Finite sample Bernstein-von Mises theorems for functionals and spectral projectors of the covariance matrix

TL;DR

This paper proves that the influence of prior distributions on the posterior of covariance matrices diminishes with larger samples, providing finite sample Bernstein-von Mises results for functionals and spectral projectors, even as dimension grows.

Contribution

It establishes finite sample Bernstein-von Mises theorems for covariance matrix functionals and spectral projectors with explicit mild prior assumptions and growing dimension.

Findings

Prior influence vanishes as sample size increases.

Finite sample Bernstein-von Mises theorem holds for eigenvalues.

Posterior distribution of spectral projector's Frobenius distance derived.

Abstract

We demonstrate that a prior influence on the posterior distribution of covariance matrix vanishes as sample size grows. The assumptions on a prior are explicit and mild. The results are valid for a finite sample and admit the dimension $p$ growing with the sample size $n$. We exploit the described fact to derive the finite sample Bernstein-von Mises theorem for functionals of covariance matrix (e.g. eigenvalues) and to find the posterior distribution of the Frobenius distance between spectral projector and empirical spectral projector. This can be useful for constructing sharp confidence sets for the true value of the functional or for the true spectral projector.