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Activity Number: 231 - SBSS Student Paper Award Session II
Type: Topic Contributed
Date/Time: Monday, July 29, 2019 : 2:00 PM to 3:50 PM
Sponsor: Section on Bayesian Statistical Science
Abstract #301852
Title: Frequentist Consistency of Variational Bayes
Author(s): Yixin Wang* and David Blei
Companies: and Columbia University
Keywords: Bernstein–-von Mises

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.

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