High-dimensional Bernstein Von-Mises theorems for covariance and precision matrices

Music City Center 

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 entire covariance/precision matrices under specific growth restrictions on $p_n$ and other mild assumptions. In particular, the limiting distribution is 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 that induce precision matrix sparsity. The posterior contraction rates and asymptotic normality of the corresponding posterior distribution are established under mild assumptions on the prior and true data-generating mechanism.

High-dimensional covariance estimation

Bernstein–von Mises theorem