Sharp Asymptotic Minimaxity for Multiple Testing Using One-Group Shrinkage Priors

TL;DR

This paper demonstrates that certain global-local shrinkage priors, including the horseshoe, achieve asymptotic minimaxity in Bayesian multiple testing for sparse Gaussian models, under both known and unknown sparsity levels.

Contribution

It establishes the first asymptotic minimaxity results for multiple testing using broad classes of global-local shrinkage priors, including empirical and fully Bayesian methods.

Findings

Horseshoe and similar priors achieve asymptotic minimaxity.

Minimaxity depends on the choice of the global shrinkage parameter.

Results apply under both known and unknown sparsity levels.

Abstract

This paper investigates asymptotic minimaxity properties of Bayesian multiple testing rules in the sparse Gaussian sequence model using a broad class of global-local scale mixtures of normals as priors for the means. Minimaxity is studied under standard misclassification loss and the composite loss given by the sum of the false discovery proportion (FDP) and false non-discovery proportion (FNP). When the sparsity level is known, we show that by suitably choosing the global shrinkage parameter based on the sparsity level, our proposed testing rule achieves the exact minimax risk asymptotically for both losses under the ''beta-min'' separation condition. When the sparsity level is unknown, both empirical Bayes and fully Bayesian adaptations of the same rule are shown to achieve exact minimax risk asymptotically under suitable assumptions on sparsity. Our results reveal that minimaxity is attained for ''horseshoe-type'' priors that are broad enough to include the horseshoe, Strawderman-Berger, standard double Pareto, and certain inverse-gamma priors, among others. For non-''horseshoe-type'' priors, minimaxity fails to hold for either loss function. To the best of our knowledge, these are the first results of their kind for multiple hypothesis testing based on global-local shrinkage priors.