Amortized Variational Inference for Joint Posterior and Predictive Distributions in Bayesian Uncertainty Quantification

TL;DR

This paper introduces an amortized variational Bayesian method that jointly approximates the posterior and predictive distributions, enabling efficient and accurate uncertainty quantification for complex models.

Contribution

It presents a novel variational framework that directly targets the joint posterior-predictive distribution, reducing computational costs in Bayesian inference.

Findings

Achieves more accurate predictive distributions than traditional two-stage methods.

Substantially reduces online inference computational cost.

Demonstrated effectiveness on analytical and finite-element models.

Abstract

Bayesian predictive inference propagates parameter uncertainty to quantities of interest through the posterior-predictive distribution. In practice, this is typically performed using a two-stage procedure: first approximating the posterior distribution of model parameters, and then propagating posterior samples through the predictive model via Monte Carlo simulation. This sequential workflow can be computationally demanding, particularly for high-fidelity models such as those governed by partial differential equations. We propose a variational Bayesian framework that directly targets the posterior-predictive distribution and jointly learns variational approximations of both the posterior and the corresponding predictive distribution. The formulation introduces a variational upper bound on the Kullback--Leibler divergence together with moment-based regularization terms. The variational distributions are trained in an amortized manner, shifting computational effort to an offline stage and enabling efficient online inference. Numerical experiments ranging from analytical benchmarks to a finite-element solid mechanics problem demonstrate that the proposed method achieves more accurate predictive distributions than conventional two-stage variational inference, while substantially reducing the cost of online predictive inference.