文件名称：Frequentist Consistency of Variational Bayes
文件大小：756KB
文件格式：PDF
更新时间：2021-12-13 06:19:56
VB
A key challenge for modern Bayesian statistics is how to perform scalable inference of pos- terior 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 esti- mates based on variational approximations, called frequentist variational approximations, and we use the connection to prove a variational Bernstein–von Mises theorem. The theorem lever- ages 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.