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.

翻译：贝叶斯预测推断通过后验预测分布将参数不确定性传播至目标量。实际应用中，通常采用两阶段流程：首先近似模型参数的后验分布，然后通过蒙特卡洛模拟将后验样本传播至预测模型。这种顺序计算流程对高保真模型（如偏微分方程控制的模型）而言计算成本高昂。我们提出一种直接针对后验预测分布的变分贝叶斯框架，联合学习后验分布及其对应预测分布的变分近似。该公式引入了基于矩正则化项的库尔贝克-莱布勒散度变分上界。变分分布以摊销方式训练，将计算负担转移至离线阶段，从而实现高效在线推断。从解析基准问题到有限元固体力学问题的数值实验表明，所提方法相比传统两阶段变分推断能获得更精确的预测分布，同时显著降低在线预测推断的计算成本。