  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.
