Bayesian Neural Networks for Functional ANOVA model

With the increasing demand for interpretability in machine learning, functional ANOVA decomposition has gained renewed attention as a principled tool for breaking down high-dimensional function into low-dimensional components that reveal the contributions of different variable groups. Recently, Tensor Product Neural Network (TPNN) has been developed and applied as basis functions in the functional ANOVA model, referred to as ANOVA-TPNN. A disadvantage of ANOVA-TPNN, however, is that the components to be estimated must be specified in advance, which makes it difficult to incorporate higher-order TPNNs into the functional ANOVA model due to computational and memory constraints. In this work, we propose Bayesian-TPNN, a Bayesian inference procedure for the functional ANOVA model with TPNN basis functions, enabling the detection of higher-order components with reduced computational cost compared to ANOVA-TPNN. We develop an efficient MCMC algorithm and demonstrate that Bayesian-TPNN performs well by analyzing multiple benchmark datasets. Theoretically, we prove that the posterior of Bayesian-TPNN is consistent.

💡 Research Summary

The paper addresses a central challenge in interpretable machine learning: how to decompose a high‑dimensional predictive function into low‑dimensional components (the functional ANOVA or Sobol decomposition) while still being able to capture higher‑order interactions without prohibitive computational cost. Existing work, notably ANOVA‑TPNN, uses Tensor‑Product Neural Networks (TPNNs) as basis functions for each component, but it requires the user to pre‑specify the maximum interaction order d and the number of basis networks K_S for every subset S of variables. Because the number of subsets grows combinatorially with p, practical implementations are limited to main effects and pairwise interactions, leaving many real‑world problems under‑modeled.

The authors propose Bayesian‑TPNN, a fully Bayesian treatment of the functional ANOVA model in which the architecture itself—i.e., the number of hidden nodes K and the variable sets S_k attached to each node—is treated as a random variable. The model can be written as

\