Improved minimax predictive densities under Kullback--Leibler loss

Let $X\mathrm{\mid }\mu \sim N_p(\mu ,v_{x}I)$ and $Y\mathrm{\mid }\mu \sim N_p(\mu ,v_{y}I)$ be independent p-dimensional multivariate normal vectors with common unknown mean $\mu$. Based on only observing $X=x$, we consider the problem of obtaining a predictive density $\widehat{p}(y\mathrm{\mid }x)$ for $Y$ that is close to $p(y\mathrm{\mid }\mu )$ as measured by expected Kullback–Leibler loss. A natural procedure for this problem is the (formal) Bayes predictive density ${\widehat{p}}_{\mathrm{U}}(y\mathrm{\mid }x)$ under the uniform prior ${\pi }_{\mathrm{U}}(\mu )\equiv 1$, which is best invariant and minimax. We show that any Bayes predictive density will be minimax if it is obtained by a prior yielding a marginal that is superharmonic or whose square root is superharmonic. This yields wide classes of minimax procedures that dominate ${\widehat{p}}_{\mathrm{U}}(y\mathrm{\mid }x)$, including Bayes predictive densities under superharmonic priors. Fundamental similarities and differences with the parallel theory of estimating a multivariate normal mean under quadratic loss are described.