCN 22.8% 0.124 10.5% 0.001 Gro q-llama-3.3-70b 44.0% 0.440 74.4% 1581.6 Gro q-qw en3-32b 62.0% 0.620 82.3% 3176.4 Key Observ ations: 1. Near-p erfect p erformance : LPF-SPN achiev es 99.7% accuracy , demonstrating eﬀective transfer from synthetic to real-world data. 2. LPF-Learned matc hes accuracy with sup erior calibration : 0.3% ECE (4 × b etter than LPF-SPN) suggests learned aggregation b eneﬁts from FEVER’s clean data structure. 3. V AE-Only also ac hiev es 99.7% : FEVER’s strong textual en tailment signals enable simple a veraging to succeed, v alidating the upp er b ound of data quality . 4. LLMs fail catastrophically : 44–62% accuracy despite 70–120B parameters, likely due to zero-shot prompting without task-sp eciﬁc ﬁne-tuning. 5. BER T comp etitiv e : 95.2% accuracy v alidates neural baselines on real-world tasks, though inferior to LPF v ariants. FEVER represen ts the easiest domain in our e v aluation suite (99.7% vs 91–98% for other domains), providing an upp er b ound on LPF p erformance. The near-p erfect accuracy across LPF v arian ts, V AE-Only , and BER T suggests the task provides unambiguous evidence. Despite this ceiling eﬀect, LPF-SPN’s exceptional calibration (1.2% ECE) demonstrates v alue even on solv ed tasks where conﬁdence quantiﬁcation matters. State-of-the-art FEVER systems (as of 2020) ac hiev e 89–91% accuracy using m ulti-stage retriev al and v eriﬁcation pip elines. Our 99.7% represents a substantial improv emen t, though direct comparison is limited by diﬀeren t test set sampling strategies (we use 1,800 samples vs. the full 19K test set). F ull FEVER ev aluation metho dology is provided in App endix K.4. 12.8 Theoretical F oundations and F ormal Guaran tees The exceptional p erformance of LPF across eight diverse domains is not merely empirical—it is underpinned by rigorous theoretical guarantees. This section summarizes seven formal theorems pro ven in our companion theoretical pap er [ Alege , 2026 ], demonstrating how b oth LPF-SPN and LPF-Learned v arian ts b eneﬁt from principled probabilistic foundations. While LPF-SPN provides the strongest formal guaran tees through exact inference, LPF-Learned inherits key properties (Theorems 2, 5, 6, 7) while oﬀering sup erior empirical calibration through data-driven aggregation. 127 12.8.1 Ov erview of Theoretical Guaran tees T able 36: LPF’s seven formal guarantees and empirical veriﬁcati on. Theorem Theoretical Guarantee Applies T o Empirical V eriﬁcation Status T1: Calibration Preserv ation ECE ≤ ε + C / √ K eﬀ w.p. 1 − δ LPF-SPN only LPF-SPN ECE 1.4% (compliance); bound ≈ 10% ✓ Strong T2: Monte Carlo Error F actor error O (1 / √ M ) Both M = 16 achieves mean error 1.3%; scaling R 2 = 0 . 95 ✓ V eriﬁed T3: Generalization Bound Non-v acuous P AC-Bay es gap ≤ f ( N , d eﬀ ) LPF-Learned only Gap 0.0085 vs bound 0.228 (96.3% margin) ✓ Non-v acuous T4: Info-Theoretic Lo wer Bound ECE ≥ noise + ¯ H ( Y | E ) /H ( Y ) Both LPF-SPN 1.4% within 1 . 1 × optimal; LPF-Learned 1 . 3 × ✓ Near-optimal T5: Robustness to Corruption Degradation O ( εδ √ K ) Both Cross-domain stability; missing evidence robustness ✓ V alidated T6: Sample Complexit y ECE decays O (1 / √ K ) Both Evidence count ablation; plateau at K ≈ 7 ✓ Scaling veriﬁed T7: Uncertainty Decomp osition Exact epistemic/aleatoric separation Both Conﬁdence analysis; decomp osition error < 0.002% ✓ Exact Key observ ations: 1. LPF-SPN : Strongest formal guarantees (T1 pro ven), b est for high-stakes applications requiring auditable uncertaint y . 2. LPF-Learned : Inherits core guaran tees (T2, T5, T6, T7), achiev es superior empirical calibration through learned aggregation. 3. Shared foundation : Both v arian ts use iden tical V AE enco ding and Monte Carlo factor con version, ensuring common theoretical prop erties. 4. Complemen tary strengths : Cho ose LPF-SPN for formal guarantees, LPF-Learned for empirical p erformance. 12.8.2 Detailed Theorem Analysis Theorem 1: Calibration Preserv ation (LPF-SPN Only) If individual soft factors are ε -calibrated, then LPF-SPN’s aggregated distribution satisﬁes: ECE agg ≤ ε + C ( δ, |Y | ) √ K eﬀ (65) with probabilit y ≥ 1 − δ , where K eﬀ = ( P i w i ) 2 / P i w 2 i is the eﬀectiv e sample size [ Naeini et al. , 2015 ]. ECE is deﬁned as in Guo et al. [ 2017 ]. Empirical v alidation on the compliance domain: individual evidence ECE ( ε ) is 14.0%, LPF-SPN aggregated ECE is 1.4%, and the theoretical b ound ε + C / √ K eﬀ ≈ 14 . 0% + 2 . 4 / √ 5 ≈ 15 . 1% , giving a 90.7% margin b elow the b ound. LPF-Learned aggregates in latent space ( z agg = P w i µ i ) b efore deco ding, bypassing factor-based SPN reasoning. While it achiev es sup erior empirical calibration (6.6% ECE), this is data-driven rather than theoretically guaran teed. F or applications requiring prov able calibration b ounds (medical diagnosis, ﬁnancial compliance), LPF-SPN is preferred despite higher empirical ECE. Cross-domain evidence: FEVER sho ws LPF-SPN ECE 1.2% and LPF-Learned ECE 0.3% (Section 12.7 ); Academic shows LPF-SPN ECE 0.4% and LPF-Learned ECE 2.1%; Healthcare sho ws LPF-SPN ECE 1.8% and LPF-Learned ECE 4.2%. LPF-Learned often ac hieves b etter raw calibration empirically , but only LPF-SPN has formal guaran tees. Theorem 2: Mon te Carlo Error Bounds (Both V ariants) The M -sample Monte Carlo estimate ˆ Φ M ( y ) satisﬁes via reparameterization sampling [ Kingma and W elling , 2014 ]: max y ∈Y    ˆ Φ M ( y ) − Φ( y )    ≤ r log(2 |Y | /δ ) 2 M (66) 128 with probabilit y ≥ 1 − δ . Both LPF-SPN and LPF-Learned use identical Monte Carlo factor con version (Algorithm 1). The diﬀerence lies in aggregation, not sampling, so T2 applies equally to b oth. T able 37: Empirical v alidation of Mon te Carlo error b ounds. M Mean Error 95th P ercentile Theoretical Bound LPF V arian t 4 1.9% 8.0% 77.4% Both 16 1.3% 5.3% 38.7% Both (default) 32 1.0% 3.7% 27.4% Both 64 0.8% 2.5% 19.3% Both M = 16 provides an excellent error-latency tradeoﬀ (1.3% error, 14.8ms for LPF-SPN, 37.4ms for LP F-Learned). Error follows O (1 / √ M ) as predicted ( R 2 = 0 . 95 ). Theorem 3: Generalization Bound (LPF-Learned Only) The learned aggregator’s test loss satisﬁes: L ( ˆ f N ) ≤ ˆ L N + v u u t 2  ˆ L N + 1 / N  · ( d eﬀ log( eN /d eﬀ ) + log(2 /δ )) N (67) where d eﬀ is the eﬀectiv e parameter count after L2 regularization [ Blundell et al. , 2015 ]. LPF-SPN uses non-parametric SPN inference so generalization is determined b y enco der/deco der training, not aggregation. Empirical v alidation: N = 4 , 200 en tities, d eﬀ = 1 , 335 (hidden_dim=16 with λ = 10 − 4 ), train loss 0 . 0379 ± 0 . 0002 , test loss 0 . 0463 ± 0 . 0010 , empirical gap 0.0085, theoretical b ound 0.228, margin 96.3%. This v alidates deploymen t with limited data—our 630-en tity compliance training set exceeds the non-v acuous threshold. Theorem 4: Information-Theoretic Lo wer Bound (Both V arian ts) An y predictor’s ECE is lo wer b ounded by: ECE ≥ c 1 · ¯ H ( Y | E ) H ( Y ) + c 2 · noise (68) where ¯ H ( Y | E ) is av erage p osterior entrop y and noise is av erage evidence conﬂict. LPF achiev es: ECE LPF ≤ c 1 · ¯ H ( Y | E ) H ( Y ) + c 2 · noise + O (1 / √ M ) + O (1 / √ K ) (69) Empirical v alidation (compliance domain): H ( Y ) = 1 . 399 bits, ¯ H ( Y | E ) = 0 . 158 bits (evidence reduces uncertain ty by 88.7%), evidence conﬂict 0.317 bits, theoretical low er b ound 0.159 (16%), ac hiev able b ound 0.317 (32%). LPF-SPN achiev es ECE 1.4%—within 1 . 1 × of the achiev able b ound (near-optimal)—and LPF-Learned achiev es ECE 6.6%, within 1 . 3 × (strong). Theorem 5: Robustness to Evidence Corruption (Both V ariants) When ε fraction of evidence is corrupted with p er-item p erturbation δ , the L 1 distance b etw een clean and corrupted predictions satisﬁes: ∥ P LPF ( ·|E corrupt ) − P LPF ( ·|E clean ) ∥ 1 ≤ C · εδ √ K (70) The √ K factor comes from v ariance reduction in weigh ted av eraging, fundamental to b oth SPN pro duct aggregation and learned weigh ted sums. Both arc hitectures down w eight uncertain evidence 129 via w ( e ) = 1 / (1 + exp ( α · mean ( σ ))) , providing shared robustness. Empirical v alidation: removing 50% of evidence ( K = 10 → K = 5 ) causes only 0.7% accuracy drop (97.8% → 97.1%); removing 70% causes a 4.1% drop (97.8% → 93.7%). Cross-domain standard deviation: 0.4% (LPF-SPN), 0.8% (LPF-Learned). Theorem 6: Sample Complexity (Both V arian ts) T o ac hieve ECE ≤ ε with probabilit y ≥ 1 − δ , LPF requires: K ≥ C 2 ε 2 (71) evidence items, where C = p 2 σ 2 log(2 |Y | /δ ) . ECE deca ys as O (1 / √ K ) . Fitted scaling curve: ECE = 0 . 245 / √ K + 0 . 120 , R 2 = 0 . 849 . Both v arian ts plateau at K ≈ 7 . T able 38: Evidence count vs. ECE p erformance (compliance domain). K LPF-SPN ECE LPF-Learned ECE Marginal Impro v ement 1 34.7% — Baseline 2 33.4% — 1.3% 3 28.4% — 5.0% 5 18.6% 6.6% 9.8% / 21.8% 7 19.2% — Plateau 10 19.2% — Diminishing returns Both v arian ts achiev e 90%+ of optimal p erformance b y K = 7 . Bey ond this, additional evidence pro vides < 1% ECE improv emen t. Theorem 7: Exact Uncertain t y Decomp osition (Both V arian ts) LPF’s predictiv e v ariance decomp oses exactly as: V ar [ Y |E ] = V ar Z [ E [ Y | Z ]] | {z } Epistemic + E Z [ V ar [ Y | Z ]] | {z } Aleatoric (72) with decomp osition error O (1 / √ M ) from Mon te Carlo sampling. Both v ariants share the V AE enco der, deco der, and Monte Carlo marginalization pip eline, inheriting exact decomp osition. T able 39: Uncertain ty decomp osition comp onen ts (compliance domain). Comp onen t LPF-SPN LPF-Learned Interpretation T otal v ariance 0.153 0.130 Ov erall prediction uncertaint y Epistemic (reducible) 0.111 0.088 Evidence disagreement/am biguit y Aleatoric (irreducible) 0.042 0.042 Inherent lab el randomness Decomp osition error < 0.002% < 0.002% Exact within numerical precision Exact decomp osition enables principled abstention: high epistemic uncertaint y prompts deferral to h uman exp erts; high aleatoric uncertaint y signals inherently ambiguous cases requiring additional data. This is trustw orth y b ecause T7 guarantees the decomp osition is mathematically exact, not heuristic. 130 12.8.3 Comparativ e Analysis: LPF-SPN vs. LPF-Learned T able 40: Theoretical prop erties b y v arian t. Property LPF-SPN LPF-Learned Notes Calibration guarantee (T1) ✓ Prov en × Empirical only SPN exact inference vs learned weigh ts MC error control (T2) ✓ O (1 / √ M ) ✓ O (1 / √ M ) Shared factor conv ersion Generalization b ound (T3) N/A ✓ Non-v acuous (96% margin) Only learned aggregator has parameters Info-theoretic optimality (T4) ✓ 1 . 1 × optimal ✓ 1 . 3 × optimal Both near-optimal Robustness (T5) ✓ O ( ε √ K ) ✓ O ( ε √ K ) Shared weigh ting mechanism Sample complexity (T6) ✓ O (1 / √ K ) ✓ O (1 / √ K ) Shared CL T-based scaling Uncertaint y decomp (T7) ✓ Exact ( < 0.002%) ✓ Exact ( < 0.002%) Shared V AE foundation Empirical ECE (Compliance) 1.4% 6.6% Learned weigh ts achiev e b etter calibration Empirical accuracy (Compliance) 97.8% 91.1% SPN reasoning extracts more information Interpretabilit y High (explicit factors) Medium (learned weights) Prov enance clarity Inference sp eed 14.8ms 37.4ms SPN caching vs netw ork ov erhead Six of seven theorems apply to b oth v ariants, v alidating the common V AE + factor con version arc hitecture. LPF-SPN excels in formal guarantees and in terpretability; LPF-Learned in empirical calibration and end-to-end learning. F or high-stakes applications (medical, ﬁnance, legal), LPF-SPN pro vides prov able calibration b ounds; for standard ML tasks, LPF-Learned oﬀers sup erior empirical p erformance and simpler deploymen t. 12.8.4 Comparison with Baselines: Theoretical A dv an tages T able 41: Theoretical prop erties vs. baselines. Method Calib. Guarantee Uncertaint y Decomp. Robustness Gen. Bound Multi-Evidence LPF-SPN ✓ T1 (proven) ✓ Exact (T7) ✓ O ( ε √ K ) N/A ✓ Purp ose-built LPF-Learned × Empirical ✓ Exact (T7) ✓ O ( ε √ K ) ✓ Non-vacuous (T3) ✓ Purpose-built BER T × None × Heuristic × Unknown × V acuous × Single-input adapted EDL × Single-input only × Approximate × Unknown × Unknown × Catastrophic (43%) R-GCN × None × None × Unknown × Unknown × Unsuitable (15.6%) V AE-Only × None × None × Unknown × Unknown × No aggregation LLMs × Severe miscal (80% ECE) × No access × Unknown × Unknown × Zero-shot, no guarantees Both LPF v arian ts are the only metho ds with formal multi-evidence aggregation guarantees. No baseline pro vides T7’s exact epistemic/aleatoric separation. LPF’s O ( ε √ K ) robustness is pro v ably sup erior to baselines’ susp ected O ( εK ) linear degradation. 12.8.5 Practical Implications of Theoretical Guarantees F or High-Stakes Deploymen t In medical diagnosis (Healthcare domain: 99.3% accuracy , 1.8% ECE), T1 lets do ctors trust conﬁdence scores for triage, T7 distinguishes inheren tly ambiguous symptoms from contradictory test results, and T5 ensures graceful degradation when imaging is una v ailable. In ﬁnancial compliance, T1 pro vides formal calibration b ounds for regulatory audit defense, T7 ﬂags cases requiring manual review, and T3 v alidates deplo yment with only 630 training en tities. In legal case prediction, T7 enables exp ert witnesses to explain uncertaint y from conﬂicting preceden ts, and T5 ensures v alid predictions even with 30% of do cumen ts redacted. F or Resource Allo cation T6 shows ECE plateaus at K ≈ 7 evidence items across all domains, so collecting more than 7 items yields diminishing returns ( < 1% ECE impro vemen t). If each evidence 131 item costs $50, the optimal budget is $350 (7 items) rather than $1,000 (20 items). T3 requires N ≥ 1 . 5 × d eﬀ training en tities for non-v acuous generalization; for a new domain, this means collecting at least 1,500 lab eled en tities. T2 prov es M = 16 samples suﬃcient (1.3% error), so pro duction deplo yment with M = 16 is b oth theoretically grounded and computationally eﬃcient (14.8ms LPF-SPN, 37.4ms LPF-Learned). F or Model T rust and Interpretabilit y Combining pro venance records (Section 12.5 ) with T7’s exact decomp osition enables fully quantiﬁed audit trails: “Prediction: HIGH (conﬁdence 0.93) based on evidence E1 (w eight 0.35), E3 (w eight 0.28), E5 (w eight 0.20). Epistemic uncertaint y: 0.08 (evidence mostly agrees). Aleatoric: 0.04 (inherent ambiguit y).” T7 also enables principled selectiv e classiﬁcation: automating predictions with conﬁdence > 0 . 9 yields 100% accuracy on 69.6% of cases, with humans reviewing the remaining 30.4% where epistemic uncertaint y reﬂects genuine evidence disagreemen t. 12.8.6 Limitations of Curren t Theory While our guarantees are strong, they rely on assumptions that may not hold p erfectly in real-w orld deplo yments. Assumption Violations A1: Conditional Indep endence. Evidence items are assumed conditionally indep enden t given entit y and predicate. Real-world evidence ma y share systematic biases (correlated sources, sensor errors). Our companion pap er [ Alege , 2026 ] measures av erage pairwise correlation ρ = 0 . 12 (weak dep endence), well within the safe regime prov en by T5. A2: Bounded Enco der V ariance. Our V AE suﬀers from posterior collapse at K = 1 (artiﬁcially low epistemic v ariance 0.034), inﬂating individual ECE ( ε = 14 . 0% in T1). T1’s b ound incorp orates this ε term, so violations are accounted for. Mitigation: β -V AE [ Higgins et al. , 2017 ] or normalizing ﬂows. A3: Calibrated Deco der. The deco der p θ ( y | z ) has individual ECE=14.0%. T1’s b ound scales linearly with ε ; improving deco der calibration (e.g., temp erature scaling [ Guo et al. , 2017 ]) w ould tighten aggregation b ounds. Despite ε = 14 . 0% , aggregated ECE=1.4% demonstrates that aggregation substantially reduces miscalibration. T able 42: Empirical results vs. theoretical b ounds. Theorem Empirical Result Theoretical Bound Gap T1 (Calibration) 1.4% ECE ∼ 15% 90.7% b elo w b ound T2 (MC Error, M = 16 ) 1.3% error 38.7% 96.6% b elo w b ound T3 (Generalization) 0.0085 gap 0.228 96.3% b elo w b ound T5 (Robustness, ε = 0 . 5 ) 12% L 1 316% 96.2% b elo w b ound Tigh tness of Bounds All b ounds are non-v acuous and correctly predict qualitative scaling b eha vior, but w orst-case analysis yields conserv ativ e b ounds. F uture w ork should develop data- dep enden t b ounds (e.g., Bernstein inequalities with empirical v ariance) to tighten the gaps. Scop e Limitations The curren t theory is limited to classiﬁcation with categorical predicates ( |Y | < ∞ ); regression and structured prediction are not co v ered. Cross-domain generalization b ey ond 132 classiﬁcation is empirically v alidated (Section 12.2 ) but not theoretically prov en. Evidence coun ts b ey ond K = 20 are not exp erimen tally veriﬁed, though T6’s plateau at K ≈ 7 suggests diminishing returns. Baseline theoretical characterization is limited to simple uniform av eraging; state-of-the-art m ulti-evidence metho ds (attention-based fusion, transformers) are not theoretically analyzed. 12.8.7 Summary: Why Theoretical Guaran tees Matter LPF is the only multi-evidence framework providing formal reliability guarantees across calibration, robustness, sample complexity , and uncertaint y decomp osition. Three core b eneﬁts justify this emphasis. T rust w orthiness for High-Stakes Applications. Neural baselines achiev e comp etitiv e accuracy but lac k calibration guarantees: BER T shows 3.2% ECE ( 2 . 3 × w orse) and Qwen3-32B sho ws 79.7% ECE ( 57 × w orse) despite 98% accuracy . T1 and T7 ensure that LPF’s conﬁdence scores are statistically rigorous—not heuristic—enabling medical, ﬁnancial, and legal deploymen t where mistakes carry serious consequences. Principled System Design. T2 prov es M = 16 suﬃcient, T6 sho ws K ≈ 7 optimal, and T3 sp eciﬁes minimum training data ( N ≥ 1 . 5 × d eﬀ ). This conv erts h yp erparameter tuning from ad-ho c trial and error into theory-guided decisions, reducing inference time from 50ms to 15ms without sacriﬁcing guaranteed p erformance. Diﬀeren tiation from Blac k-Box Methods. BER T, EDL, R-GCN, and LLMs hav e zero formal guarantees for multi-evidence aggregation. LPF pro vides seven theorems with empirical v alidation across 8 domains—enabling deploym ent with formal reliability certiﬁcates rather than empirical hop e. T able 43: Theoretical prop erties across all metho ds. Prop ert y BER T EDL LLMs LPF-SPN LPF-Learned Multi-evidence design × Adapted × F ails × Zero-shot ✓ Purp ose-built ✓ Purp ose-built Calibration guarantee × None × None × None ✓ T1 (prov en) × Empirical (6.6%) Exact uncertaint y decomp × Heuristic × Appro x × No access ✓ T7 ( < 0.002%) ✓ T7 ( < 0.002%) Robustness guarantee × Unkno wn × Unkno wn × Unkno wn ✓ T5 ✓ T5 Pro v able generalization × V acuous × Unknown × Unkno wn N/A ✓ T3 (96% margin) Near-optimal calibration × No × No × No ✓ T4 ( 1 . 1 × ) ✓ T4 ( 1 . 3 × ) Summary of V eriﬁcation ✓ T1 : Calibration preserved (1.4% ≤ 15% b ound, 90% margin) ✓ T2 : MC error controlled (1.3% follows O (1 / √ M ) , R 2 = 0 . 95 ) ✓ T3 : Generalization non-v acuous (gap 0.0085 vs b ound 0.228, 96% margin) ✓ T4 : Near-optimal calibration ( 1 . 1 × info-theoretic limit) ✓ T5 : Graceful robustness ( O ( ε √ K ) v alidated cross-domain) ✓ T6 : Sample complexity ( O (1 / √ K ) scaling, R 2 = 0 . 85 ) ✓ T7 : Exact uncertaint y decomp osition ( < 0.002% error) F or complete pro ofs, detailed assumptions, and extended analysis, see the companion theoretical pap er [ Alege , 2026 ]. LPF works b ecause its guaran tees are prov en, not merely observ ed. 133 13 Discussion 13.1 Wh y LPF W orks: Architectural Insigh ts The exceptional performance of LPF across eight div erse domains (mean accuracy 94.6%, ECE 3.5%) stems from three synergistic design principles that address fundamen tal gaps in existing approac hes. 13.1.1 Explicit Uncertain ty Propagation Unlik e neural aggregation metho ds (BER T, attention mechanisms) that pro duce p oin t predictions without calibrated conﬁdence, LPF maintains probabilistic seman tics throughout the inference pip eline: Evidence → V AE Posterior → Soft F actor → Aggregated Distribution Eac h transformation preserves uncertaint y information. V AE Enco ding: The p osterior v ariance σ 2 captures epistemic uncertaint y ab out evidence meaning. Am biguous or con tradictory evidence pro duces high-v ariance p osteriors ( σ 2 > 0 . 3 ), while clear evidence yields p eaked distributions ( σ 2 < 0 . 1 ). F actor Con version: Monte Carlo integration explicitly marginalizes ov er latent uncertain ty , pro ducing soft factors that reﬂect b oth seman tic conten t (via deco ded distributions) and reliability (via credibility weigh ts). Structured Aggregation: SPN inference main tains exact probabilistic seman tics, while learned aggregation preserves uncertaint y through qualit y and consistency netw orks. Empirical V alidation: Our ablation studies demonstrate that removing any of these uncertaint y mec hanisms degrades both accuracy and calibration. Replacing soft factors with deterministic predictions (SPN-Only baseline) reduces accuracy from 97.8% to 94.8% and increases ECE from 1.4% to 30.9%. 13.1.2 Multi-Evidence Arc hitectural Design The catastrophic failure of EDL adaptations (28.1–56.3% accuracy vs. LPF-SPN’s 97.8%) reveals a fundamen tal truth: uncertaint y quantiﬁcation alone is insuﬃcient for m ulti-evidence reasoning. The task requires sp ecialized architectures that mo del evidence interactions. T raining-Inference Distribution Mismatch: EDL-Individual treats each evidence piece as an indep enden t training example with the entit y’s lab el, creating severe label noise. A single piece of evidence stating “minor audit discrepancies found” may appear with b oth “high compliance” and “low compliance” lab els across diﬀerent en tities, preven ting the mo del from learning meaningful patterns. Loss of Relational Structure: EDL-Aggregated av erages evidence embeddings b efore pre- diction, collapsing the distributional information EDL is designed to capture. This fails to mo del con tradictions, corrob orations, or v arying evidence quality . LPF’s Solution: Purp ose-built multi-evidence handling provides evidence-level enco ding (V AE pro cesses each piece indep enden tly , capturing p er-evidence uncertaint y), explicit aggregation (SPN factor-based or learned qualit y/consistency netw ork mechanisms designed for combining m ultiple uncertain signals), and prov enance preserv ation (every prediction traces back to source evidence with interpretable weigh ts). Cross-Domain V alidation: The consistent + 2.4% impro vemen t ov er b est baselines across sev en diverse domains demonstrates that this arc hitectural adv an tage generalizes broadly , not just to compliance assessment. 134 13.1.3 Calibration b y Design LPF achiev es sup erior calibration (mean ECE 3.5% vs. 12.1% for BER T) through architectural c hoices rather than p ost-ho c correction. Principled Probabilistic Semantics: Soft factors represent likelihoo d p oten tials with v alid probabilit y interpretations, SPN inference computes exact marginals without approximation error, and credibility weigh ts are deriv ed from p osterior uncertain ty rather than learned heuristics. Mon te Carlo A v eraging: Explicitly marginalizing o v er laten t uncertaint y via sampling produces w ell-calibrated factors even with mo derate sample counts ( M = 16 achiev es ECE 1.4%). V ariance-Based W eigh ting: The sigmoid p enalt y w ( e ) = 1 / (1 + exp ( α · mean ( σ ))) pro vides a principled mechanism to down w eight uncertain evidence without requiring calibration-sp eciﬁc tuning. T emp erature as Reﬁnement: While LPF ac hieves strong calibration without temp erature scaling (ECE 1.4% at T = 1 . 0 ), optional tuning pro vides further impro v ement (ECE 1.3% at T = 0 . 8 ) for domains requiring precise calibration. Con trast with Neural Baselines: BER T requires careful p ost-ho c temp erature tuning to ac hieve ECE 8.9%—still 6.3 × w orse than LPF’s default conﬁguration—demonstrating that arc hitectural design, not hyperparameter optimization, drives calibration qualit y . 13.2 Arc hitectural Comparison: LPF-SPN vs. LPF-Learned The dual-arc hitecture design of LPF enables a controlled comparison of reasoning paradigms under iden tical evidence enco ding. 13.2.1 P erformance T radeoﬀs A ccuracy: LPF-SPN achiev es sup erior accuracy (97.8% vs. 91.1% on compliance, 99.3% vs. 98.0% mean across domains) through exact probabilistic inference. The pro duct op eration in SPNs ampliﬁes agreemen t among evidence pieces, pro ducing sharp er predictions when evidence conv erges. Calibration: LPF-SPN demonstrates exceptional calibration (ECE 1.4% vs. 6.6%) b ecause SPN marginals are exact probability distributions. LPF-Learned relies on learned aggregation, which ma y pro duce slightly ov erconﬁden t or underconﬁden t predictions dep ending on training data distribution. Sp eed: LPF-SPN is faster (14.8ms vs. 37.4ms) despite requiring 80 deco der calls (5 evidence × 16 MC samples). This coun ter-intuitiv e result stems from cached SPN structures (compilation ov erhead amortized across queries), batch deco ding (GPU parallelization of 80 sim ultaneous forward passes), and aggregator ov erhead (quality/consistency net works add computational cost in LPF-Learned). In terpretabilit y: LPF-SPN provides explicit soft factors with probabilistic semantics, enabling ﬁne-grained prov enance analysis. LPF-Learned’s aggregation weigh ts are less transparent—neural net works learn implicit combination rules that are harder to in terpret. 13.2.2 When to Use Each V arian t Cho ose LPF-SPN when: • Calibration is critical: medical diagnosis, ﬁnancial risk, and legal decisions requiring well- calibrated conﬁdence estimates • In terpretability matters: regulatory compliance, scien tiﬁc discov ery , and high-stakes decisions demanding audit trails 135 • Structured reasoning is av ailable: domain knowledge suggests sp eciﬁc factor independence assumptions • Exact inference is feasible: n umber of evidence items and domain sizes p ermit tractable SPN marginals Cho ose LPF-Learned when: • Arc hitectural simplicit y is prioritized: deploymen t scenarios fa voring end-to-end neural pip elines • T raining data is abundant: suﬃcien t entit y-lev el lab els to train aggregator netw orks eﬀectively • Evidence correlations are complex: learned aggregation may capture non-linear dep endencies b etter than indep endence assumptions • Sligh t calibration degradation is acceptable: ECE 6.6% is still strong compared to neural baselines (12.1% for BER T) Hybrid Approach: F or pro duction systems, we recommend deplo ying LPF-SPN as the primary inference engine with LPF-Learned as a fallback for edge cases (e.g., missing SPN structure for rare predicates, computational constrain ts). The shared V AE encoder enables seamless switc hing b et w een v arian ts. 13.2.3 Theoretical Implications The p erformance gap b et ween v arian ts (6.7% accuracy , 4.7 × calibration diﬀerence) provides empirical evidence for a theoretical claim: structured probabilistic reasoning outp erforms learned aggregation when indep endence assumptions hold. Prop osition: F or evidence sets satisfying conditional indep endence given entit y and predicate, SPN-based aggregation achiev es low er generalization error than neural aggregation with b ounded capacit y . In tuition: SPNs enco de prior knowledge (indep endence structure) that neural netw orks must learn from data. When priors are correct, explicit structure dominates pure learning. When violated, learned aggregation may comp ensate through ﬂexibilit y . Empirical Supp ort: Cross-domain results suggest compliance, ﬁnance, and healthcare satisfy indep endence assumptions (LPF-SPN margin: 6–8%), while legal and academic reasoning ma y in volv e complex evidence interactions (LPF-SPN margin: 0–2%). F uture W ork: F ormal analysis of when each approach dominates, p ossibly through P AC-Ba y es b ounds relating domain structure to aggregation p erformance. 13.3 The Multi-Evidence P aradigm Shift LPF addresses a problem class underexplored in mac hine learning literature: aggregating multiple noisy , p otentially contradictory pieces of evidence to make calibrated predictions with limited training data. 13.3.1 Con trast with Standard ML Standard sup ervised learning op erates on a single data p oin t (image, sentence, measuremen t), with thousands to millions of lab eled examples, aiming to maximize predictive accuracy . 136 Multi-evidence reasoning (LPF’s domain) op erates on a set of heterogeneous evidence pieces (a vg. 8.3 p er entit y in our exp eriments), with hundreds of lab eled en tities (630 in the compliance domain), aiming for calibrated uncertaint y quantiﬁcation with pro venance. Man y real-world decision-making scenarios follow the multi-evidence paradigm: knowledge base completion (aggregate web evidence to p opulate KB facts), medical diagnosis (com bine patien t history , lab results, imaging rep orts, symptoms), legal case assessment (synthesize briefs, precedents, witness statements, exhibits), corp orate compliance (merge regulatory ﬁlings, audit rep orts, news articles, internal do cumen ts), and scientiﬁc literature review (in tegrate ﬁndings across m ultiple pap ers). Standard ML approac hes fail in this regime due to data ineﬃciency (neural metho ds require large training sets; LPF works with 630 en tities), lack of uncertain ty quantiﬁcation (p oin t predictions are inadequate for high-stakes decisions), and absence of prov enance (black-box aggregation preven ts auditing). 13.3.2 P ositioning Against Existing P aradigms vs. Probabilistic Soft Logic / Mark ov Logic Net w orks: PSL/MLN require manual rule engineering and assume discrete symbolic predicates. LPF learns from unstructured evidence (text) without manual rules, providing scalability to real-world text data without kno wledge engineering. vs. Neural Aggregation (T ransformers, A ttention): T ransformers employ implicit learned aggregation with p o or calibration (BER T ECE: 12.1%). LPF uses explicit probabilistic reasoning with sup erior calibration (ECE: 1.4%), providing interpretabilit y and trustw orthiness for high-stakes applications. vs. Eviden tial Deep Learning: EDL p erforms single-input uncertain ty quantiﬁcation and struggles with multi-evidence (56.3% accuracy). LPF is purp ose-built for multi-evidence scenarios (97.8% accuracy), representing a fundamental architectural match to the problem structure. vs. Kno wledge Graph Completion: K G metho ds rely on sym b olic en tities and relations without uncertain ty ov er facts. LPF pro cesses unstructured evidence in to probabilistic b eliefs ab out facts, handling ambiguous and contradictory evidence with calibrated conﬁdence. Unique Con tribution: LPF is the ﬁrst framework combining neural p erception of unstruc- tured evidence (V AE), structured probabilistic reasoning (SPN), explicit uncertaint y quantiﬁcation (p osterior v ariance → credibilit y weigh ts), and nativ e prov enance trac king (immutable audit trails). 13.4 Lessons from Cross-Domain Ev aluation Our ev aluation across eight domains rev eals insights ab out LPF’s strengths, limitations, and the nature of multi-evidence reasoning. 13.4.1 Domain Characteristics and Performance Easiest Domain — FEVER (99.7% accuracy , ECE 0.3%): Clean, well-structured textual en tailment signals drawn from Wikip edia provide authoritativ e, unambiguous information. LPF ac hieves near-p erfect p erformance when evidence is high-qualit y and consistent. Hardest Domain — Legal (83.6% v alidation accuracy): Subtle distinctions b et ween case outcomes require nuanced reasoning ov er legal briefs containing in tricate argumen ts with m ulti-faceted precedents. P erformance ceiling reﬂects inheren t task diﬃculty , not mo del limitations. Most V ariance — Materials (std 0.5%): A highly technical domain with quan titative evidence (thermo dynamic stability scores, DFT calculations) leads to initialization sensitivity in 137 ho w the enco der learns to w eight numerical vs. textual features. Domains with mixed mo dalities ma y require domain-sp eciﬁc architecture tuning. Best Generalization — Compliance, Finance, Healthcare (train-v al gap: − 3.3%, − 1.2%, + 0.4% ): Structured evidence patterns with consistent lab el distributions, combined with LPF’s β -V AE regularization (KL weigh t 0.01), preven t o verﬁtting even with limited data. Negative gaps (v alidation outp erforms training in compliance/ﬁnance) suggest eﬀective regularization. 13.4.2 Evidence Characteristics Ablation studies rev eal diminishing returns b ey ond k = 5 evidence pieces: k = 1 yields 79.3% accuracy (single evidence insuﬃcient), k = 5 yields 97.0% accuracy ( + 17.7% absolute gain), and k = 20 yields 97.8% accuracy (only + 0.8% ov er k = 5 ). Most en tities hav e 1–3 highly informative evidence pieces; additional items pro vide redundant information. This v alidates the default top_k=5 setting and suggests active learning could reduce evidence collection costs. Evidence credibility scores (mean 0.87, std 0.08) show a strong correlation with p erformance: high-qualit y evidence ( > 0.9 credibilit y) yields 99.1% accuracy and 0.8% ECE; medium-quality (0.7– 0.9) yields 97.2% accuracy and 1.5% ECE; low-qualit y ( < 0.7) yields 91.3% accuracy and 3.4% ECE. System p erformance degrades gracefully with evidence quality , v alidating the credibility weigh ting mec hanism. 13.4.3 Hyp erparameter Consistency Remark ably , the same h yp erparameter conﬁguration achiev es strong p erformance across all domains: n_samples of 4–16 (domain-dep enden t: 4 for simple tasks, 16 for complex), temp erature 0.8 (sligh t sharp ening universally b eneﬁcial), alpha 0.1 (minimal uncertaint y p enalt y for calibration), and top_k 5 (suﬃcient evidence cov erage across domains). This suggests LPF’s fundamental arc hitecture captures domain-agnostic principles of multi-evidence reasoning that are robust to hyperparameter c hoices. One exception: FEVER b eneﬁts from higher n_samples (32) due to its massive scale (145K training claims), suggesting sample count should scale with dataset size. 13.5 Practical Deplo ymen t Considerations Based on our implementation exp erience and exp erimen tal results, w e provide concrete guidance for practitioners. 13.5.1 When LPF is a Go o d Fit LPF is well suited when multiple evidence sources p er en tity (3–20 pieces) are av ailable, lab eled training data is limited (h undreds to lo w thousands of entities), decisions are high-stak es and require calibrated uncertain ty (medical, ﬁnancial, legal), regulatory or scientiﬁc requiremen ts demand pro venance, and evidence types are heterogeneous (rep orts, ﬁlings, certiﬁcations, news articles). Concrete example scenarios include: healthcare diagnosis of rare diseases from patient history , lab results, and imaging rep orts (limited training cases, high stakes); ﬁnancial credit risk assessmen t for small businesses from ﬁnancial statemen ts, news, and so cial signals (sparse lab els, regulatory requiremen ts); legal con tract dispute outcome prediction from case do cuments, preceden ts, and exhibits (interpretabilit y critical); and scientiﬁc hypothesis v alidation from literature evidence across m ultiple pap ers (prov enance for citations). 138 13.5.2 When LPF Ma y Not Be Optimal LPF is a p o or ﬁt for single-input classiﬁcation (images, sentences), scenarios with massive training data (millions of samples) where large T ransformers ma y suﬃce, real-time latency requiremen ts b elo w 1ms, and evidence collection that is trivial or free (no need for aggregation optimization). Preferred alternatives in these scenarios include Evidential Deep Learning or Deep Ensem bles for single-input uncertain t y , pre-trained T ransformers (BER T, RoBER T a) for large-scale text, kno wledge graph em b eddings (T ransE, ComplEx) for structured kno wledge, and distilled or quantized neural net works for real-time inference. 13.5.3 Implemen tation Best Practices Data Preparation: Use F AISS for eﬃcien t similarity search (sub-millisecond retriev al for millions of items). Pre-compute and store all evidence embeddings to av oid rep eated Sen tence-BER T calls. Apply entit y-based stratiﬁed splits to preven t data leak age across evidence. T raining Proto col: T est 7–15 random seeds and select the b est by v alidation accuracy (provides 0.5–2% improv emen t). Apply early stopping with patience of 5 ep o c hs (most mo dels conv erge by ep och 15). Start with β = 0 . 01 for KL w eight; increase if p osterior collapses ( σ → 0 ) or decrease if reconstruction fails. Hyp erparameter Selection: Start with n_samples=16; reduce to 4 if latency is critical, increase to 32 if calibration is paramoun t. Default T = 1 . 0 ; tune on v alidation ECE if calibration is critical. Default α = 0 . 1 for strong calibration; increase to 1.0 if accuracy is the sole metric. Default top_k=5; p erform an ablation study to v alidate for your domain. Deplo yment: Cac he authoritative facts in a canonical database for a sub-millisecond fast path. Pre-compile SPN structures p er predicate to a void rep eated construction. Process multiple queries sim ultaneously for GPU eﬃciency . Use write-ahead logging for pro venance records to enable async hronous writes without blo c king inference. 13.5.4 Computational Costs T raining (compliance domain, 630 entities, 7 seeds): appro ximately 6 hours on an 8-core CPU (parallelizable across seeds to 1 hour on 64-core); 6 GB p eak memory during enco der training; 1.6 GB storage p er trained mo del (enco der + deco der + aggregator). Inference: 14.8ms latency (LPF-SPN), 37.4ms (LPF-Learned); 68 queries/second/core (LPF- SPN); 1.2 GB memory (mo del + SPN cac he); a 64-core mac hine handles 4,352 queries/second (376M/da y), far exceeding typical workloads. Cost Comparison vs. LLM Baselines: LPF-SPN incurs $0/query for self-hosted inference vs. $200–600/million queries for Gro q LLMs. LPF is 60–200 × faster (14.8ms vs. 1500–3000ms). LPF is cost-eﬀectiv e for pro duction deploymen t at low-to-medium query volume ( < 10M/da y). F or massive scale ( > 100M/day), distributed deploymen t or appro ximate inference should b e considered. 13.6 Limitations and F ailure Mo des Honest assessment of LPF’s limitations guides appropriate application and future research. 13.6.1 Arc hitectural Limitations Discrete Predicates Only: The curren t implementation handles categorical outputs (compliance ∈ {lo w, medium, high}) but not contin uous regression (e.g., predicting exact compliance score ∈ [0 , 1] ). 139 A work around is to discretize contin uous targets into bins, though this loses granularit y . The natural future direction is to extend the deco der to Gaussian mixture outputs for contin uous predictions while preserving uncertaint y quantiﬁcation. Conditional Indep endence Assumption: LPF-SPN assumes evidence pieces are conditionally indep enden t given entit y and predicate. This fails when evidence has causal dep endencies (e.g., audit rep ort A triggers inv estigation B). The legal domain shows the smallest LPF-SPN adv an tage (0.7% ov er LPF-Learned), suggesting complex evidence interactions, while most domains show a 2–6% adv an tage that v alidates the indep endence assumption holds broadly . LPF-Learned explicitly mo dels consistency (capturing dep endencies), though at the cost of interpretabilit y . Static Evidence: The current system treats evidence as ﬁxed at query time, not supp orting temp oral dynamics or evidence decay . F or example, compliance prediction using a 5-year-old audit rep ort will b e ov erconﬁden t if regulations ha ve changed. T emp oral weigh ting is a natural future direction to address this. 13.6.2 Data Requiremen ts Evidence-Lev el Lab els: Enco der-deco der training requires eac h evidence piece lab eled with ground truth, creating annotation burden (t ypically 4,500 lab eled evidence items: 900 en tities × 5 evidence each). W eak sup ervision approaches using silver lab els from entit y-level signals could reduce annotation costs. En tit y-Level Lab els (LPF-Learned only): The aggregator requires en tit y-level ground truth, limiting applicability to scenarios with entit y lab els. LPF-SPN requires only evidence-level lab els (no entit y lab els needed), making it more suitable for sparse sup ervision. 13.6.3 Scalabilit y Constraints SPN inference complexity scales as O ( N × | domain | 2 ) where N is the evidence count. F or N > 50 , exact inference b ecomes slow. Observed limits: N ≤ 20 supp orts real-time ( < 50ms); N ≤ 50 supp orts interactiv e use ( < 200ms); N > 50 requires batch pro cessing. Appro ximate inference (top- k factor selection, b eam searc h) trades exactness for sp eed. Curren t implemen tation is tested on 3-class predicates; scaling to 10+ classes increases deco der output dimensionality and SPN complexit y . F or massiv e state spaces, approximate metho ds are required. 13.6.4 Observ ed F ailure Mo des Analysis rev eals 3 high-conﬁdence errors ( > 0.8 conﬁdence) in the compliance domain test set (2.2% error rate). The ro ot cause is mixed evidence with balanced supp ort for multiple classes (3 pieces supp orting “high”, 2 supp orting “medium”): when con tradictory evidence has similar credibility , aggregation may amplify the minority signal. Example (C0089): 60% of evidence supp orts “high”, 40% supp orts “medium”, but the mo del predicted “medium” (conﬁdence 0.89) because medium-supporting evidence had sligh tly higher a verage credibilit y (0.795 vs. 0.883), o verriding the ma jority . The system correctly identiﬁed uncertain ty (0.89 conﬁdence is appropriately cautious), but the ﬁnal prediction follow ed a misleading statistical pattern. A p oten tial ﬁx is to incorp orate evidence count as a prior (Bay esian correction for imbalanced evidence) or add meta-reasoning ab out evidence correlation. P erformance degrades gracefully with evidence quality—99.1% accuracy for high-quality ( > 0.9 credibilit y) vs. 91.3% for lo w-quality ( < 0.7)—though calibration remains acceptable (ECE 3.4% ev en with low-qualit y evidence), conﬁrming that the system requires reasonably reliable evidence sources but is not brittle to imp erfect inputs. 140 13.7 Broader Impact and Ethical Considerations As a framework designed for high-stakes decision-making, LPF carries signiﬁcant so cietal implications. 13.7.1 T rust w orthy AI Beneﬁts Calibrated Uncertain t y: ECE 1.4% (LPF-SPN) enables do ctors to trust conﬁdence scores when triaging patien ts, credit decisions based on reliable risk estimates, and judges informed b y well- calibrated case outcome predictions. In con trast, large language mo dels ac hieve 44–62% accuracy with catastrophic miscalibration (ECE 74–87%); deplo ying such systems in high-stak es domains risks ov erconﬁdent errors. Pro venance and Auditability: The imm utable ledger with cryptographic hashing enables ﬁnancial institutions to satisfy audit requirements, researchers to cite exact evidence chains, and complete reasoning trails for court pro ceedings. Human-AI Collab oration: W ell-calibrated predictions enable selective classiﬁcation: automate predictions with conﬁdence > 0.9 (99.2% accuracy on 87.4% of cases) while routing uncertain cases (12.6%) to h uman review. This maximizes eﬃciency while maintaining human ov ersight for am biguous decisions. 13.7.2 P otential Risks and Mitigation Bias Ampliﬁcation: If training data contains biases (e.g., historical discrimination in loan appro v als), LPF may propagate or amplify them through evidence weigh ting. Mitigation strategies include regular fairness assessmen ts on protected attributes, aggregating from diverse evidence sources to counter single-source bias, and constrained training ob jectives that satisfy fairness criteria. Ov er-Reliance: Users ma y trust well-calibrated predictions without verifying underlying evidence, creating automation bias (e.g., a ph ysician accepting a “sev ere” diagnosis at 0.95 conﬁdence without reviewing patient symptoms). Mitigation includes mandatory h uman review proto cols for high-stak es decisions, highlighting k ey evidence through prov enance records, and conﬁdence threshold ﬂags near decision b oundaries. A dversarial Manipulation: A ttack ers could craft misleading evidence with high credibilit y scores to inﬂuence predictions—for example, publishing a fak e audit report with professional formatting to manipulate a compliance prediction. Defenses include cryptographic signatures on evidence from trusted sources, adv ersarially robust enco der training, and anomaly detection to ﬂag evidence inconsistent with historical patterns. Priv acy Violations: The pro venance ledger stores complete reasoning chains that may exp ose sensitiv e information (e.g., medical diagnosis pro venance revealing patien t symptoms and lab results). Mitigation includes homomorphic encryption for stored prov enance, role-based access controls for ledger queries, and diﬀerential priv acy for aggregated statistics. 13.7.3 So cietal Applications P ositiv e use cases include healthcare equit y (aggregating evidence from underserved p opulations to impro ve diagnosis for rare diseases), ﬁnancial inclusion (credit assessment for unbank ed p opulations using alternative evidence such as utility pa yments and education history), legal aid (pro-b ono case outcome predictions for resource allo cation), scientiﬁc disco very (literature syn thesis for drug repurp osing and materials design), and climate mo deling (evidence aggregation from div erse sensors for p olicy decisions). 141 Inappropriate use cases that w e explicitly discourage include surveillance (aggregating evidence for p opulation monitoring), discrimination (biased decision-making in hiring, lending, or housing), autonomous w eap ons (target selection based on evidence aggregation), and mass manipulation (propaganda synthesis from selective evidence). W e adv o cate for resp onsible deplo yment guidelines analogous to medical device regulation: requiring v alidation studies, transparent do cumen tation, and ongoing monitoring for LPF systems deplo yed in high-stakes domains. 13.8 Key T akea w a ys F or ML Practitioners: Multi-evidence reasoning is a distinct problem class requiring sp ecialized arc hitectures, not adaptation of single-input metho ds. Uncertain ty quantiﬁcation matters: ECE 1.4% vs. 80% (LLMs) is the diﬀerence betw een trustw orthy and dangerous in high-stak es applications. Calibration b y design b eats p ost-hoc correction: arc hitectural choices (V AE v ariance, Monte Carlo a veraging, SPN inference) pro duce well-calibrated outputs without tuning. Lo w-data regimes reward principled design: LPF achiev es 97.8% accuracy with 630 training entities by enco ding inductive biases. F or Researc hers: Structured reasoning com bined with neural p erception is pow erful: combining V AE (unstructured text) with SPN (probabilistic logic) ac hieves the beneﬁts of both paradigms. Dual arc hitectures enable controlled comparison: LPF-SPN vs. LPF-Learned isolates the v alue of structured reasoning (6.7% accuracy gain, 4.7 × calibration improv emen t). Cross-domain ev aluation is essential: single-domain results do not v alidate broad applicability; our 8-domain study demonstrates true generalization. Pro venance is a ﬁrst-class citizen: nativ e audit trails enable scien tiﬁc repro ducibility , regulatory compliance, and human-AI collab oration. F or Domain Exp erts: Aggregation quality exceeds individual evidence quality: LPF achiev es 97.0% accuracy with imperfect evidence (a vg. credibility 0.87) through principled aggregation. Calibrated conﬁdence enables selective automation: automate high-conﬁdence predictions ( > 0.9), defer uncertain cases to h umans. Prov enance traces enable debugging: when predictions fail, insp ect soft factors to identify misleading evidence. Domain-sp eciﬁc tuning is often unnecessary: default h yp erparameters (n_samples=16, T = 1 . 0 , α = 0 . 1 , k = 5 ) work across div erse domains. 14 F uture W ork The LPF framework op ens numerous a ven ues for extending its capabilities and exploring nov el applications. W e organize future directions b y their p oten tial impact and technical feasibilit y , noting that the theoretical foundations of LPF ha ve b een extensiv ely developed in companion work [ Alege , 2026 ] establishing formal guarantees for calibration preserv ation, robustness, sample complexit y , and uncertain ty quantiﬁcation. 14.1 Immediate Extensions (6–12 Mon ths) 14.1.1 Con tinuous Predicate Outputs Curren t Limitation: LPF handles categorical predicates (compliance ∈ {lo w, medium, high}) but not contin uous regression (e.g., compliance_score ∈ [0 , 1] ). Prop osed Solution: Replace the categorical decoder with a Gaussian mixture output to pro duce con tinuous distributions. F actor conv ersion would extend Monte Carlo in tegration to con tinuous 142 distributions: Φ e ( y ) = Z p θ ( y | z ) q ϕ ( z | e ) dz ≈ 1 M M X m =1 N  y ; µ ( m ) , σ ( m )2  (73) SPN in tegration would use contin uous leaf distributions (Gaussians) instead of categorical. Exp ected Beneﬁts: Preserving full distributional information (not discretized in to bins), main- taining uncertain ty quantiﬁcation o ver contin uous v alues, and enabling regression tasks (predicting exact compliance scores, risk v alues, etc.). Challenges: Mixture mo del training stability (mo de collapse), con tinuous SPN inference algorithms (less mature than discrete), and ev aluation metrics for contin uous predictions with uncertain ty . V alidation Study: T est on ﬁnancial risk prediction (credit scores ∈ [300 , 850] ) or materials prop ert y prediction (bandgap ∈ [0 , 5] e V). 14.1.2 A ctive Evidence Collection Motiv ation: The curren t system retrieves top- k evidence passively . Activ e learning could reduce evidence collection costs while maintaining accuracy . Core Idea: Iterativ ely select evidence that maximally reduces p osterior uncertaint y . A t eac h step, estimate the information gain from each candidate evidence item and select the one with the highest exp ected reduction in prediction entrop y . Exp ected Results: A chiev e 97% accuracy with 3–4 evidence items (vs. 5 baseline), reducing retriev al cost by 20–40%. This is particularly v aluable when evidence acquisition is exp ensiv e (API calls, human annotation). Under submo dularit y assumptions, greedy information-gain selection can b e prov en to achiev e near-optimal p erformance. Comparison Baselines: Random selection, uncertaint y sampling (select evidence with highest p osterior v ariance), and diversit y-based selection (maximize co verage of evidence types). 14.1.3 Con trastive Explanations Motiv ation: Pro venance records sho w what evidence was used; contr astive explanations answ er why a particular prediction was made. Core Idea: Generate explanations by con trasting the actual prediction with counterfactual alternativ es. The system iden tiﬁes evidence pieces that discriminate b et w een the predicted class and the next-most-likely alternative, then presents these highlights in natural language. Example Output: Entity: C0042 Predicted: HIGH compliance (not MEDIUM) because: Supporting Evidence: - E12 (weight=0.15): "Company demonstrates excellent record-keeping" -> 82% confidence in HIGH vs. 15% for MEDIUM - E20 (weight=0.12): "Consistently meets all regulatory requirements" -> 79% confidence in HIGH Contradicting Evidence: - E7 (weight=0.03): "Minor discrepancies found in Q2 filing" -> 61% confidence in MEDIUM, but low weight due to 143 high uncertainty (sigma=0.34) Overall: 3 strong pieces support HIGH vs. 1 weak piece for MEDIUM. Ev aluation: Human study with domain exp erts rating explanation qualit y on correctness (do es the explanation accurately reﬂect mo del reasoning?), usefulness (do es it help the user under- stand/trust the prediction?), and actionabilit y (can the user iden tify evidence to v erify or c hallenge?). 14.2 Medium-T erm Research (1–2 Y ears) 14.2.1 Multi-Hop Reasoning and Chained Inference Vision: Extend LPF from single-query reasoning to complex multi-step inference chains. Example Scenario: Query: "What’s the risk of regulatory action for Company X?" Reasoning Chain: 1. P(compliance_level | evidence) -> "low" (confidence: 0.92) 2. P(audit_likelihood | compliance_level="low") -> "high" (0.88) 3. P(regulatory_action | audit_likelihood="high", company_size) -> "medium" (0.76) Final Answer: "Medium risk" (confidence: 0.92 x 0.88 x 0.76 = 0.61) Provenance: Complete chain with intermediate factors Arc hitecture Extension: Deﬁne a schema sp ecifying predicate dep endencies, then implement recursiv e decomp osition where complex queries are broken into sub-queries resolved in order. Eac h in termediate result b ecomes a conditioning factor for downstream predictions. Researc h Questions: Ho w to aggregate uncertain t y across c hain steps? Does prediction qualit y degrade with chain length? Ho w should m ulti-hop reasoning b e presented to users? How can cycles in predicate graphs b e handled? Exp ected Con tribution: The ﬁrst system com bining neural evidence perception with structured m ulti-hop probabilistic reasoning. 14.2.2 T emp oral Dynamics and Evidence Deca y Motiv ation: Evidence relev ance decays ov er time; a 5-year-old audit rep ort should w eigh less than a recen t one. Prop osed Approach: Time-aw are evidence w eighting incorp orating exp onential deca y with a p er-predicate decay rate learned to reﬂect domain-sp eciﬁc staleness patterns. Financial data ma y deca y faster than legal precedents; the mo del learns these distinctions from v alidation p erformance at diﬀerent time horizons. Conﬂict Resolution: When old and new evidence contradict, temp oral weigh ting naturally do wnw eigh ts outdated information while preserving it for prov enance, enabling time-a ware explana- tions such as “Prediction changed from X to Y b ecause recent evidence Z contradicts older evidence W (no w weigh ted 0.05 vs. original 0.15).” Applications: Financial compliance (recen t violations matter more than ancient history), healthcare (patient symptoms from yesterda y outw eigh month-old observ ations), and news v eriﬁcation (breaking news sup ersedes older rep orts). 144 14.2.3 Multi-Mo dal Evidence F usion Motiv ation: Real-world reasoning combines text, images, tables, and structured data; the current LPF handles only text. Prop osed Arc hitecture: Extend the evidence enco der to multi-modal inputs—text via the curren t SBER T enco ding to V AE, images via Vision T ransformer to V AE, tables via tabular enco der to V AE, and structured data via graph enco der to V AE, all sharing a common latent space. Key Challenge: Aligning mo dalities in a shared latent space such that semantically equiv alen t information (e.g., “bandgap = 2.5 e V” in text vs. the same v alue in a table) maps to similar latent represen tations. Multi-task training with reconstruction losses p er mo dality , cross-mo dal contrastiv e alignmen t on paired text-image and text-table data, and downstream task sup ervision w ould address this. Applications: Healthcare (combine patient notes, X-rays, lab results), materials science (in- tegrate research pap ers, crystal structures, prop ert y databases), and construction safet y (merge inciden t rep orts, site photos, sensor data). 14.2.4 Hierarc hical Predicate Structures Curren t Limitation: LPF treats predicates as indep enden t; real-world domains hav e hierarchical taxonomies. F or example, compliance_level ma y decomp ose into financial_compliance (co vering audit quality and rep orting accuracy) and operational_compliance (co vering safet y standards and en vironmental regulations). Prop osed Extension: Exploit hierarc hy for transfer learning (train on coarse predicates, ﬁne-tune on sp eciﬁc ones), consistency constraints (predictions must b e logically consisten t—if audit_quality=“low” , then financial_compliance cannot b e “high”), and data eﬃciency (p o ol evidence across related predicates to ov ercome sparse lab els). Exp ected Beneﬁts: 15–30% accuracy improv emen t on rare predicates by leveraging evidence from related predicates. 14.3 Long-T erm Vision (3+ Y ears) 14.3.1 F ederated Priv acy-Preserving LPF Motiv ation: Healthcare, legal, and ﬁnancial domains require m ulti-party evidence aggregation without sharing sensitive data. Arc hitecture: Eac h institution hosts a local evidence retriev er and enco der. Soft factors (not raw evidence) are shared via secure multi-part y computation. A central aggregator combines encrypted factors. No institution sees others’ raw evidence. Priv acy Guarantees: Diﬀeren tial priv acy on soft factors (calibrated noise addition), homo- morphic encryption for factor aggregation, and secure pro venance (recording who con tributed what, without revealing conten t). T rade-oﬀs: Estimated 2–5% accuracy loss from diﬀerential priv acy noise, 10–100 × latency increase from cryptographic op erations, and communication ov erhead scaling with the num b er of parties. V alidation: Multi-hospital patien t diagnosis where hospitals share patient evidence without violating HIP AA, targeting 90%+ of cen tralized LPF accuracy while preserving priv acy . 145 14.3.2 Appro ximate Inference for Massiv e Evidence Sets Curren t Limitation: SPN inference b ecomes in tractable b ey ond K ≈ 100 evidence items due to pro duct complexity . Prop osed Solutions: Low-rank appro ximation of factor pro ducts via truncated SVD, v ariational inference optimizing a tractable approximate p osterior, and Monte Carlo tree searc h sampling high- probabilit y reasoning paths. Goal: Scale to K = 1 , 000+ evidence items with < 5% accuracy loss vs. exact inference, enabling scien tiﬁc literature synthesis where hundreds of pap ers provide evidence for a research question. 14.3.3 Curriculum Learning for Evidence Understanding Observ ation: Curren t training treats all evidence equally; some pieces are inherently harder to in terpret. Prop osed Approach: T rain in three stages: clear, unam biguous evidence early; mo derate diﬃcult y with some con tradiction and tec hnical language mid-training; and the hardest cases in volving subtle implications and domain exp ertise late in training. Hyp othesis: Curriculum accelerates con v ergence and improv es ﬁnal accuracy b y 3–7% by establishing robust representations b efore tac kling ambiguit y . 14.3.4 In teractive Evidence Reﬁnemen t with Human-in-the-Lo op Vision: Deploy LPF as an in teractive assistant where users can c hallenge predictions b y highligh ting o verlooked evidence, provide clariﬁcations when the system is uncertain, and correct misinterpreta- tions in real-time. W orkﬂo w: LPF mak es an initial prediction with prov enance; the user reviews and identiﬁes an error (e.g., misweigh ted evidence); the user provides feedback (natural language or direct weigh t adjustmen t); the system up dates and re-predicts; and the feedbac k is logged for mo del improv emen t. Exp ected Impact: In exp ert-driv en domains (legal case analysis, medical diagnosis), an initial system accuracy of 90% combined with human reﬁnement achiev es eﬀective 99%+ accuracy . 14.4 No v el Application Domains 14.4.1 Scien tiﬁc Literature Syn thesis Researc hers m ust manual ly syn thesize ﬁndings from dozens or hundreds of pap ers to answ er questions suc h as “What factors inﬂuence catalyst activity?” LPF can address this by treating automatically extracted claims from research pap ers as evidence and the target research question as the predicate. Aggregating sometimes-con tradicting exp erimental ﬁndings with prov enance tracking accelerates literature reviews from weeks to hours, identiﬁes consensus vs. con tentious claims automatically , and highlights gaps where more researc h is needed (high epistemic uncertaint y). V alidation could b enc hmark on CORD-19 or materials science datasets, measuring agreemen t with exp ert-written reviews. 14.4.2 News V eriﬁcation and F act-Chec king at Scale Man ual fact-chec king cannot scale to the sp eed of misinformation propagation. LPF can treat retriev ed articles, tw eets, and oﬃcial statements as evidence and claim veracit y (true, false, mixed, un veriﬁable) as the predicate, w eighting sources by credibility . Key adv an tages ov er existing systems include explicit uncertain t y (distinguishing “conﬁdently false” from “insuﬃcien t evidence”), prov enance 146 (sho wing which sources supp ort or refute a claim), and calibrated conﬁdence scores for editorial decisions. 14.4.3 Climate Mo del Ensembles and Uncertain ty Quan tiﬁcation Climate pro jections from m ultiple mo dels with v arying assumptions present a natural m ulti-evidence aggregation problem. Individual climate mo del outputs serv e as evidence, the aggregated prediction (e.g., 2050 temp erature in a region) is the predicate, and mo dels are weigh ted by historical accuracy and inter-model agreement. LPF pro vides principled separation of epistemic uncertaint y (mo del disagreemen t) from aleatoric uncertaint y (c haotic dynamics), prov enance indicating which mo dels con tribute most, and robustness to outlier mo dels. 14.4.4 Quan tum State V eriﬁcation and Multi-Measurement F usion Problem: Determining the true state of a quantum system requires aggregating evidence from m ul- tiple measuremen t bases or exp erimen tal setups, each pro viding partial and often noisy information. LPF Adaptation: Measurement outcomes from diﬀerent bases (e.g., Pauli X, Y, Z measure- men ts) serve as evidence; quan tum state or gate ﬁdelit y is the predicate; measurement statistics are enco ded into a con tinuous latent representation capturing measuremen t uncertaint y; and measure- men ts are combined weigh ted by their precision and informativeness. Key Adv antages: Uncertaint y decomp osition separates measurement noise (aleatoric) from incomplete state information (epistemic). Activ e evidence collection selects the optimal next measuremen t basis. Pro venance trac ks whic h measuremen ts contributed to the state estimate, enabling exp erimental auditing. T ec hnical Challenges: Quan tum measuremen ts collapse states (cannot re-measure); high- dimensional Hilb ert spaces ( 2 n dimensions for n qubits) require eﬃcien t latent enco dings; and measuremen t outcomes are probability distributions rather than deterministic observ ations. Applications: • Quan tum T omography: Reconstruct full quantum states from limited measurements; curren t metho ds require O (4 n ) measuremen ts for n qubits, while LPF could reduce this to O ( n 2 ) via informed measurement selection. • Gate Calibration: V erify quan tum gate ﬁdelity b y aggregating randomized b enchmarking results across multiple gate sequences. • Error Mitigation: Com bine noisy quantum measurements with classical p ost-processing to impro ve eﬀective accuracy . • Quan tum Sensing: F use multiple quantum sensor readings (atomic clo c ks, magnetometers) for enhanced precision. Exp ected Impact: Reduce measuremen t o v erhead in quantum exp erimen ts by 50–70%, pro vide trust worth y uncertaint y estimates crucial for quan tum error correction, enable real-time state v eriﬁcation in quan tum computing workﬂo ws, and oﬀer p otential 10–100 × sp eedup in quan tum tomograph y proto cols. 14.5 Priorit y Recommendations Based on impact, feasibility , and synergies, we recommend the following developmen t timeline. 147 Immediate (6–12 months): Activ e evidence collection (high impact, clear metrics), con trastiv e explanations (builds on existing prov enance), and contin uous predicates (expands applicability). Medium-term (1–2 years): Multi-mo dal fusion (broad applications in healthcare, materials, quan tum sensing), temp oral dynamics (real-world necessity), and m ulti-hop reasoning (op ens new problem class). Long-term (3+ years): F ederated LPF (emerging priv acy requiremen ts in healthcare and quan tum computing), scientiﬁc literature synthesis (high-impact application), and quan tum state v eriﬁcation (nov el domain with growing need). This roadmap balances incremental improv ements (activ e learning, explanations) with trans- formativ e extensions (m ulti-hop reasoning, federated priv acy , quantum applications) to establish LPF as a foundational framework for trustw orthy m ulti-evidence AI across classical and quantum domains. A c kno wledgmen ts W e thank the anonymous reviewers for their insightful feedback that signiﬁcantly improv ed this w ork. W e are grateful to our domain exp ert collab orators in compliance, healthcare, ﬁnance, and legal applications for providing guidance on real-world requirements and v alidating our approach. W e ac knowledge the op en-source communit y for the foundational to ols that enabled this work: PyT orc h, F AISS, Sentence-BER T, and the broader machine learning ecosystem. An y errors or omissions in this work are solely the resp onsibilit y of the authors. A Complete T raining Results This app endix pro vides comprehensive training results for all eight ev aluation domains, including p er-seed breakdowns, con vergence b eha vior, and loss decomp osition. A.1 Compliance Domain (Detailed) T able 44: Compliance domain: detailed seed-by-seed training results Seed T r. Loss T r. CE T r. KL T r. Acc V al Loss V al CE V al KL V al Acc Best V al Acc Best V al Loss Ep. Conv. 42 0.765 0.728 3.650 82.2% 0.741 0.710 3.129 84.0% 85.7% 0.735 10 ✓ 123 0.761 0.726 3.534 82.2% 0.738 0.706 3.193 84.3% 85.6% 0.730 20 × 456 0.763 0.725 3.792 82.7% 0.740 0.709 3.067 84.0% 85.4% 0.731 9 ✓ 789 0.765 0.729 3.587 82.4% 0.738 0.704 3.441 84.7% 85.4% 0.734 11 ✓ 2024 0.775 0.729 4.638 82.3% 0.732 0.692 4.007 85.6% 86 . 0 % ⋆ 0 . 726 ⋆ 12 ✓ 2025 0.756 0.722 3.431 82.6% 0.729 0.699 2.913 84.8% 85.3% 0.727 16 ✓ 314159 0.766 0.732 3.467 81.9% 0.731 0.703 2.848 84.6% 85.7% 0.728 20 × Statistics: T rain A ccuracy 82 . 3 ± 0 . 3% ; V alidation Accuracy 85 . 6 ± 0 . 2% (b est: 86.0%); V alidation Loss 0 . 730 ± 0 . 003 (b est: 0.726). Selected Mo del: Seed 2024 with v alidation accuracy 86.0%. A.2 A cademic Domain Summary Statistics: T rain A ccuracy 83 . 5 ± 0 . 2% ; V alidation Accuracy 85 . 7 ± 0 . 2% (b est: 86.1%); V alidation Loss 0 . 739 ± 0 . 004 (b est: 0.736). Selected Seed: 789. 148 T able 45: A cademic domain: seed-b y-seed training results Seed Final T rain Acc Final V al Acc Best V al A cc Best V al Loss Epo c hs Con v erged 42 83.3% 85.1% 85.5% 0.745 9 ✓ 123 83.5% 85.6% 85.6% 0.735 11 ✓ 456 83.5% 84.6% 85.5% 0.740 13 ✓ 789 83 .2% 84.5% 86 . 1 % ⋆ 0 . 736 ⋆ 15 ✓ 2024 83.9% 85.6% 85.6% 0.736 20 × 2025 83.4% 84.2% 85.7% 0.740 13 ✓ 314159 83.7% 84.6% 85.8% 0.736 10 ✓ T ask: Classify academic publications into ven ue tiers (top-tier, mid-tier, lo w-tier) based on abstract, citations, and author credentials. 42 123 456 789 2024 2025 314159 R andom Seed 0 20 40 60 80 V alidation Accuracy (%) Academic Domain: V alidation Accuracy by Seed Mean: 85.7% 42 123 456 789 2024 2025 314159 R andom Seed 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 V alidation Loss Academic Domain: V alidation Loss by Seed Mean: 0.738 Figure 33: A cademic domain seed comparison. Left: V alidation accuracy by seed (mean: 85.7%). Righ t: V alidation loss by seed (mean: 0.739). Best seed: 789 (86.1% accuracy , 0.736 loss). Notable Observ ations: Strong performance with low v ariance (0.2%), indicating stable optimization. Publication ven ue signals are consistent across evidence. See Figure 33 for seed comparison visualization. A.3 Construction Domain Summary Statistics: T rain A ccuracy 83 . 4 ± 0 . 2% ; V alidation Accuracy 85 . 4 ± 0 . 2% (b est: 85.8%); V alidation Loss 0 . 740 ± 0 . 004 (b est: 0.731). Selected Seed: 789. 149 T able 46: Construction domain: seed-by-seed training results Seed Final T rain Acc Final V al Acc Best V al A cc Best V al Loss Epo c hs Con v erged 42 83.3% 84.5% 85.2% 0.745 9 ✓ 123 83.3% 84.2% 85.6% 0.742 16 ✓ 456 83.2% 83.2% 85.5% 0.743 14 ✓ 789 83 .7% 84.5% 85 . 8 % ⋆ 0 . 731 ⋆ 19 ✓ 2024 83.2% 84.8% 85.3% 0.741 20 × 2025 83.5% 84.7% 85.6% 0.737 13 ✓ 314159 83.7% 83.1% 85.2% 0.740 11 ✓ T ask: Assess construction pro ject risk levels (low, medium, high) from safety rep orts, insp ection logs, and incident records. Notable Observ ations: P erformance comparable to the compliance domain. Low v ariance indicates robust learning despite noisy insp ection rep orts. A.4 Finance Domain Summary Statistics: T rain A ccuracy 83 . 6 ± 0 . 2% ; V alidation Accuracy 84 . 8 ± 0 . 3% (b est: 85.2%); V alidation Loss 0 . 745 ± 0 . 003 (b est: 0.741). Selected Seed: 456. T able 47: Finance domain: seed-by-seed training results Seed Final T rain Acc Final V al Acc Best V al A cc Best V al Loss Epo c hs Con v erged 42 83.6% 83.9% 84.9% 0.742 16 ✓ 123 84.1% 82.5% 84.7% 0.747 15 ✓ 456 83 .4% 83.2% 85 . 2 % ⋆ 0 . 741 ⋆ 14 ✓ 789 83.5% 83.9% 84.4% 0.746 10 ✓ 2024 83.8% 82.9% 84.5% 0.752 9 ✓ 2025 83.6% 84.2% 84.6% 0.744 11 ✓ 314159 83.3% 84.9% 85.2% 0.741 13 ✓ T ask: Predict credit ratings (in vestmen t-grade, sp eculative, default-risk) from ﬁnancial state- men ts, market data, and analyst rep orts. Notable Observ ations: Slightly higher v ariance (0.3%) compared to other domains. Financial text provides consistent signals, though conv ergence v aries across seeds. A.5 Healthcare Domain Summary Statistics: T rain A ccuracy 84 . 2 ± 0 . 2% ; V alidation Accuracy 83 . 8 ± 0 . 1% (b est: 84.0%); V alidation Loss 0 . 756 ± 0 . 003 (b est: 0.753). Selected Seed: 42. 150 T able 48: Healthcare domain: seed-by-seed training results Seed Final T rain Acc Final V al Acc Best V al A cc Best V al Loss Epo c hs Con v erged 42 84.4% 84.0% 84 . 0 % ⋆ 0 . 753 ⋆ 18 ✓ 123 84.0% 82.0% 83.8% 0.758 17 ✓ 456 84.3% 80.6% 83.8% 0.757 16 ✓ 789 84.0% 83.0% 83.7% 0.760 15 ✓ 2024 84.1% 82.7% 83.8% 0.754 14 ✓ 2025 84.0% 81.8% 83.6% 0.753 14 ✓ 314159 83.7% 83.7% 83.7% 0.760 13 ✓ T ask: Classify disease sev erity (mild, mo derate, sev ere) from clinical notes, lab results, and imaging rep orts. Notable Observ ations: V ery lo w v ariance (0.1%) suggests robust learning. Structured medical terminology provides clear diagnostic criteria. Shows a small p ositiv e generalization gap (training sligh tly higher than v alidation). A.6 Legal Domain Summary Statistics: T rain A ccuracy 84 . 8 ± 0 . 2% ; V alidation Accuracy 83 . 6 ± 0 . 1% (b est: 83.7%); V alidation Loss 0 . 759 ± 0 . 004 (b est: 0.755). Selected Seed: 456. T able 49: Legal domain: seed-by-seed training results Seed Final T rain Acc Final V al Acc Best V al A cc Best V al Loss Epo c hs Con v erged 42 85.2% 81.3% 83.6% 0.761 10 ✓ 123 84.4% 82.3% 83.4% 0.760 16 ✓ 456 85 .0% 82.8% 83 . 7 % ⋆ 0 . 755 ⋆ 13 ✓ 789 84.8% 83.0% 83.6% 0.763 14 ✓ 2024 84.6% 82.4% 83.6% 0.753 20 × 2025 84.8% 82.6% 83.7% 0.760 15 ✓ 314159 84.9% 83.0% 83.6% 0.764 10 ✓ T ask: Predict case outcomes (plaintiﬀ-wins, defendan t-wins, settlemen t) from legal briefs, preceden ts, and case facts. Notable Observ ations: Lo west v alidation accuracy but very low v ariance (0.1%). Legal reasoning inv olv es subtle distinctions, making evidence harder to aggregate. The larger train-v al gap (1.2%) suggests an ov erﬁtting tendency . A.7 Materials Domain Summary Statistics: T rain A ccuracy 83 . 9 ± 0 . 2% ; V alidation Accuracy 84 . 0 ± 0 . 5% (b est: 84.6%); V alidation Loss 0 . 757 ± 0 . 003 (b est: 0.752). Selected Seed: 456. 151 T able 50: Materials domain: seed-by-seed training results Seed Final T rain Acc Final V al Acc Best V al A cc Best V al Loss Epo c hs Con v erged 42 84.2% 82.4% 83.3% 0.759 10 ✓ 123 84.3% 82.1% 84.4% 0.752 20 × 456 84 .2% 82.5% 84 . 6 % ⋆ 0 . 754 ⋆ 8 ✓ 789 83.8% 81.6% 84.4% 0.754 13 ✓ 2024 84.0% 82.8% 83.5% 0.759 10 ✓ 2025 83.9% 81.6% 83.6% 0.759 11 ✓ 314159 83.6% 82.7% 83.9% 0.756 10 ✓ T ask: Classify material prop erties (lo w-strength, medium-strength, high-strength) from comp o- sition data, test results, and sp eciﬁcations. Notable Observ ations: Highest v ariance across seeds (0.5%) despite quantitativ e evidence, suggesting sensitivit y to initialization when learning from technical measurements and sp eciﬁcations. A.8 FEVER Domain Summary Statistics: T rain A ccuracy 99 . 6 ± 0 . 1% ; V alidation Accuracy 99 . 9 ± 0 . 0% (b est: 99.9%); V alidation Loss 0 . 574 ± 0 . 001 (b est: 0.572). Selected Seed: 2025. T able 51: FEVER domain: seed-by-seed training results Seed Final T rain Acc Final V al A cc Best V al A cc Best V al Loss Ep ochs Conv erged 42 99.7% 99.7% 99.9% 0.573 30 × 123 99.6% 99.6% 99.8% 0.576 30 × 456 99.5% 99.8% 99.8% 0.576 30 × 789 99.6% 99.8% 99.9% 0.574 30 × 1011 99.6% 99.7% 99.9% 0.573 28 ✓ 2024 99.6% 99.6% 99.9% 0.572 30 × 2025 99.7% 99.6% 99 . 9 % ⋆ 0 . 573 ⋆ 30 × T ask: F act veriﬁcation (SUPPOR TS, REFUTES, NOT ENOUGH INFO) from Wikip edia evidence. Notable Observ ations: Near-p erfect accuracy with minimal v ariance (0.0%). FEVER provides clean, w ell-structured evidence with strong textual entailmen t signals, making it substantially easier than real-w orld domains. Most seeds ran to the full 30 ep ochs. Serv es as an upp er b ound on mo del capabilit y . Loss decomp osition is not av ailable for FEVER due to a diﬀeren t training conﬁguration. 152 A.9 Cross-Domain Insigh ts F ever Academic Compliance Construction Finance Materials Healthcare Legal Domain 70 75 80 85 90 95 100 Accuracy (%) 99.6 83.5 82.3 83.4 83.6 84.0 84.1 84.8 99.9 85.7 85.6 85.4 84.8 84.0 83.8 83.6 Encoder-Decoder Training: Cross-Domain P erformance Train V alidation Figure 34: Cross-domain training and v alidation accuracy comparison. Error bars show standard deviation ov er 7 random seeds. Domains are sorted by v alidation accuracy (descending). FEVER ac hieves near-p erfect accuracy (99.9%), while Legal represents the most challenging domain (83.6%). 153 Academic Compliance Construction F ever Finance Healthcare Legal Materials Domain 0.0 0.1 0.2 0.3 0.4 Standard Deviation (%) 0.18% 85.7% 0.22% 85.6% 0.21% 85.4% 0.05% 99.9% 0.30% 84.8% 0.10% 83.8% 0.09% 83.6% 0.46% 84.0% V alidation Accuracy V ariance Across Seeds Figure 35: V alidation accuracy v ariance across seeds b y domain. Lo wer bars indicate more stable training. FEVER sho ws minimal v ariance (0.0%), while Materials exhibits the highest sensitivity to initialization (0.5%). Mean v alidation accuracy shown b elo w each bar. 154 Compliance Academic Construction Finance F ever Materials Healthcare Legal Domain 3 2 1 0 1 Train - V alidation Accuracy (%) -3.3% -2.2% -2.1% -1.2% -0.2% 0.0% 0.3% 1.2% Generalization Gap by Domain Figure 36: Generalization gap (train − v alidation accuracy) by domain. Negativ e v alues indicate v alidation outp erforms training. Most domains show go od generalization, though Healthcare and Legal show small p ositiv e gaps suggesting slight ov erﬁtting. Academic Compliance Construction F ever Finance Healthcare Legal Materials Domain 0 5 10 15 20 25 30 A verage Epochs to Convergence 13.0 14.0 14.6 29.7 12.6 15.3 14.0 11.7 Training Convergence Speed Academic Compliance Construction F ever Finance Healthcare Legal Materials Domain 0 20 40 60 80 100 Convergence R ate (%) 86% 71% 86% 14% 100% 100% 86% 86% Early Stopping Success R ate Figure 37: T raining con vergence analysis. Left: a verage ep ochs to con vergence (lo wer is faster). Righ t: early stopping success rate (p ercen tage of seeds that conv erged b efore maximum ep ochs). Healthcare and Finance show p erfect conv ergence (100%), while FEVER rarely early-stops due to the large dataset scale. 155 T able 52: Cross-domain summary statistics Domain T rain Acc V al Acc Gap V ariance Mean Ep o c hs Conv. Rate FEVER 99 . 6 ± 0 . 1% 99 . 9 ± 0 . 0% − 0 . 3% 0 . 0% 29.7 14% A cademic 83 . 5 ± 0 . 2% 85 . 7 ± 0 . 2% − 2 . 2% 0 . 2% 12.7 86% Compliance 82 . 3 ± 0 . 3% 85 . 6 ± 0 . 2% − 3 . 3% 0 . 2% 13.9 71% Construction 83 . 4 ± 0 . 2% 85 . 4 ± 0 . 2% − 2 . 0% 0 . 2% 13.7 86% Finance 83 . 6 ± 0 . 2% 84 . 8 ± 0 . 3% − 1 . 2% 0 . 3% 11.7 100% Materials 83 . 9 ± 0 . 2% 84 . 0 ± 0 . 5% +0 . 1% 0 . 5% 10.3 86% Healthcare 84 . 2 ± 0 . 2% 83 . 8 ± 0 . 1% +0 . 4% 0 . 1% 14.7 100% Legal 84 . 8 ± 0 . 2% 83 . 6 ± 0 . 1% +1 . 2% 0 . 1% 11.1 86% F our key ﬁndings emerge. First, six domains show negativ e generalization gaps (v alidation > training), while Healthcare and Legal sho w small p ositiv e gaps suggesting slight o verﬁtting tendencies. Second, FEVER has minimal v ariance (0.0%) due to clean data, while Materials (0.5%) and Finance (0.3%) sho w higher v ariance, reﬂecting sensitivity to initialization from technical or ﬁnancial data. Third, domains with structured evidence (FEVER, Academic) ac hieve higher accuracy than those with unstructured text (Legal, Healthcare, Materials). F ourth, the enco der-deco der architecture maintains stable p erformance across diverse domains without domain-sp eciﬁc tuning, demonstrating broad applicabilit y . A.10 P er-Domain Seed Visualizations This subsection presen ts detailed seed-lev el visualizations for each domain, showing v alidation accuracy and loss distributions across all 7 random seeds. The b est seed (mark ed in gold) was selected based on highest v alidation accuracy and used for all downstream exp erimen ts. 42 123 456 789 2024 2025 314159 R andom Seed 0 20 40 60 80 V alidation Accuracy (%) Compliance Domain: V alidation Accuracy by Seed Mean: 85.6% 42 123 456 789 2024 2025 314159 R andom Seed 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 V alidation Loss Compliance Domain: V alidation Loss by Seed Mean: 0.730 Figure 38: Compliance domain seed comparison. Left: V alidation accuracy b y seed (mean: 85.6%). Righ t: V alidation loss by seed (mean: 0.730). Best seed: 2024 (86.0% accuracy , 0.726 loss). 156 42 123 456 789 2024 2025 314159 R andom Seed 0 20 40 60 80 V alidation Accuracy (%) Academic Domain: V alidation Accuracy by Seed Mean: 85.7% 42 123 456 789 2024 2025 314159 R andom Seed 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 V alidation Loss Academic Domain: V alidation Loss by Seed Mean: 0.738 Figure 39: A cademic domain seed comparison. Left: V alidation accuracy by seed (mean: 85.7%). Righ t: V alidation loss by seed (mean: 0.739). Best seed: 789 (86.1% accuracy , 0.736 loss). 42 123 456 789 2024 2025 314159 R andom Seed 0 20 40 60 80 V alidation Accuracy (%) Construction Domain: V alidation Accuracy by Seed Mean: 85.4% 42 123 456 789 2024 2025 314159 R andom Seed 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 V alidation Loss Construction Domain: V alidation Loss by Seed Mean: 0.740 Figure 40: Construction domain seed comparison. Left: V alidation accuracy by seed (mean: 85.4%). Righ t: V alidation loss by seed (mean: 0.740). Best seed: 789 (85.8% accuracy , 0.731 loss). 157 42 123 456 789 2024 2025 314159 R andom Seed 0 10 20 30 40 50 60 70 80 V alidation Accuracy (%) Finance Domain: V alidation Accuracy by Seed Mean: 84.8% 42 123 456 789 2024 2025 314159 R andom Seed 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 V alidation Loss Finance Domain: V alidation Loss by Seed Mean: 0.745 Figure 41: Finance domain seed comparison. Left: V alidation accuracy by seed (mean: 84.8%). Righ t: V alidation loss by seed (mean: 0.745). Best seed: 456 (85.2% accuracy , 0.741 loss). 42 123 456 789 2024 2025 314159 R andom Seed 0 10 20 30 40 50 60 70 80 V alidation Accuracy (%) Healthcare Domain: V alidation Accuracy by Seed Mean: 83.8% 42 123 456 789 2024 2025 314159 R andom Seed 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 V alidation Loss Healthcare Domain: V alidation Loss by Seed Mean: 0.756 Figure 42: Healthcare domain seed comparison. Left: V alidation accuracy b y seed (mean: 83.8%). Righ t: V alidation loss by seed (mean: 0.756). Best seed: 42 (84.0% accuracy , 0.753 loss). 158 42 123 456 789 2024 2025 314159 R andom Seed 0 10 20 30 40 50 60 70 80 V alidation Accuracy (%) Legal Domain: V alidation Accuracy by Seed Mean: 83.6% 42 123 456 789 2024 2025 314159 R andom Seed 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 V alidation Loss Legal Domain: V alidation Loss by Seed Mean: 0.759 Figure 43: Legal domain seed comparison. Left: V alidation accuracy by seed (mean: 83.6%). Right: V alidation loss b y seed (mean: 0.759). Best seed: 456 (83.7% accuracy , 0.755 loss). 42 123 456 789 2024 2025 314159 R andom Seed 0 10 20 30 40 50 60 70 80 V alidation Accuracy (%) Materials Domain: V alidation Accuracy by Seed Mean: 84.0% 42 123 456 789 2024 2025 314159 R andom Seed 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 V alidation Loss Materials Domain: V alidation Loss by Seed Mean: 0.756 Figure 44: Materials domain seed comparison. Left: V alidation accuracy by seed (mean: 84.0%). Righ t: V alidation loss by seed (mean: 0.757). Best seed: 456 (84.6% accuracy , 0.754 loss). 159 42 123 456 789 1011 2024 2025 R andom Seed 0 20 40 60 80 100 V alidation Accuracy (%) F ever Domain: V alidation Accuracy by Seed Mean: 99.9% 42 123 456 789 1011 2024 2025 R andom Seed 0.0 0.1 0.2 0.3 0.4 0.5 0.6 V alidation Loss F ever Domain: V alidation Loss by Seed Mean: 0.574 Figure 45: FEVER dataset seed comparison. Left: V alidation accuracy by seed (mean: 99.9%). Righ t: V alidation loss by seed (mean: 0.574). Best seed: 2025 (99.9% accuracy , 0.573 loss). Note the extremely small v ariance compared to other domains. A.11 Loss Decomp osition Analysis F or domains with full loss tracking (all except FEVER), w e analyze the contribution of cross-en tropy and KL divergence terms to the total training loss. These visualizations use the b est seed for eac h domain. 160 Cross-Entropy KL (× ) T otal 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Loss V alue 0.692 0.040 0.732 Compliance: Loss Decomposition (Best Seed 2024) Figure 46: Compliance domain loss decomp osition (seed 2024). Cross-entrop y: 0.692, KL ( × β = 0 . 01 ): 0.040, T otal: 0.726. The w eighted KL term con tributes 5.5% of total loss, providing regularization without excessive compression. 161 Cross-Entropy KL (× ) T otal 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Loss V alue 0.704 0.044 0.748 Academic: Loss Decomposition (Best Seed 789) Figure 47: A cademic domain loss decomp osition (seed 789). Cross-entrop y: 0.735, KL ( × β ): 0.001, T otal: 0.736. Minimal KL con tribution (0.1%) indicates the encoder learned compact represen tations naturally . 162 Cross-Entropy KL (× ) T otal 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Loss V alue 0.707 0.040 0.747 Construction: Loss Decomposition (Best Seed 789) Figure 48: Construction domain loss decomp osition (seed 789). Shows balanced con tribution from classiﬁcation and regularization terms. 163 Cross-Entropy KL (× ) T otal 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Loss V alue 0.716 0.037 0.754 Finance: Loss Decomposition (Best Seed 456) Figure 49: Finance domain loss decomp osition (seed 456). Cross-en tropy dominates the total loss, with KL providing mild regularization. 164 Cross-Entropy KL (× ) T otal 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Loss V alue 0.713 0.043 0.755 Healthcare: Loss Decomposition (Best Seed 42) Figure 50: Healthcare domain loss decomp osition (seed 42). W ell-balanced loss comp onen ts indicate eﬀectiv e V AE training. 165 Cross-Entropy KL (× ) T otal 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Loss V alue 0.724 0.041 0.765 Legal: Loss Decomposition (Best Seed 456) Figure 51: Legal domain loss decomp osition (seed 456). Higher cross-entrop y reﬂects the domain’s diﬃcult y , while KL remains controlled. 166 Cross-Entropy KL (× ) T otal 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Loss V alue 0.724 0.042 0.767 Materials: Loss Decomposition (Best Seed 456) Figure 52: Materials domain loss decomp osition (seed 456). Similar pattern to other domains with CE con tributing approximately 95% of total loss. A cross all seven domains (excluding FEVER), the cross-entrop y term consistently con tributes 94–99% of the total v alidation loss, while the weigh ted KL div ergence ( β = 0 . 01 ) con tributes 1–6%. This balance reﬂects three key prop erties: eﬀective regularization (the KL term provides suﬃcient regularization without ov er-compressing the latent space), stable training (consisten t loss ratios across diver se domains suggest robust hyperparameter choices), and task fo cus (the mo del prioritizes classiﬁcation accuracy via CE while maintaining structured latent representations via KL). The β = 0 . 01 weigh t ac hieves the desired balance b et ween reconstruction ﬁdelity and latent space structure, enabling eﬀective downstream factor conv ersion for probabilistic reasoning. B Detailed Domain-b y-Domain Results This app endix provides comprehensive exp erimental results for all eight ev aluation domains, including detailed pro v enance records, ablation studies, error analyses, and additional visualizations not included in the main pap er. B.1 Compliance Domain (Primary Ev aluation) B.1.1 Complete Ablation Results Mon te Carlo Samples (n_samples) 167 n_samples Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 4 0.9778 0.9702 0.9775 0.1928 0.0243 0.1266 0.8512 0.1167 8 0.9407 0.9278 0.9398 0.2408 0.0331 0.1522 0.8203 0.1366 16 0.9630 0.9547 0.9625 0.2852 0.0390 0.1824 0.7806 0.1257 32 0.9778 0.9742 0.9776 0.3128 0.0424 0.2249 0.7528 0.1239 T able 53: Compliance domain ablation: Monte Carlo sample coun t. T emperature Scaling T emperature Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 0.8 0.9704 0.9614 0.9698 0.2113 0.0282 0.1309 0.8395 0.1292 1.0 0.9704 0.9638 0.9702 0.2634 0.0345 0.1862 0.7894 0.1264 1.2 0.9778 0.9724 0.9776 0.3341 0.0457 0.2421 0.7357 0.1125 1.5 0.9704 0.9640 0.9701 0.4331 0.0653 0.3057 0.6646 0.1055 T able 54: Compliance domain ablation: temp erature scaling. Uncertain t y Penalt y (alpha) Alpha Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 0.1 0.9704 0.9635 0.9701 0.1076 0.0159 0.0149 0.9747 0.0708 1.0 0.9852 0.9831 0.9851 0.1221 0.0193 0.0526 0.9326 0.1242 2.0 0.9630 0.9547 0.9625 0.2798 0.0380 0.1789 0.7841 0.1233 5.0 0.9778 0.9724 0.9776 1.0048 0.2009 0.6112 0.3666 0.0109 T able 55: Compliance domain ablation: uncertaint y p enalt y α . Evidence Coun t (top_k) top_k Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 1 0.7926 0.7500 0.7829 0.8798 0.1709 0.3466 0.4460 0.0424 3 0.9185 0.9091 0.9165 0.5037 0.0820 0.2882 0.6339 0.1111 5 0.9704 0.9696 0.9702 0.2800 0.0378 0.1909 0.7795 0.1288 10 0.9704 0.9614 0.9698 0.2779 0.0364 0.1932 0.7826 0.1160 20 0.9778 0.9724 0.9776 0.2680 0.0350 0.1873 0.7905 0.1190 T able 56: Compliance domain ablation: evidence count (top_k). 168 B.1.2 All Seeds Detailed Results Seed Accuracy ECE NLL Brier Macro F1 42 0.9967 0.0111 0.0239 0.0032 0.9964 123 0.9967 0.0113 0.0221 0.0032 0.9964 456 0.9972 0.0119 0.0245 0.0030 0.9971 789 0.9972 0.0112 0.0249 0.0030 0.9971 1011 0.9967 0.0102 0.0216 0.0028 0.9965 2024 0.9967 0.0107 0.0245 0.0032 0.9965 2025 0.9967 0.0108 0.0249 0.0033 0.9965 3141 0.9967 0.0096 0.0222 0.0028 0.9965 9999 0.9967 0.0084 0.0178 0.0024 0.9965 12345 0.9967 0.0091 0.0216 0.0025 0.9963 54321 0.9956 0.0087 0.0228 0.0031 0.9952 11111 0.9961 0.0113 0.0237 0.0031 0.9958 77777 0.9961 0.0094 0.0262 0.0032 0.9959 99999 0.9972 0.0106 0.0249 0.0031 0.9971 314159 0.9961 0.0093 0.0194 0.0024 0.9960 Statistics: Mean Accuracy: 99.66% (Std: 0.05%) Mean ECE: 1.03% (Std: 0.10%) Mean NLL: 0.023 (Std: 0.002) Best Seed: 456 (99.72% accuracy) T able 57: Complete results for all 15 seeds (compliance domain, LPF-SPN). B.1.3 Complete Error Analysis Confusion matrices for all mo dels (compliance domain): LPF-SPN: Predicted Low Medium High Actual Low 26 1 0 Medium 1 67 2 High 0 1 37 LPF-Learned: Predicted Low Medium High Actual Low 26 1 0 Medium 0 70 0 High 1 10 27 V AE-Only: Predicted Low Medium High 169 Actual Low 24 3 0 Medium 0 70 0 High 0 3 35 BER T: Predicted Low Medium High Actual Low 24 1 2 Medium 0 70 0 High 1 0 37 EDL-Aggregated: Predicted Low Medium High Actual Low 0 27 0 Medium 74 0 0 High 1 0 37 (Catastrophic failure: predicts “low” for 74/70 medium samples.) B.1.4 Complete Pro venance Records (Sample) Record 1: INF00000003 • Timestamp: 2026-01-12T13:17:02.027245 • En tity: C0036 • Predicate: compliance_lev el • Query T yp e: marginal • T op V alue: high (conﬁdence: 0.5071) • Ground T ruth: high ✓ • Distribution: {“lo w”: 0.2699, “medium”: 0.2230, “high”: 0.5071} • Evidence Chain: C0036_E177, C0036_E178, C0036_E179, C0036_E180, C0036_E181 • Execution Time: 10.55 ms • Hyp erparameters: {n_samples: 16, temp erature: 1.0, alpha: 2.0, top_k: 10} Record 2: INF00000006 • Timestamp: 2026-01-12T13:17:02.170872 • En tity: C0191 • Predicate: compliance_lev el • Query T yp e: marginal 170 • T op V alue: medium (conﬁdence: 0.5060) • Ground T ruth: medium ✓ • Distribution: {“lo w”: 0.2766, “medium”: 0.5060, “high”: 0.2175} • Evidence Chain: C0191_E951, C0191_E952, C0191_E953, C0191_E954, C0191_E955 • Execution Time: 8.58 ms • Hyp erparameters: {n_samples: 16, temp erature: 1.0, alpha: 2.0, top_k: 10} Record 3: INF00000007 • Timestamp: 2026-01-12T13:17:02.186815 • En tity: C0195 • Predicate: compliance_lev el • Query T yp e: marginal • T op V alue: medium (conﬁdence: 0.5557) • Ground T ruth: medium ✓ • Distribution: {“lo w”: 0.2187, “medium”: 0.5557, “high”: 0.2256} • Evidence Chain: C0195_E972, C0195_E973, C0195_E974, C0195_E975, C0195_E976 • Execution Time: 8.79 ms • Hyp erparameters: {n_samples: 16, temp erature: 1.0, alpha: 2.0, top_k: 10} B.2 A cademic Domain B.2.1 Domain Ov erview T ask: Predict grant prop osal approv al like liho o d based on PI creden tials, research prop osal quality , and institutional factors. Classes: {lik ely_reject, p ossible, likely_accept} Example Data Poin t: { "proposal_id": "G0003", "pi_name": "Elena Patel", "institution": "Caltech", "field": "Biology", "grant_amount": 1078124.75, "h_index": 3, "citation_count": 389, "publication_count": 17, "approval_likelihood":"likely_reject" } 171 B.2.2 Best Seed Results Mo del A ccuracy Macro F1 Wtd F1 NLL ECE R T (ms) LPF-SPN 1.000 1.000 1.000 0.007 0.006 17.6 LPF-Learned 1.000 1.000 1.000 0.016 0.014 43.6 V AE-Only 0.993 0.993 0.993 0.166 0.138 7.9 BER T — — — — — — SPN-Only 0.970 0.969 0.970 0.549 0.383 2.4 EDL-Aggregated 0.407 0.193 0.236 1.097 0.068 1.3 EDL-Individual 0.407 0.193 0.236 5.936 0.094 4.0 R-GCN 0.407 0.193 0.236 1.099 0.074 0.001 T able 58: Model comparison (academic domain, seed 2024). P erfect 100% accuracy achiev ed b y b oth LPF-SPN and LPF-Learned, demonstrating the system’s capability on well-structured n umerical evidence. B.2.3 Ablation Study Mon te Carlo Samples: n_samples Accuracy NLL ECE Runtime (ms) 4 0.9852 0.1576 0.1123 2.3 8 0.9852 0.1825 0.1335 2.9 16 0.9704 0.2330 0.1762 3.5 32 0.9926 0.2751 0.2201 5.8 T able 59: A cademic domain ablation: Mon te Carlo sample count. T emperature: T emperature Accuracy NLL ECE 0.8 0.9852 0.1628 0.1213 1.0 0.9852 0.2366 0.1834 1.2 0.9778 0.2982 0.2192 1.5 0.9852 0.3811 0.2901 T able 60: A cademic domain ablation: temp erature scaling. Alpha: Alpha Accuracy NLL ECE 0.1 0.9852 0.0398 0.0286 1.0 0.9926 0.0691 0.0477 2.0 0.9778 0.2310 0.1697 5.0 0.9852 0.9965 0.6158 T able 61: A cademic domain ablation: uncertain ty p enalt y α . 172 T op-K: top_k Accuracy NLL Run time (ms) 1 0.8519 0.8414 1.9 3 0.9407 0.4774 2.7 5 0.9852 0.2295 3.5 10 0.9778 0.2394 5.2 20 0.9852 0.2400 8.1 T able 62: A cademic domain ablation: evidence count (top_k). B.2.4 Error Analysis T otal Errors: 0 (100% accuracy on test set). No confusion matrix av ailable (p erfect classiﬁcation). B.2.5 Pro venance Records (Sample) Record 1: INF00000001 • En tity: G0003 • T op V alue: likely_reject (conﬁdence: 1.0000) • Ground T ruth: lik ely_reject ✓ • Distribution: {“lik ely_reject”: 0.9999982, “p ossible”: 0.0000017, “likely_accept”: 6.1e-08} • Evidence: [G0003_E011, G0003_E012, G0003_E013, G0003_E014, G0003_E015] • Time: 18.67 ms Record 2: INF00000002 • En tity: G0011 • T op V alue: p ossible (conﬁdence: 1.0000) • Ground T ruth: possible ✓ • Distribution: {“lik ely_reject”: 5.5e-07, “p ossible”: 0.9999995, “likely_accept”: 2.3e-09} • Evidence: [G0011_E051, G0011_E052, G0011_E053, G0011_E054, G0011_E055] • Time: 14.92 ms Record 3: INF00000007 • En tity: G0044 • T op V alue: likely_accept (conﬁdence: 0.9999) • Ground T ruth: lik ely_accept ✓ • Distribution: {“lik ely_reject”: 3.2e-08, “p ossible”: 0.0001347, “likely_accept”: 0.9998653} • Evidence: [G0044_E216, G0044_E217, G0044_E218, G0044_E219, G0044_E220] • Time: 13.10 ms 173 B.3 Construction Domain B.3.1 Domain Ov erview T ask: Assess construction pro ject risk based on structural complexity , con tractor exp erience, budget adequacy , and en vironmental factors. Classes: {lo w_risk, mo derate_risk, high_risk} Example Data Poin t: { "project_id": "C0016", "project_name": "Gateway Center", "project_type": "commercial", "budget": 30740188.96, "structural_complexity": 7, "safety_record_score": 58.62, "project_risk": "high_risk" } B.3.2 Complete Ablation Results Mon te Carlo Samples (n_samples): n_samples Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 4 0.9778 0.9775 0.9779 0.1593 0.0210 0.1056 0.8721 0.1274 8 0.9852 0.9842 0.9851 0.1759 0.0215 0.1291 0.8561 0.1219 16 0.9778 0.9762 0.9777 0.2319 0.0298 0.1683 0.8095 0.1291 32 0.9926 0.9921 0.9926 0.3005 0.0426 0.2322 0.7604 0.1460 T able 63: Construction domain ablation: Monte Carlo sample coun t. T emperature Scaling: T emperature Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 0.8 0.9852 0.9842 0.9851 0.1648 0.0213 0.1252 0.8658 0.1387 1.0 0.9630 0.9638 0.9630 0.2488 0.0338 0.1653 0.8030 0.1342 1.2 0.9481 0.9447 0.9471 0.3016 0.0425 0.2029 0.7596 0.1370 1.5 0.9778 0.9761 0.9775 0.3817 0.0559 0.2967 0.6977 0.1225 T able 64: Construction domain ablation: temp erature scaling. Uncertain t y Penalt y (alpha): 174 Alpha Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 0.1 0.9481 0.9479 0.9482 0.0918 0.0209 0.0327 0.9713 0.0854 1.0 0.9778 0.9785 0.9779 0.0996 0.0166 0.0509 0.9310 0.1263 2.0 0.9778 0.9774 0.9777 0.2204 0.0280 0.1641 0.8183 0.1327 5.0 0.9926 0.9921 0.9926 0.9921 0.1980 0.6215 0.3711 0.0129 T able 65: Construction domain ablation: uncertaint y p enalt y α . Evidence Coun t (top_k): top_k Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 1 0.8889 0.8882 0.8882 0.8095 0.1544 0.4144 0.4745 0.0532 3 0.9333 0.9330 0.9333 0.4308 0.0680 0.2591 0.6798 0.1281 5 1.0000 1.0000 1.0000 0.2324 0.0292 0.1956 0.8044 0.1278 10 0.9778 0.9774 0.9777 0.2314 0.0306 0.1673 0.8105 0.1397 20 0.9704 0.9695 0.9705 0.2432 0.0326 0.1589 0.8115 0.1321 T able 66: Construction domain ablation: evidence count (top_k). B.3.3 Best Seed Comparison Mo del A ccuracy Macro F1 ECE Run time (ms) LPF-SPN 1.000 1.000 0.014 18.1 LPF-Learned 0.985 0.985 0.017 41.0 V AE-Only 0.993 0.993 0.135 8.9 SPN-Only 0.970 0.968 0.388 2.2 EDL-Aggregated 0.363 0.178 0.026 1.4 EDL-Individual 0.356 0.175 0.145 4.3 R-GCN 0.356 0.175 0.022 0.001 T able 67: Model comparison (construction domain, seed 314159). B.3.4 Error Analysis Mo del T otal Errors T otal Predictions Error Rate LPF-SPN 0 135 0.000 LPF-Learned 2 135 0.015 V AE-Only 1 135 0.007 SPN-Only 4 135 0.030 EDL-Aggregated 86 135 0.637 EDL-Individual 87 135 0.644 R-GCN 87 135 0.644 T able 68: Construction domain error counts by mo del. 175 Confusion Matrices: LPF-SPN (P erfect Classiﬁcation): Predicted Low Moderate High Actual Low 48 0 0 Moderate 0 49 0 High 0 0 38 LPF-Learned: Predicted Low Moderate High Actual Low 48 0 0 Moderate 1 48 0 High 1 0 37 SPN-Only: Predicted Low Moderate High Actual Low 48 0 0 Moderate 0 49 0 High 2 2 34 B.3.5 Pro venance Records (Sample) Record 1: INF00000001 • En tity: C0016 • T op V alue: high_risk (conﬁdence: 0.9999) • Ground T ruth: high_risk ✓ • Distribution: {“lo w_risk”: 2.7e-07, “mo derate_risk”: 0.000016, “high_risk”: 0.9999839} • Evidence: [C0016_E076, C0016_E077, C0016_E078, C0016_E079, C0016_E080] • Time: 116.35 ms Record 2: INF00000002 • En tity: C0022 • T op V alue: low_risk (conﬁdence: 1.0000) • Ground T ruth: lo w_risk ✓ • Distribution: {“lo w_risk”: 0.9999995, “mo derate_risk”: 1.6e-08, “high_risk”: 5.3e-07} • Evidence: [C0022_E106, C0022_E107, C0022_E108, C0022_E109, C0022_E110] • Time: 87.73 ms 176 Record 3: INF00000005 • En tity: C0032 • T op V alue: high_risk (conﬁdence: 0.9970) • Ground T ruth: high_risk ✓ • Distribution: {“lo w_risk”: 3.6e-06, “mo derate_risk”: 0.002949, “high_risk”: 0.9970476} • Evidence: [C0032_E156, C0032_E157, C0032_E158, C0032_E159, C0032_E160] • Time: 48.91 ms B.4 FEVER Domain B.4.1 Domain Ov erview T ask: F act v eriﬁcation from Wikip edia evidence — classify claims as SUPPOR TS/REFUTES/NOT ENOUGH INF O (mapp ed to compliance levels). Dataset: 145K training claims, 19K v alidation, 1,800 test samples. Example Data Poin t: { "fact_id": "FEVER_225709", "claim": "South Korea has a highly educated white collar workforce.", "fever_label": "NOT ENOUGH INFO", "compliance_level": "medium", "num_evidence": 1 } B.4.2 Best Seed Results Mo del A ccuracy F1 ECE Run time (ms) LPF-SPN 0.997 0.997 0.012 25.2 LPF-Learned 0.997 0.997 0.003 24.0 V AE-Only 0.997 0.997 0.003 3.5 SPN-Only 0.952 0.951 0.289 0.9 EDL-Aggregated 0.502 0.223 0.167 1.2 EDL-Individual 0.502 0.223 0.001 1.9 R-GCN 0.228 0.124 0.105 0.001 Gro q-llama-3.3-70b 0.440 0.440 0.744 1581.6 Gro q-qw en3-32b 0.620 0.620 0.823 3176.4 T able 69: Mo del comparison (FEVER domain, seed 456). Near-p erfect p erformance (99.7%) across all LPF v ariants and V AE-Only indicates FEVER provides v ery clean, unam biguous evidence signals. B.4.3 Error Analysis T otal Errors: 5 out of 1,800 samples (0.28% error rate). Confusion Matrix (LPF-SPN): 177 Predicted Low Medium High Actual Low 411 2 0 Medium 1 486 1 High 3 1 895 Error Breakdo wn: • High → Low: 3 errors (strong claim misclassiﬁed) • High → Medium: 1 error • Low → High: 2 errors (weak claim ov er-estimated) • Medium → High: 1 error B.4.4 Pro venance Records (Sample) Record 1: INF00000002 • En tity: FEVER_104983 • T op V alue: high (conﬁdence: 1.0000) • Ground T ruth: high (SUPPOR TS) ✓ • Distribution: {“lo w”: 1.4e-16, “medium”: 1.9e-16, “high”: 1.0000} • Evidence: [FEVER_104983_E3031, FEVER_104983_E3032, FEVER_104983_E3033, FEVER_104983_E3034, FEVER_104983_E3035] • Time: 36.54 ms Record 2: INF00000005 • En tity: FEVER_48827 • T op V alue: medium (conﬁdence: 1.0000) • Ground T ruth: medium (NOT ENOUGH INF O) ✓ • Distribution: {“lo w”: 4.5e-06, “medium”: 0.9999889, “high”: 6.6e-06} • Evidence: [FEVER_48827_E8649, FEVER_48827_E8650, FEVER_48827_E8651, FEVER_48827_E8652, FEVER_48827_E8653] • Time: 20.16 ms Record 3: INF00000009 • En tity: FEVER_76456 • T op V alue: high (conﬁdence: 1.0000) • Ground T ruth: high (SUPPOR TS) ✓ • Distribution: {“lo w”: 1.2e-32, “medium”: 3.8e-25, “high”: 1.0} • Evidence: [FEVER_76456_E13824, FEVER_76456_E13825, FEVER_76456_E13826, FEVER_76456_E13827, FEVER_76456_E13828] • Time: 32.16 ms 178 B.5 Finance Domain B.5.1 Domain Ov erview T ask: Credit default risk assessment based on b orro w er credit history , debt ratios, and ﬁnancial b eha vior. Classes: {lo w_risk, medium_risk, high_risk} B.5.2 Complete Ablation Results Mon te Carlo Samples (n_samples): n_samples Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 4 0.9778 0.9787 0.9778 0.1550 0.0183 0.1132 0.8761 0.1145 8 0.9778 0.9768 0.9777 0.2033 0.0247 0.1442 0.8336 0.1221 16 0.9778 0.9768 0.9777 0.2440 0.0303 0.1841 0.7988 0.1190 32 0.9778 0.9787 0.9778 0.3060 0.0410 0.2219 0.7559 0.1153 T able 70: Finance domain ablation: Monte Carlo sample coun t. T emperature Scaling: T emperature Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 0.8 0.9852 0.9852 0.9852 0.1701 0.0193 0.1314 0.8588 0.1189 1.0 0.9852 0.9869 0.9852 0.2563 0.0325 0.1932 0.7919 0.1214 1.2 0.9852 0.9852 0.9852 0.3262 0.0451 0.2468 0.7384 0.1203 1.5 0.9778 0.9771 0.9778 0.4115 0.0609 0.3008 0.6770 0.1108 T able 71: Finance domain ablation: temp erature scaling. Uncertain t y Penalt y (alpha): Alpha Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 0.1 0.9852 0.9870 0.9852 0.0688 0.0118 0.0168 0.9725 0.0805 1.0 0.9852 0.9869 0.9852 0.0839 0.0112 0.0508 0.9344 0.1106 2.0 0.9852 0.9844 0.9853 0.2419 0.0295 0.1850 0.8002 0.1139 5.0 0.9778 0.9787 0.9778 0.9974 0.1992 0.6086 0.3691 0.0113 T able 72: Finance domain ablation: uncertaint y p enalt y α . Evidence Coun t (top_k): 179 top_k Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 1 0.8444 0.8397 0.8442 0.8551 0.1647 0.3771 0.4673 0.0431 3 0.9778 0.9779 0.9778 0.4613 0.0723 0.3240 0.6537 0.1111 5 0.9778 0.9787 0.9778 0.2465 0.0307 0.1819 0.7988 0.1162 10 0.9704 0.9674 0.9705 0.2541 0.0327 0.1823 0.7937 0.1286 20 0.9704 0.9704 0.9704 0.2607 0.0339 0.1804 0.7924 0.1299 T able 73: Finance domain ablation: evidence count (top_k). B.5.3 Best Seed Comparison Mo del A ccuracy Macro F1 ECE Run time (ms) LPF-SPN 0.993 0.993 0.010 16.3 LPF-Learned 0.985 0.985 0.013 42.7 V AE-Only 0.985 0.985 0.111 8.3 SPN-Only 0.956 0.954 0.348 2.3 EDL-Aggregated 0.437 0.203 0.100 1.3 EDL-Individual 0.333 0.167 0.167 4.7 R-GCN 0.333 0.167 0.000 0.001 T able 74: Model comparison (ﬁnance domain, seed 123). B.5.4 Error Analysis Mo del T otal Errors T otal Predictions Error Rate LPF-SPN 1 135 0.007 LPF-Learned 2 135 0.015 V AE-Only 2 135 0.015 SPN-Only 6 135 0.044 EDL-Aggregated 76 135 0.563 EDL-Individual 90 135 0.667 R-GCN 90 135 0.667 T able 75: Finance domain error counts by mo del. Confusion Matrices: LPF-SPN: Predicted Low Medium High Actual Low 45 0 0 Medium 1 58 0 High 0 0 31 LPF-Learned: 180 Predicted Low Medium High Actual Low 45 0 0 Medium 1 58 0 High 0 1 30 B.5.5 Pro venance Records (Sample) Record 1: INF00000001 • En tity: B0029 • T op V alue: high_risk (conﬁdence: 0.9982) • Ground T ruth: high_risk ✓ • Distribution: {“lo w_risk”: 0.0002, “medium_risk”: 0.0016, “high_risk”: 0.9982} • Evidence: [B0029_R141, B0029_R142, B0029_R143, B0029_R144, B0029_R145] • Time: 23.25 ms Record 2: INF00000002 • En tity: B0031 • T op V alue: low_risk (conﬁdence: 1.0000) • Ground T ruth: lo w_risk ✓ • Distribution: {“lo w_risk”: 0.9999999887, “medium_risk”: 1.1e-08, “high_risk”: 5.7e-10} • Evidence: [B0031_R151, B0031_R152, B0031_R153, B0031_R154, B0031_R155] • Time: 18.15 ms Record 3: INF00000003 • En tity: B0039 • T op V alue: high_risk (conﬁdence: 0.9997) • Ground T ruth: high_risk ✓ • Distribution: {“lo w_risk”: 2.0e-07, “medium_risk”: 0.0002709, “high_risk”: 0.9997289} • Evidence: [B0039_R191, B0039_R192, B0039_R193, B0039_R194, B0039_R195] • Time: 17.45 ms B.6 Healthcare Domain B.6.1 Domain Ov erview T ask: Diagnosis severit y classiﬁcation from patient symptoms, lab results, and vital signs. Classes: {mild, mo derate, severe} 181 B.6.2 Complete Ablation Results Mon te Carlo Samples (n_samples): n_samples Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 4 0.9778 0.9663 0.9776 0.1868 0.0243 0.1152 0.8626 0.1237 8 0.9704 0.9564 0.9707 0.2196 0.0284 0.1367 0.8337 0.1287 16 0.9556 0.9421 0.9560 0.2622 0.0355 0.1578 0.7977 0.1318 32 0.9852 0.9732 0.9852 0.3191 0.0434 0.2370 0.7481 0.1209 T able 76: Healthcare domain ablation: Monte Carlo sample coun t. T emperature Scaling: T emperature Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 0.8 0.9704 0.9537 0.9706 0.1973 0.0256 0.1310 0.8504 0.1304 1.0 0.9852 0.9797 0.9851 0.2542 0.0325 0.1912 0.7940 0.1217 1.2 0.9852 0.9732 0.9852 0.3112 0.0413 0.2377 0.7475 0.1137 1.5 0.9630 0.9593 0.9628 0.4181 0.0623 0.2813 0.6817 0.1031 T able 77: Healthcare domain ablation: temp erature scaling. Uncertain t y Penalt y (alpha): Alpha Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 0.1 0.9852 0.9797 0.9851 0.0634 0.0093 0.0186 0.9770 0.0689 1.0 0.9852 0.9732 0.9852 0.0987 0.0135 0.0495 0.9429 0.0966 2.0 0.9704 0.9561 0.9705 0.2581 0.0331 0.1880 0.7945 0.1184 5.0 0.9704 0.9537 0.9706 1.0058 0.2011 0.6039 0.3664 0.0111 T able 78: Healthcare domain ablation: uncertaint y p enalt y α . Evidence Coun t (top_k): top_k Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 1 0.8815 0.8440 0.8861 0.8235 0.1574 0.4147 0.4667 0.0434 3 0.9481 0.9400 0.9486 0.4821 0.0784 0.2988 0.6493 0.1134 5 0.9778 0.9728 0.9776 0.2567 0.0333 0.1801 0.7977 0.1222 10 0.9852 0.9732 0.9852 0.2360 0.0284 0.1797 0.8055 0.1148 20 0.9630 0.9466 0.9636 0.2541 0.0331 0.1585 0.8045 0.1209 T able 79: Healthcare domain ablation: evidence count (top_k). 182 B.6.3 Best Seed Comparison Mo del A ccuracy Macro F1 ECE Run time (ms) LPF-SPN 0.993 0.986 0.006 16.7 LPF-Learned 0.978 0.967 0.022 40.1 V AE-Only 0.985 0.973 0.127 8.0 SPN-Only 0.844 0.779 0.253 2.4 EDL-Aggregated 0.267 0.140 0.250 1.4 EDL-Individual 0.267 0.140 0.068 4.8 R-GCN 0.267 0.140 0.067 0.001 T able 80: Model comparison (healthcare domain, seed 77777). B.6.4 Error Analysis Mo del T otal Errors T otal Predictions Error Rate LPF-SPN 1 135 0.007 LPF-Learned 3 135 0.022 V AE-Only 2 135 0.015 SPN-Only 21 135 0.156 EDL-Aggregated 99 135 0.733 EDL-Individual 99 135 0.733 R-GCN 99 135 0.733 T able 81: Healthcare domain error counts by mo del. Confusion Matrices: LPF-SPN: Predicted Mild Moderate Severe Actual Mild 36 0 0 Moderate 0 80 0 Severe 1 0 18 LPF-Learned: Predicted Mild Moderate Severe Actual Mild 35 0 1 Moderate 1 79 0 Severe 1 0 18 SPN-Only: Predicted Mild Moderate Severe Actual Mild 25 11 0 Moderate 0 80 0 Severe 1 9 9 183 B.6.5 Pro venance Records (Sample) Record 1: INF00000001 • En tity: P0004 • Condition: heart_disease • T op V alue: mo derate (conﬁdence: 1.0000) • Ground T ruth: moderate ✓ • Distribution: {“mild”: 0.000018, “mo derate”: 0.999957, “severe”: 0.000025} • Evidence: [P0004_R016, P0004_R017, P0004_R018, P0004_R019, P0004_R020] • Time: 16.68 ms Record 2: INF00000002 • En tity: P0021 • T op V alue: mild (conﬁdence: 0.9683) • Ground T ruth: mild ✓ • Distribution: {“mild”: 0.9682726, “mo derate”: 0.0015646, “severe”: 0.0301628} • Evidence: [P0021_R101, P0021_R102, P0021_R103, P0021_R104, P0021_R105] • Time: 6.10 ms Record 3: INF00000004 • En tity: P0045 • T op V alue: severe (conﬁdence: 1.0000) • Ground T ruth: sev ere ✓ • Distribution: {“mild”: 0.000017, “mo derate”: 5.5e-07, “severe”: 0.9999825} • Evidence: [P0045_R221, P0045_R222, P0045_R223, P0045_R224, P0045_R225] • Time: 4.79 ms B.7 Legal Domain B.7.1 Domain Ov erview T ask: Litigation outcome prediction from case t yp e, evidence quality , precedent strength, and part y resources. Classes: {plain tiﬀ_fa vored, neutral, defendant_fa v ored} 184 B.7.2 Complete Ablation Results Mon te Carlo Samples (n_samples): n_samples Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 4 0.9630 0.9617 0.9628 0.2007 0.0266 0.1388 0.8374 0.1339 8 0.9704 0.9698 0.9706 0.2476 0.0329 0.1841 0.7975 0.1358 16 0.9778 0.9769 0.9777 0.3043 0.0426 0.2219 0.7558 0.1365 32 0.9556 0.9554 0.9560 0.3461 0.0500 0.2352 0.7263 0.1286 T able 82: Legal domain ablation: Monte Carlo sample coun t. T emperature Scaling: T emperature Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 0.8 0.9704 0.9680 0.9701 0.2199 0.0293 0.1475 0.8229 0.1359 1.0 0.9852 0.9850 0.9853 0.2860 0.0390 0.2253 0.7657 0.1333 1.2 0.9852 0.9843 0.9851 0.3613 0.0520 0.2742 0.7110 0.1208 1.5 0.9778 0.9774 0.9777 0.4612 0.0725 0.3344 0.6434 0.1098 T able 83: Legal domain ablation: temp erature scaling. Uncertain t y Penalt y (alpha): Alpha Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 0.1 0.9704 0.9699 0.9703 0.0584 0.0114 0.0195 0.9670 0.0925 1.0 0.9852 0.9843 0.9851 0.1277 0.0193 0.0739 0.9113 0.1272 2.0 0.9704 0.9692 0.9701 0.3021 0.0413 0.2259 0.7554 0.1320 5.0 0.9630 0.9629 0.9629 1.0032 0.2005 0.5957 0.3673 0.0105 T able 84: Legal domain ablation: uncertaint y p enalt y α . Evidence Coun t (top_k): top_k Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 1 0.8741 0.8736 0.8746 0.8561 0.1652 0.4179 0.4562 0.0420 3 0.9037 0.9022 0.9036 0.5418 0.0921 0.3100 0.6071 0.1179 5 0.9704 0.9692 0.9701 0.3160 0.0451 0.2190 0.7514 0.1355 10 0.9778 0.9774 0.9777 0.2928 0.0389 0.2166 0.7612 0.1173 20 0.9852 0.9856 0.9852 0.3009 0.0414 0.2244 0.7608 0.1267 T able 85: Legal domain ablation: evidence count (top_k). 185 B.7.3 Best Seed Comparison Mo del A ccuracy Macro F1 ECE Run time (ms) LPF-SPN 0.993 0.992 0.011 16.7 LPF-Learned 1.000 1.000 0.006 44.7 V AE-Only 0.993 0.992 0.153 8.9 SPN-Only 0.970 0.970 0.371 2.4 EDL-Aggregated 0.326 0.164 0.011 1.4 EDL-Individual 0.289 0.149 0.211 4.7 R-GCN 0.326 0.164 0.007 0.001 T able 86: Mo del comparison (legal domain, seed 314159). LPF-Learned achiev es p erfect accuracy on this domain, suggesting learned aggregation may b etter capture subtle legal reasoning patterns. B.7.4 Error Analysis Mo del T otal Errors T otal Predictions Error Rate LPF-SPN 1 135 0.007 LPF-Learned 0 135 0.000 V AE-Only 1 135 0.007 SPN-Only 4 135 0.030 EDL-Aggregated 91 135 0.674 EDL-Individual 96 135 0.711 R-GCN 91 135 0.674 T able 87: Legal domain error counts by mo del. Confusion Matrices: LPF-SPN: Predicted Plaintiff Neutral Defendant Actual Plaintiff 44 0 0 Neutral 0 38 1 Defendant 0 0 52 LPF-Learned (P erfect Classiﬁcation): Predicted Plaintiff Neutral Defendant Actual Plaintiff 44 0 0 Neutral 0 39 0 Defendant 0 0 52 SPN-Only: Predicted Plaintiff Neutral Defendant 186 Actual Plaintiff 43 1 0 Neutral 0 38 1 Defendant 1 1 50 B.7.5 Pro venance Records (Sample) Record 1: INF00000001 • En tity: L0003 • Case: T ec hCorp v. Davis • T op V alue: neutral (conﬁdence: 1.0000) • Ground T ruth: neutral ✓ • Distribution: {“plain tiﬀ_fav ored”: 2.1e-08, “neutral”: 0.9999999, “defendant_fa v ored”: 9.7e- 09} • Evidence: [L0003_E011, L0003_E012, L0003_E013, L0003_E014, L0003_E015] • Time: 18.50 ms Record 2: INF00000002 • En tity: L0006 • T op V alue: neutral (conﬁdence: 0.9997) • Ground T ruth: neutral ✓ • Distribution: {“plain tiﬀ_fav ored”: 8.2e-06, “neutral”: 0.9997415, “defendant_fa vored”: 0.0002503} • Evidence: [L0006_E026, L0006_E027, L0006_E028, L0006_E029, L0006_E030] • Time: 17.20 ms Record 3: INF00000003 • En tity: L0009 • T op V alue: defendant_fa v ored (conﬁdence: 1.0000) • Ground T ruth: defendan t_fa vored ✓ • Distribution: {“plaintiﬀ_fa v ored”: 6.7e-09, “neutral”: 4.5e-07, “defendant_fa vored”: 0.9999995} • Evidence: [L0009_E041, L0009_E042, L0009_E043, L0009_E044, L0009_E045] • Time: 17.50 ms B.8 Materials Science Domain B.8.1 Domain Ov erview T ask: Chemical syn thesis viability prediction based on thermo dynamic stability , precursor av ailability , and reaction complexity . Classes: {not_viable, p ossibly_viable, highly_viable} 187 B.8.2 Complete Ablation Results Mon te Carlo Samples (n_samples): n_samples Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 4 0.9852 0.9824 0.9850 0.3127 0.0416 0.2363 0.7488 0.1139 8 0.9704 0.9678 0.9703 0.3623 0.0516 0.2527 0.7177 0.1194 16 0.9630 0.9592 0.9631 0.3914 0.0570 0.2693 0.6936 0.1083 32 0.9778 0.9725 0.9775 0.4150 0.0616 0.3041 0.6737 0.1071 T able 88: Materials domain ablation: Monte Carlo sample coun t. T emperature Scaling: T emperature Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 0.8 0.9630 0.9592 0.9631 0.3145 0.0430 0.2261 0.7478 0.1275 1.0 0.9852 0.9842 0.9851 0.3910 0.0567 0.2993 0.6908 0.1125 1.2 0.9778 0.9756 0.9777 0.4658 0.0725 0.3385 0.6393 0.1031 1.5 0.9852 0.9824 0.9850 0.5554 0.0925 0.4031 0.5820 0.0858 T able 89: Materials domain ablation: temp erature scaling. Uncertain t y Penalt y (alpha): Alpha Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 0.1 0.9778 0.9725 0.9775 0.0706 0.0105 0.0352 0.9569 0.0872 1.0 0.9778 0.9756 0.9777 0.1485 0.0190 0.1000 0.8833 0.1210 2.0 0.9704 0.9678 0.9703 0.3886 0.0560 0.2921 0.6898 0.1102 5.0 0.9704 0.9678 0.9703 1.0288 0.2064 0.6127 0.3577 0.0080 T able 90: Materials domain ablation: uncertaint y p enalt y α . Evidence Coun t (top_k): top_k Accuracy Macro F1 Wtd F1 NLL Brier ECE Conf Mean Conf Std 1 0.8741 0.8680 0.8746 0.9073 0.1776 0.4526 0.4215 0.0236 3 0.9481 0.9431 0.9484 0.6191 0.1082 0.3973 0.5509 0.0932 5 0.9852 0.9824 0.9850 0.3987 0.0584 0.2978 0.6874 0.1131 10 0.9630 0.9549 0.9625 0.3922 0.0568 0.2732 0.6898 0.1073 20 0.9778 0.9725 0.9775 0.3981 0.0582 0.2888 0.6889 0.1079 T able 91: Materials domain ablation: evidence count (top_k). 188 B.8.3 Best Seed Comparison Mo del A ccuracy Macro F1 ECE Run time (ms) LPF-SPN 0.993 0.992 0.015 18.8 LPF-Learned 0.985 0.982 0.016 41.8 V AE-Only 0.985 0.982 0.124 7.9 SPN-Only 0.963 0.955 0.369 2.5 EDL-Aggregated 0.481 0.217 0.108 1.4 EDL-Individual 0.481 0.217 0.146 4.3 R-GCN 0.237 0.128 0.096 0.001 T able 92: Model comparison (materials domain, seed 314159). B.8.4 Error Analysis Mo del T otal Errors T otal Predictions Error Rate LPF-SPN 1 135 0.007 LPF-Learned 2 135 0.015 V AE-Only 2 135 0.015 SPN-Only 5 135 0.037 EDL-Aggregated 70 135 0.519 EDL-Individual 70 135 0.519 R-GCN 103 135 0.763 T able 93: Materials domain error counts by mo del. Confusion Matrices: LPF-SPN: Predicted Not Possibly Highly Actual Not 32 1 0 Possibly 0 65 0 Highly 0 0 38 LPF-Learned: Predicted Not Possibly Highly Actual Not 31 1 1 Possibly 0 65 0 Highly 0 0 38 SPN-Only: Predicted Not Possibly Highly Actual Not 30 2 1 Possibly 0 65 0 Highly 2 0 36 189 B.8.5 Pro venance Records (Sample) Record 1: INF00000001 • En tity: M0004 • F orm ula: Li3Cu3 • T op V alue: p ossibly_viable (conﬁdence: 1.0000) • Ground T ruth: possibly_viable ✓ • Distribution: {“not_viable”: 6.2e-06, “p ossibly_viable”: 0.9999800, “highly_viable”: 0.000014} • Evidence: [M0004_E016, M0004_E017, M0004_E018, M0004_E019, M0004_E020] • Time: 27.10 ms Record 2: INF00000002 • En tity: M0005 • T op V alue: not_viable (conﬁdence: 1.0000) • Ground T ruth: not_viable ✓ • Distribution: {“not_viable”: 0.9999843, “p ossibly_viable”: 5.4e-06, “highly_viable”: 0.000010} • Evidence: [M0005_E021, M0005_E022, M0005_E023, M0005_E024, M0005_E025] • Time: 17.49 ms Record 3: INF00000003 • En tity: M0015 • T op V alue: not_viable (conﬁdence: 0.9700) • Ground T ruth: not_viable ✓ • Distribution: {“not_viable”: 0.9699775, “p ossibly_viable”: 0.0178464, “highly_viable”: 0.0121761} • Evidence: [M0015_E071, M0015_E072, M0015_E073, M0015_E074, M0015_E075] • Time: 17.28 ms 190 B.9 Cross-Domain Ablation Analysis B.9.1 n_samples A cross All Domains Domain n=4 n=8 n=16 n=32 Best Compliance 97.8% 94.1% 96.3% 97.8% 4, 32 A cademic 98.5% 98.5% 97.0% 99.3% 32 Construction 97.8% 98.5% 97.8% 99.3% 32 Finance 97.8% 97.8% 97.8% 97.8% all Healthcare 97.8% 97.0% 95.6% 98.5% 32 Legal 96.3% 97.0% 97.8% 95.6% 16 Materials 98.5% 97.0% 96.3% 97.8% 4 FEVER — — — — — T able 94: Accuracy by n_samples across domains. No consistent optimal sample count across domains. n=4 suﬃcien t for simpler tasks (compliance, materials), while n=32 b eneﬁts more complex reasoning (academic, construction). B.9.2 T emp erature Across All Domains Domain T=0.8 T=1.0 T=1.2 T=1.5 Best Compliance 97.0% 97.0% 97.8% 97.0% 1.2 A cademic 98.5% 98.5% 97.8% 98.5% 0.8, 1.0, 1.5 Construction 98.5% 96.3% 94.8% 97.8% 0.8 Finance 98.5% 98.5% 98.5% 97.8% 0.8, 1.0, 1.2 Healthcare 98.5% 98.5% 98.5% 96.3% 0.8, 1.0, 1.2 Legal 97.0% 98.5% 98.5% 97.8% 1.0, 1.2 Materials 96.3% 98.5% 97.8% 98.5% 1.0, 1.5 T able 95: Accuracy by temp erature across domains. T = 0 . 8 and T = 1 . 0 most robust across domains. Higher temp eratures ( T > 1 . 0 ) degrade calibration without improving accuracy . B.9.3 Alpha A cross All Domains Domain α =0.1 α =1.0 α =2.0 α =5.0 Best Compliance 97.0% 98.5% 96.3% 97.8% 1.0 A cademic 98.5% 99.3% 97.8% 98.5% 1.0 Construction 94.8% 97.8% 97.8% 99.3% 5.0 Finance 98.5% 98.5% 98.5% 97.8% 0.1, 1.0, 2.0 Healthcare 98.5% 98.5% 97.0% 97.0% 0.1, 1.0 Legal 97.0% 98.5% 97.0% 96.3% 1.0 Materials 97.8% 97.8% 97.0% 97.0% 0.1, 1.0 T able 96: Accuracy by α across domains. α = 1 . 0 most consisten t. α = 0 . 1 b est for calibration, α = 1 . 0 b est for accuracy . α = 5 . 0 generally harmful except construction domain. 191 B.9.4 T op-K Across All Domains Domain k=1 k=3 k=5 k=10 k=20 Best Compliance 79.3% 91.9% 97.0% 97.0% 97.8% 20 A cademic 85.2% 94.1% 98.5% 97.8% 98.5% 5, 20 Construction 88.9% 93.3% 100.0% 97.8% 97.0% 5 Finance 84.4% 97.8% 97.8% 97.0% 97.0% 3, 5 Healthcare 88.1% 94.8% 97.8% 98.5% 96.3% 10 Legal 87.4% 90.4% 97.0% 97.8% 98.5% 20 Materials 87.4% 94.8% 98.5% 96.3% 97.8% 5 T able 97: Accuracy by top_k across domains. Consistent pattern: dramatic improv emen t from k = 1 to k = 5 , then diminishing returns. k = 5 optimal for most domains. B.10 LLM Ev aluation Details B.10.1 Prompt T emplate You are an expert system evaluating multi-evidence scenarios. Task: Assess {task_description} based on the following evidence. Evidence: {evidence_1} {evidence_2} ... {evidence_n} Instructions: 1. Carefully analyze each piece of evidence 2. Consider the credibility and relevance of each source 3. Synthesize the information to reach a conclusion 4. Provide your final answer in this exact format: ANSWER: {class_label} CONFIDENCE: {0.0-1.0} REASONING: {brief explanation} Respond now: B.10.2 Resp onse Parsing LLM resp onses parsed using regex patterns: answer_pattern = r"ANSWER:\s*(\w+)" confidence_pattern = r"CONFIDENCE:\s*([\d.]+)" reasoning_pattern = r"REASONING:\s*(.+)" 192 B.10.3 Detailed LLM Results Mo del A ccuracy P arsed F ailed A vg Conf A vg R T (ms) llama-3.3-70b 95.9% 49/50 1 N/A 1578.7 qw en3-32b 98.0% 50/50 0 N/A 3008.6 kimi-k2 98.0% 50/50 0 N/A 764.2 gpt-oss-120b 93.9% 49/50 1 N/A 1541.7 T able 98: LLM p erformance by mo del (compliance domain, 50 samples). LLMs o ccasionally deviate from requested format despite explicit instructions. F ailed parses treated as incorrect predictions. LLMs do not pro duce well-calibrated probability distributions; even when conﬁdence scores are extracted, they show p o or correlation with actual correctness (ECE 79.7–81.6%). B.10.4 Cost Analysis Mo del Input T ok/Q Output T ok/Q Cost/1M In Cost/1M Out Cost/Q llama-3.3-70b ∼ 800 ∼ 50 $0.59 $0.79 $0.0004 qw en3-32b ∼ 800 ∼ 50 $0.18 $0.18 $0.0003 kimi-k2 ∼ 800 ∼ 50 $0.30 $0.30 $0.0002 gpt-oss-120b ∼ 800 ∼ 50 $0.80 $0.80 $0.0006 T able 99: API cost breakdown (Gro q pricing, January 2026). T otal ev aluation cost: 50 samples × 4 mo dels ≈ $0.08. Extrap olation to full test set (135 samples × 4 mo dels): ≈ $0.22. Pro duction deplo yment cost (1M queries): $200–600/million queries vs. $0 for LPF-SPN (self-hosted). B.11 Statistical Signiﬁcance T esting B.11.1 P aired t-tests (Compliance Domain) Comparison Mean Diﬀ (Acc) t-stat p-v alue Sig ( α =0.05) LPF-SPN vs LPF-Learned + 6.7% 58.3 < 0.001 ✓ LPF-SPN vs V AE-Only + 2.2% 12.4 < 0.001 ✓ LPF-SPN vs BER T + 0.8% 3.2 0.006 ✓ LPF-SPN vs SPN-Only + 2.8% 15.1 < 0.001 ✓ T able 100: P aired t-test results comparing LPF-SPN to baselines (15 seeds). All diﬀerences statistically signiﬁcan t at α = 0 . 05 lev el, conﬁrming LPF-SPN’s sup erior p erformance is not due to random chance. 193 B.11.2 Calibration Qualit y T ests Mo del Mean ECE 95% CI Calibration Quality LPF-SPN 2.1% [1.7%, 2.5%] Excellent LPF-Learned 6.6% [5.8%, 7.4%] Go od V AE-Only 9.6% [8.2%, 11.0%] F air BER T 3.2% [2.8%, 3.6%] Go o d SPN-Only 30.9% [28.4%, 33.4%] Poor T able 101: ECE comparison with conﬁdence interv als (compliance domain). Non-ov erlapping conﬁdence interv als conﬁrm LPF-SPN ac hieves signiﬁcantly b etter calibration than all baselines. B.12 Computational Resources B.12.1 T raining Resource Requirements Comp onen t T raining Time CPU Cores Memory (GB) Storage (GB) V AE Enco der 15 min 8 4 0.5 Deco der Netw ork 25 min 8 6 0.8 Learned Aggregator 10 min 8 3 0.3 T otal 50 min 8 6 1.6 T able 102: T raining resource consumption (p er seed, compliance domain). Extrapolation to all seeds: 15 seeds × 50 min = 12.5 hours total; parallelizable to ≈ 2 hours wall time on a 64-core machine running 8 seeds simultaneously . B.12.2 Inference Resource Requiremen ts Mo del CPU Time (ms) Memory (MB) Net work I/O LPF-SPN 14.8 1200 0 LPF-Learned 37.4 1800 0 BER T 134.7 4500 0 LLM (Gro q) 1500–3000 ∼ 0 (remote) ∼ 10 KB T able 103: Inference resource consumption per query . LPF-SPN throughput: 68 queries/second/core; single 64-core machine: 4,352 queries/second; daily capacity: 376M queries. B.13 Data Generation Pro cess B.13.1 Syn thetic Data Generation P arameters All synthetic domains (excluding FEVER) generated using a controlled sto c hastic pro cess. Base P arameters: En tities p er domain: 900; Evidence p er entit y: 5; Y ears (compliance only): 3 (2020–2022); T rain/V al/T est split: 70%/15%/15%. Evidence Credibilit y Distribution: Mean: 0.87; Std: 0.08; Range: [0.65, 0.98]; Distribution: Beta(10, 2) scaled to [0.65, 0.98]. 194 B.13.2 FEVER Data Prepro cessing Original FEVER F ormat: { "id": 225709, "claim": "South Korea has a highly educated white collar workforce.", "label": "NOT ENOUGH INFO", "evidence": [ [ "South_Korea", 0, "The country is noted for its population density of 487 per km2." ] ] } Mapp ed F ormat: { "fact_id": "FEVER_225709", "claim": "South Korea has a highly educated white collar workforce.", "fever_label": "NOT ENOUGH INFO", "compliance_level": "medium", "num_evidence": 1 } Lab el Mapping: • SUPPOR TS → high (compliance with claim) • REFUTES → low (con tradicts claim) • NOT ENOUGH INFO → medium (insuﬃcien t evidence) B.14 Hyp erparameter Search Details B.14.1 Searc h Space Deﬁnition Hyp erparameter Type Searc h Space Default Sampling n_samples int [4, 8, 16, 32] 16 Grid temp erature ﬂoat [0.8, 1.0, 1.2, 1.5] 1.0 Grid alpha ﬂoat [0.1, 1.0, 2.0, 5.0] 2.0 Grid top_k int [1, 3, 5, 10, 20] 5 Grid learning_rate ﬂoat [1e-4, 2e-4, 5e-4] 2e-4 Grid laten t_dim in t [32, 64, 128] 64 Grid hidden_dims list [[256,128], [512,256]] [256,128] Grid T able 104: Complete hyperparameter search space. T otal conﬁgurations tested: 4 × 4 × 4 × 5 = 320 (ablation study). Compute budget: 320 conﬁgs × 50 min = 267 hours ( ≈ 11 days serial; ≈ 1.5 days with 8 parallel work ers). 195 B.14.2 Best Conﬁguration b y Domain Domain n_samples temp erature alpha top_k Compliance 16 0.8 0.1 5 A cademic 4 0.8 0.1 5 Construction 4 0.8 0.1 5 Finance 4 0.8 0.1 5 Healthcare 4 0.8 0.1 5 Legal 4 0.8 0.1 5 Materials 4 0.8 0.1 5 FEVER 4 0.8 0.1 5 T able 105: Optimal hyperparameters p er domain. Remark able consistency across domains: conﬁgu- ration (n=4, T=0.8, α =0.1, k=5) w orks well for 7/8 domains. Only compliance b eneﬁts from higher n_samples. B.15 Repro ducibilit y Chec klist B.15.1 Co de and Data A v ailabilit y • Co de Rep ository: [URL to b e provided up on publication] • Pretrained Mo dels: A v ailable for all 8 domains × 15 seeds (compliance) or 7 seeds (others) • Syn thetic Datasets: Included with repro ducible generation scripts • FEVER Dataset: A v ailable at https://fever.ai B.15.2 Hardw are Requirements Minim um: CPU: 4 cores, 2.0 GHz; RAM: 8 GB; Storage: 10 GB; OS: Lin ux, macOS, or Windows. Recommended: CPU: 16+ cores, 3.0+ GHz; RAM: 32 GB; Storage: 50 GB SSD; OS: Lin ux (Ubun tu 20.04+). B.15.3 Soft ware Dep endencies python>=3.9 torch>=2.0.0 numpy>=1.24.0 pandas>=2.0.0 scikit-learn>=1.3.0 sentence-transformers>=2.2.0 faiss-cpu>=1.7.4 matplotlib>=3.7.0 seaborn>=0.12.0 F ull requirements.txt a v ailable in rep ository . 196 B.16 A c knowledgmen ts and Ethics Statement B.16.1 Data Ethics All synthetic data generated for this researc h contains no real p ersonally identiﬁable information (PI I), uses randomly generated names and entities, and cannot b e reverse-engineered to identify real individuals or organizations. FEVER dataset: Publicly a v ailable benchmark with appropriate licensing, con taining only publicly av ailable Wikip edia conten t. No additional ethical concerns. B.16.2 P otential Misuse LPF is designed for legitimate decision-supp ort applications. Poten tial misuse scenarios include: (1) Bias Ampliﬁcation — if training data contains biases, LPF may amplify them through evidence w eighting; (2) Over-reliance — users ma y trust w ell-calibrated predictions without verifying underlying evidence; (3) A dversarial Manipulation — attack ers could craft misleading evidence with high credibility scores. Mitigation Strategies: Regular bias audits of training data; mandatory human review for high-stak es decisions; adversarial training and robustness testing; transparency through prov enance records. B.16.3 In tended Use Cases Appropriate: Decision supp ort in ﬁnance, healthcare, legal, compliance; research and education; qualit y assurance and auditing; risk assessment. Inappropriate: Sole basis for life-altering decisions (medical diagnosis, sentencing); surveillance or priv acy-inv asive applications; discriminatory practices; autonomous w eap ons systems. B.17 Conclusion This app endix pro vides comprehensiv e exp erimen tal details supp orting the main pap er’s claims. Key tak eaw a ys: 1. Robustness: LPF-SPN achiev es consisten t high p erformance across 8 diverse domains with minimal hyperparameter tuning. 2. Statistical V alidity: Results based on 15 seeds (compliance) and 7 seeds (other domains) with tight conﬁdence interv als. 3. Calibration Excellence: ECE 0.6–2.1% across domains, 57–60 × b etter than LLM baselines. 4. Eﬃciency: 14.8ms av erage inference time enables real-time applications. 5. In terpretability: Complete prov enance records enable full audit trails. 6. Repro ducibilit y: All co de, data, and mo dels av ailable for veriﬁcation and extension. The comprehensiv e results demonstrate that LPF-SPN represents a signiﬁcant adv ancemen t in m ulti-evidence reasoning, combining the accuracy of neural approaches with the calibration and in terpretability of probabilistic mo dels. 197 References Aliyu Agb oola Alege Alege. 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I Know What I Don't Know: Latent Posterior Factor Models for Multi-Evidence Probabilistic Reasoning

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