High-Dimensional Multivariate Posterior Consistency Under Global-Local Shrinkage Priors

TL;DR

This paper extends scale mixture of normal shrinkage priors to multivariate linear regression, establishing posterior consistency in high-dimensional settings and providing an efficient Gibbs sampler with practical implementation.

Contribution

It introduces a multivariate Bayesian shrinkage method with proven posterior consistency for high-dimensional data, along with a new Gibbs sampling algorithm and R package.

Findings

Achieves posterior consistency even when p > n.

Demonstrates excellent finite sample performance.

Provides an efficient Gibbs sampling algorithm.

Abstract

We consider sparse Bayesian estimation in the classical multivariate linear regression model with $p$ regressors and $q$ response variables. In univariate Bayesian linear regression with a single response $y$, shrinkage priors which can be expressed as scale mixtures of normal densities are popular for obtaining sparse estimates of the coefficients. In this paper, we extend the use of these priors to the multivariate case to estimate a $p \times q$ coefficients matrix $\mathbf{B}$. We derive sufficient conditions for posterior consistency under the Bayesian multivariate linear regression framework and prove that our method achieves posterior consistency even when $p>n$ and even when $p$ grows at nearly exponential rate with the sample size. We derive an efficient Gibbs sampling algorithm and provide the implementation in a comprehensive R package called MBSP. Finally, we demonstrate through simulations and data analysis that our model has excellent finite sample performance.