Minimaxity under the half-Cauchy prior

TL;DR

This paper theoretically proves that the Bayes estimator with a half-Cauchy prior for a multivariate normal mean is minimax, confirming previous numerical suggestions and enhancing understanding of hierarchical model priors.

Contribution

It provides a rigorous theoretical proof of the minimaxity of the half-Cauchy prior-based estimator, advancing the statistical theory of hierarchical models.

Findings

The Bayes estimator with half-Cauchy prior is minimax for multivariate normal mean estimation.

Theoretical validation supports previous numerical evidence of minimaxity.

The proof uses interval arithmetic to establish minimaxity.

Abstract

This is a follow-up paper of Polson and Scott (2012, Bayesian Analysis), which claimed that the half-Cauchy prior is a sensible default prior for a scale parameter in hierarchical models. For estimation of a p-variate normal mean under the quadratic loss, they demonstrated that the Bayes estimator with respect to the half-Cauchy prior seems to be minimax through numerical experiments. In this paper, we theoretically establish the minimaxity of the corresponding Bayes estimator using the interval arithmetric.