Sharp Asymptotic Minimaxity for Multiple Testing Using One-Group Shrinkage Priors
Sayantan Paul, Prasenjit Ghosh, Arijit Chakrabarti
TL;DR
In the sparse Gaussian sequence model, the paper proves that one-group global-local shrinkage priors, and in particular horseshoe-type priors with $a=\tfrac{1}{2}$, yield sharp asymptotic minimax testing performance for both misclassification loss and the composite FDP+FNR loss. With known sparsity, suitably scaled global shrinkage $\tau$ (and $n\tau/q_n \to C$) ensures exact minimax risk; when sparsity is unknown, empirical Bayes and fully Bayesian adaptations preserve this optimality under mild growth conditions on $q_n$ and the prior on $\tau$. The results align the Bayesian one-group framework with the sharp minimax benchmarks established for BH and spike-and-slab approaches, and they demonstrate adaptivity to unknown sparsity. These findings justify the use of computationally tractable global-local priors in large-scale testing and motivate extensions to broader priors and dependence structures.
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.