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

We investigate 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 consider a recently introduced class of prior distributions, called the family of enriched standard conjugate prior distributions, and compare the Bayesian predictive distributions based on these prior distributions. Furthermore, we study the performance of the Bayesian predictive distribution based on the reference prior distribution in the family, and show that there exists a prior distribution in the family that dominates the reference prior distribution.