  We investigate predictive densities for multivariate normal models with
unknown mean vectors and known covariance matrices. Bayesian predictive
densities based on shrinkage priors often have complex representations,
although they are effective in various problems. We consider extended normal
models with mean vectors and covariance matrices as parameters, and adopt
predictive densities that belong to the extended models including the original
normal model. We adopt predictive densities that are optimal with respect to
the posterior Bayes risk in the extended models. The proposed predictive
density based on a superharmonic shrinkage prior is shown to dominate the
Bayesian predictive density based on the uniform prior under a loss function
based on the Kullback-Leibler divergence. Our method provides an alternative to
the empirical Bayes method, which is widely used to construct tractable
predictive densities.
