Diameter and Volume Minimizing Confidence Sets in Bayes and Classical Problems

Abstract

If $X \sim P_\theta, \theta \in \Omega$ and $\theta \sim G \ll \mu$, where $dG/d\mu$ belongs to the convex family $\Gamma_{L, U} = \{g: L \leq \operatorname{cg} \leq U$, for some $c > 0\}$, then the sets minimizing $\lambda(S)$ subject to $\inf_{G \in \Gamma_{L,U}} P_G(S\mid X) \geq p$ are derived, where $P_G(S\mid X)$ is the posterior probability of $S$ under the prior $G$, and $\lambda$ is any nonnegative measure on $\Omega$ such that $\mu \ll \lambda \ll \mu$. Applications are shown to several multiparameter problems and connectedness (or disconnectedness) of these sets is considered. The problem of minimizing the diameter is also considered in a general probabilistic framework. It is proved that if $\mathscr{X}$ is any finite-dimensional Banach space with a convex norm, and $\{P_\alpha\}$ is a tight family of probability measures on the Borel $\sigma$-algebra of $\mathscr{X}$, then there always exists a closed connected set minimizing the diameter under the restriction $\inf_\alpha P_\alpha(S) \geq p$. It is also proved that if $P$ is a spherical unimodal measure on $\mathbb{R}^m$, then volume (Lebesgue measure) and diameter minimizing sets are the same. A result of Borell is then used to conclude that diameter minimizing sets are spheres whenever the underlying distribution $P$ is symmetric absolutely continuous and the density $f$ is such that $f^{-1/m}$ is convex. All standard symmetric multivariate densities satisfy this condition. Applications are made to several Bayes and classical problems and admissibility implications of these results are discussed.