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

We investigate Bayesian predictions for Wishart distributions by using the Kullback-Leibler divergence. We compare between the Bayesian predictive distributions based on a recently introduced class of prior distributions, called the family of enriched standard conjugate priors, which includes the Jeffreys prior, the reference prior, and the right invariant prior. We explicitly calculate the risks of Bayesian predictive distributions without using asymptotic expansions and clarify the dependency on the sizes of current and future observations. We also construct a minimax predictive distribution with a constant risk and prove this predictive distribution is not admissible.