High-Dimensional Bernstein Von-Mises Theorems for Covariance and Precision Matrices

TL;DR

This paper proves that in high-dimensional settings, the posterior distributions of covariance and precision matrices become asymptotically normal, facilitating Bayesian inference for large-scale multivariate data.

Contribution

It establishes Bernstein-von Mises theorems for covariance and precision matrices in high-dimensional regimes, including sparse Gaussian graphical models.

Findings

Posterior distribution of covariance matrices converges to a matrix normal distribution.

Posterior distribution of precision matrices in Gaussian graphical models is asymptotically normal.

Results hold under mild growth conditions and broad prior classes.

Abstract

This paper aims to examine the characteristics of the posterior distribution of covariance/precision matrices in a "large $p$, large $n$" scenario, where $p$ represents the number of variables and $n$ is the sample size. Our analysis focuses on establishing asymptotic normality of the posterior distribution of the entire covariance/precision matrices under specific growth restrictions on $p_n$ and other mild assumptions. In particular, the limiting distribution turns out to be a symmetric matrix variate normal distribution whose parameters depend on the maximum likelihood estimate. Our results hold for a wide class of prior distributions which includes standard choices used by practitioners. Next, we consider Gaussian graphical models which induce sparsity in the precision matrix. Asymptotic normality of the corresponding posterior distribution is established under mild assumptions on the prior and true data-generating mechanism.