Frequentist Consistency of Variational Bayes

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 article, 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. Supplementary materials for this article are available online.

KEYWORDS:

• Asymptotic normality
• Bayesian inference
• Bernstein–von Mises
• Consistency
• Statistical computing
• Variational methods

Supplementary Materials

The online supplementary materials contain the appendices for the article.

Acknowledgments

The authors thank the associate editor and two anonymous reviewers for their constructive comments. The authors thank Adji Dieng, Christian Naesseth, and Dustin Tran for their valuable feedback on their manuscript. The authors also thank Richard Nickl for pointing them to a key reference.

Additional information

Funding

This work is supported by ONR N00014-11-1-0651, DARPA PPAML FA8750-14-2-0009, the Alfred P. Sloan Foundation, and the John Simon Guggenheim Foundation.