# Frequentist Consistency of Variational Bayes

## TL;DR

This paper establishes the theoretical foundations of variational Bayes (VB), proving its consistency and asymptotic normality, and connects VB to classical frequentist estimators, thus enhancing its credibility for scalable Bayesian inference.

## Contribution

We provide the first rigorous proof of frequentist consistency and asymptotic normality for VB methods, linking them to variational Bernstein-von Mises theorems and applying results to various Bayesian models.

## Key findings

- VB posterior converges to the KL minimizer centered at the truth
- Variational expectation of parameters is consistent and asymptotically normal
- Theoretical results are supported by simulation studies

## Abstract

A key challenge for modern Bayesian statistics is how to perform scalable inference of posterior distributions. To address this challenge, variational Bayes (VB) methods have emerged as a popular alternative to the classical Markov chain Monte Carlo (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 Kullback-Leibler (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.