Post-Processed Posteriors for Banded Covariances

TL;DR

This paper introduces a computationally efficient post-processing method for Bayesian inference of banded covariance matrices, achieving near-optimal rates and reliable credible intervals, validated through simulations and real data.

Contribution

It proposes a novel two-step post-processing approach for covariance matrix inference that ensures structural constraints and maintains statistical optimality.

Findings

Nearly optimal minimax rates for banded covariances.

Asymptotic coverage probability of 1-lpha for credible sets.

Validated effectiveness through simulations and real data analysis.

Abstract

We consider Bayesian inference of banded covariance matrices and propose a post-processed posterior. The post-processing of the posterior consists of two steps. In the first step, posterior samples are obtained from the conjugate inverse-Wishart posterior which does not satisfy any structural restrictions. In the second step, the posterior samples are transformed to satisfy the structural restriction through a post-processing function. The conceptually straightforward procedure of the post-processed posterior makes its computation efficient and can render interval estimators of functionals of covariance matrices. We show that it has nearly optimal minimax rates for banded covariances among all possible pairs of priors and post-processing functions. Furthermore, we prove that the expected coverage probability of the $(1-\alpha)100\%$ highest posterior density region of the post-processed posterior is asymptotically $1-\alpha$ with respect to a conventional posterior distribution. It implies that the highest posterior density region of the post-processed posterior is, on average, a credible set of a conventional posterior. The advantages of the post-processed posterior are demonstrated by a simulation study and a real data analysis.