Proper Bayes and Minimax Predictive Densities for a Matrix-variate Normal Distribution

TL;DR

This paper develops Bayesian predictive densities for matrix-variate normal distributions, demonstrating that hierarchical priors can produce admissible and minimax estimators with strong theoretical and numerical properties.

Contribution

It introduces hierarchical priors for predictive densities that are proven to be admissible and minimax, advancing Bayesian methods for matrix-variate normal models.

Findings

Hierarchical priors yield admissible and minimax predictive densities.

Superharmonicity of priors improves numerical performance.

Bayesian predictive densities outperform traditional methods.

Abstract

This paper deals with the problem of estimating predictive densities of a matrix-variate normal distribution with known covariance matrix. Our main aim is to establish some Bayesian predictive densities related to matricial shrinkage estimators of the normal mean matrix. The Kullback-Leibler loss is used for evaluating decision-theoretical optimality of predictive densities. It is shown that a proper hierarchical prior yields an admissible and minimax predictive density. Also, superharmonicity of prior densities is paid attention to for finding out a minimax predictive density with good numerical performance.