Bayesian Multiple Testing Under Sparsity for Polynomial-Tailed Distributions

TL;DR

This paper analyzes Bayesian multiple testing procedures for polynomial-tailed distributions, establishing their asymptotic optimality and highlighting the importance of choosing appropriate false discovery rate levels.

Contribution

It provides general asymptotic optimality results for multiple testing under polynomial-tailed distributions and evaluates existing procedures through simulations.

Findings

Bayesian procedures are asymptotically optimal under certain conditions.

Incorrect FDR levels can lead to high Bayes risk.

Benjamini-Hochberg procedure's performance depends on FDR level choice.

Abstract

This paper considers Bayesian multiple testing under sparsity for polynomial-tailed distributions satisfying a monotone likelihood ratio property. Included in this class of distributions are the Student's t, the Pareto, and many other distributions. We prove some general asymptotic optimality results under fixed and random thresholding. As examples of these general results, we establish the Bayesian asymptotic optimality of several multiple testing procedures in the literature for appropriately chosen false discovery rate levels. We also show by simulation that the Benjamini-Hochberg procedure with a false discovery rate level different from the asymptotically optimal one can lead to high Bayes risk.