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

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.