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

2026-05-05

AI Summary

This work proposes a variational Bayesian framework that jointly models the posterior and predictive distributions, circumventing the conventional two-stage pipeline of posterior approximation followed by Monte Carlo propagation. By leveraging amortized variational inference, the method completes training offline, thereby eliminating the need for repeated online sampling and significantly reducing computational overhead during prediction. The approach introduces a variational upper bound on the KL divergence together with a moment-matching regularization term to enable efficient and accurate uncertainty quantification. Evaluated on both analytical benchmarks and finite-element-based solid mechanics problems, the proposed method achieves substantially higher predictive accuracy compared to traditional approaches while drastically lowering online computational costs, making it well-suited for high-fidelity models where conventional Bayesian inference is prohibitively expensive.

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.

Problem

• Bayesian uncertainty quantification
• posterior-predictive distribution
• amortized variational inference
• computational efficiency
• high-fidelity models

Innovation

• amortized variational inference
• posterior-predictive distribution
• Bayesian uncertainty quantification
• joint variational approximation
• moment-based regularization