High-dimensional Bayesian Regression: Asymptotics via the Naive Mean-field Approximation
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 (Columbia University).