A key challenge for modern Bayesian statistics is how to perform scalable inference of posterior distributions. To address this challenge, VB methods have emerged as a popular alternative to the classical MCMC methods. VB methods tend to be faster while achieving comparable predictive performance. However, there are few theoretical results around VB. In this paper, we establish frequentist consistency and asymptotic normality of VB methods. Specifically, we connect VB methods to point estimates based on variational approximations, called frequentist variational approximations, and we use the connection to prove a variational Bernstein-von-Mises theorem. The theorem leverages the theoretical characterizations of frequentist variational approximations to understand asymptotic properties of VB. In summary, we prove that (1) the VB posterior converges to the KL minimizer of a normal distribution, centered at the truth and (2) the corresponding variational expectation of the parameter is consistent and asymptotically normal. As applications of the theorem, we derive asymptotic properties of VB posteriors in Bayesian mixture models, Bayesian generalized linear mixed models, and Bayesian stochastic block models. We conduct a simulation study to illustrate these theoretical results.

本文证明了变分贝叶斯法在频率学意义下是稳健的，它可以通过极小化KL散度来估计后验分布，并且其对应的参数的变分期望是一致的和渐近正态的。此理论应用于贝叶斯混合模型、Bayesian广义线性混合模型和贝叶斯随机块模型，并通过模拟研究进行了验证。

变分贝叶斯方法的频率派一致性