High-dimensional Bayesian regression: Asymptotics via the naive mean-field approximation

Tue March 16th 2021, 4:30pm
Subhabrata Sen, Harvard University

Abstract:   Variational approximations provide an attractive computational alternative to MCMC-based strategies for approximating the posterior distribution in Bayesian inference. Despite their popularity in applications, supporting theoretical guarantees are limited, particularly in high-dimensional settings. We study Bayesian inference in the context of a linear model with product priors, and derive sufficient conditions for the correctness (to leading order) of the naive mean-field approximation. To this end, we utilize recent advances in the theory of non-linear large deviations (Chatterjee and Dembo, 2014). Next, we analyze the naive mean-field variational problem, and precisely characterize the asymptotic properties of the posterior distribution in this setting.

This is based on joint work with Sumit Mukherjee of Columbia University.

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