Predictive variational inference: Learn the predictively optimal posterior distribution

Vanilla variational inference finds an optimal approximation to the Bayesian posterior distribution, but even the exact Bayesian posterior is often not meaningful under model misspecification. We propose predictive variational inference (PVI): a general inference framework that seeks and samples from an optimal posterior density such that the resulting posterior predictive distribution is as close to the true data generating process as possible, while this closeness is measured by multiple scoring rules. By optimizing the objective, the predictive variational inference is generally not the same as, or even attempting to approximate, the Bayesian posterior, even asymptotically. Rather, we interpret it as implicit hierarchical expansion. Further, the learned posterior uncertainty detects heterogeneity of parameters among the population, enabling automatic model diagnosis. This framework applies to both likelihood-exact and likelihood-free models. We demonstrate its application in real data examples.

💡 Research Summary

Predictive Variational Inference (PVI) is introduced as a principled alternative to traditional variational inference (VI) that focuses on approximating the Bayesian posterior. Instead of minimizing the Kullback–Leibler (KL) divergence between an approximate distribution (q(\theta)) and the exact posterior (p(\theta\mid y)), PVI directly optimizes the quality of the posterior predictive distribution with respect to the true data‑generating process. The authors formalize this by selecting a proper scoring rule (S) (e.g., log‑score, CRPS, Brier score, interval score) and defining a divergence (D) on the outcome space that corresponds to the chosen score. The objective becomes

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