Asymptotically minimax Bayes predictive densities

TL;DR

This paper investigates the asymptotic minimax properties of Bayesian predictive densities under Kullback-Leibler loss, identifying optimal priors and revealing dimension-dependent admissibility and minimaxity results.

Contribution

It characterizes asymptotically least favorable priors for Bayesian predictive densities and extends Stein's paradox to the multivariate setting with new minimax and admissibility findings.

Findings

Jeffreys prior is minimax in dimensions ≥3

Jeffreys prior is admissible and minimax in 1- and 2-dimensional cases

Identifies asymptotically least favorable predictive densities

Abstract

Given a random sample from a distribution with density function that depends on an unknown parameter $\theta$, we are interested in accurately estimating the true parametric density function at a future observation from the same distribution. The asymptotic risk of Bayes predictive density estimates with Kullback--Leibler loss function $D(f_{\theta}||{\hat{f}})=\int{f_{\theta} \log{(f_{\theta}/ hat{f})}}$ is used to examine various ways of choosing prior distributions; the principal type of choice studied is minimax. We seek asymptotically least favorable predictive densities for which the corresponding asymptotic risk is minimax. A result resembling Stein's paradox for estimating normal means by the maximum likelihood holds for the uniform prior in the multivariate location family case: when the dimensionality of the model is at least three, the Jeffreys prior is minimax, though inadmissible. The Jeffreys prior is both admissible and minimax for one- and two-dimensional location problems.