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

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.