Enriched standard conjugate priors and the right invariant prior for Wishart distributions

TL;DR

This paper explores Bayesian predictive distributions for Wishart distributions, analyzing the impact of enriched conjugate priors and the right invariant prior, providing new insights into multivariate analysis with ordered variables.

Contribution

It introduces and compares enriched standard conjugate priors and the right invariant prior for Wishart distributions, revealing dominance relationships and risk decomposition.

Findings

Identified a prior in the family that dominates the reference prior.

Decomposed the risk of Bayesian predictive distributions using conditional reducibility.

Provided new insights into multivariate analysis with ordered importance of variables.

Abstract

The prediction of the variance-covariance matrix of the multivariate normal distribution is important in the multivariate analysis. We investigated Bayesian predictive distributions for Wishart distributions under the Kullback-Leibler divergence. The conditional reducibility of the family of Wishart distributions enables us to decompose the risk of a Bayesian predictive distribution. We considered a recently introduced class of prior distributions, which is called the family of enriched standard conjugate prior distributions, and compared the Bayesian predictive distributions based on these prior distributions. Furthermore, we studied the performance of the Bayesian predictive distribution based on the reference prior distribution in the family and showed that there exists a prior distribution in the family that dominates the reference prior distribution. Our study provides new insight into the multivariate analysis when there exists an ordered inferential importance for the independent variables.