Bernstein-von Mises Theorems for Functionals of Covariance Matrix

TL;DR

This paper develops a comprehensive theoretical framework for Bernstein-von Mises theorems applicable to various functionals of covariance matrices, facilitating Bayesian inference in high-dimensional settings.

Contribution

It introduces explicit, easy-to-verify conditions for Bernstein-von Mises theorems for matrix functionals, covering covariance, precision matrices, and related statistical procedures.

Findings

The framework applies to entries of covariance and precision matrices.

Results include asymptotic normality for quadratic forms and eigenvalues.

Applicable to Bayesian Gaussian covariance/precision estimation and discriminant analysis.

Abstract

We provide a general theoretical framework to derive Bernstein-von Mises theorems for matrix functionals. The conditions on functionals and priors are explicit and easy to check. Results are obtained for various functionals including entries of covariance matrix, entries of precision matrix, quadratic forms, log-determinant, eigenvalues in the Bayesian Gaussian covariance/precision matrix estimation setting, as well as for Bayesian linear and quadratic discriminant analysis.