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

Abstract

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 when the variance of future samples is sufficiently large. Our method provides an alternative to the empirical Bayes method, which is widely used to construct tractable predictive densities.