Post-Processed Posteriors for Banded Covariances

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. Additionally, we provide a theorem on the credible set of the post-processed posterior under the finite dimension assumption. We prove that the expected coverage probability of the $100(1-\mathit{\alpha })\mathrm{\%}$ highest posterior density region of the post-processed posterior is asymptotically $1-\mathit{\alpha }$ with respect to any conventional posterior distribution. It implies that the highest posterior density region of the post-processed posterior is, on average, a credible set of conventional posterior. The advantages of the post-processed posterior are demonstrated by a simulation study and a real data analysis.