Improved Minimax Prediction Under Kullback-Leibler Loss

Let X | μ ∼ Np(μ, vxI) and Y | μ ∼ Np(μ, vyI) be independent p-dimensional multivariate normal vectors with common unknown mean μ, and let p(x|μ) and p(y |μ) denote the conditional densities of X and Y . Based on only observing X = x, we consider the problem of obtaining a predictive distribution p̂(y |x) for Y that is close to p(y |μ) as measured by Kullback-Leibler loss. The natural straw man for this problem is the best invariant predictive distribution, the Bayes rule pU (y | x) under the uniform prior πU (μ) ≡ 1, which is seen to be minimax. We show that pU (y |x) is dominated by any Bayes rules for which the square root of the marginal distribution is superharmonic. This yields wide classes of dominating predictive distributions including Bayes rules under superharmonic priors. These dominating predictive shrinkage distributions can be constructed to adaptively shrink pU (y | x) towards arbitrary points or subspaces. Those procedures corresponding to superharmonic priors can be further combined to obtain minimax multiple shrinkage predictive distributions that adaptively shrink pU (y |x) towards an arbitrary number of points or subspaces. Fundamental similarities and differences with the parallel theory of estimating a multivariate normal mean under quadratic loss are described throughout.