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The prediction of the variance-covariance matrix of the multivariate normaldistribution is important in the multivariate analysis. We investigatedBayesian predictive distributions for Wishart distributions under theKullback-Leibler divergence. The conditional reducibility of the family ofWishart distributions enables us to decompose the risk of a Bayesian predictivedistribution. 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 priordistributions. Furthermore, we studied the performance of the Bayesianpredictive distribution based on the reference prior distribution in the familyand showed that there exists a prior distribution in the family that dominatesthe reference prior distribution. Our study provides new insight into themultivariate analysis when there exists an ordered inferential importance forthe independent variables.

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