Exact Minimax Predictive Density Estimation and MDL

The problems of predictive density estimation with Kullback-Leibler loss, optimal universal data compression for MDL model selection, and the choice of priors for Bayes factors in model selection are interrelated. Research in recent years has identified procedures which are minimax for risk in predictive density estimation and for redundancy in universal data compression. Here, after reviewing some of the general story, we focus on the case of location families. The exact minimax procedures use an improper uniform prior on the location parameter. We illustrate use of the minimax optimal procedures with data previously used in a study of robustness of location estimates. Plus we discuss applications of minimax MDL criteria to variable selection problems in regression.