Minimaxity under half-Cauchy type priors

TL;DR

This paper extends the understanding of half-Cauchy priors in hierarchical models, providing theoretical conditions for minimaxity of associated estimators and developing algorithms for practical implementation.

Contribution

It establishes sufficient conditions for minimaxity of generalized Bayes estimators with U-shaped priors, broadening the theoretical foundation beyond half-Cauchy priors.

Findings

Half-Cauchy prior exhibits minimaxity under certain conditions.

Developed an efficient posterior sampling algorithm.

Numerical results support theoretical findings.

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 normal mean vector under the quadratic loss, they showed that the Bayes estimator with respect to the half-Cauchy prior seems to be minimax through numerical experiments. In terms of the shrinkage coefficient, the half-Cauchy prior has a U-shape and can be interpreted as a continuous spike and slab prior. In this paper, we consider a general class of priors with U-shapes and theoretically establish sufficient conditions for the minimaxity of the corresponding (generalized) Bayes estimators. We also develop an algorithm for posterior sampling and present numerical results.