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

Predictive densities for multivariate normal models based on extended models and shrinkage Bayes methods