Post-Processed Posteriors for Sparse Covariances and Its Application to Global Minimum Variance Portfolio

TL;DR

This paper introduces a novel Bayesian method for estimating sparse covariance matrices using post-processing of inverse-Wishart samples, achieving optimal rates and improving global minimum variance portfolio estimation.

Contribution

It proposes a post-processed posterior approach that enforces sparsity in covariance estimation and demonstrates optimal minimax rates for both covariance matrices and portfolios.

Findings

Post-processed posterior attains optimal minimax rates for sparse covariance estimation.

Method improves global minimum variance portfolio estimation under sparsity assumptions.

Simulation and real data analyses validate the effectiveness of the approach.

Abstract

We consider Bayesian inference of sparse covariance matrices and propose a post-processed posterior. This method consists of two steps. In the first step, posterior samples are obtained from the conjugate inverse-Wishart posterior without considering the sparse structural assumption. The posterior samples are transformed in the second step to satisfy the sparse structural assumption through the hard-thresholding function. This non-traditional Bayesian procedure is justified by showing that the post-processed posterior attains the optimal minimax rates. We also investigate the application of the post-processed posterior to the estimation of the global minimum variance portfolio. We show that the post-processed posterior for the global minimum variance portfolio also attains the optimal minimax rate under the sparse covariance assumption. The advantages of the post-processed posterior for the global minimum variance portfolio are demonstrated by a simulation study and a real data analysis with S&P 400 data.