A Unified View of False Discovery Rate Control: Reconciliation of Bayesian and Frequentist Approaches

Abstract

This paper explores the intrinsic connections between the Bayesian false discovery rate (FDR) control procedures and their counterpart of frequentist procedures. We attempt to offer a unified view of FDR control within and beyond the setting of testing exchangeable hypotheses. Under the standard two-groups model and the Oracle condition, we show that the Bayesian and the frequentist methods can achieve asymptotically equivalent FDR control at arbitrary levels. Built on this result, we further illustrate that rigorous post-fitting model diagnosis is necessary and effective to ensure robust FDR controls for parametric Bayesian approaches. Additionally, we show that the Bayesian FDR control approaches are coherent and naturally extended to the setting beyond testing exchangeable hypotheses. Particularly, we illustrate that $p$-values are no longer the natural statistical instruments for optimal frequentist FDR control in testing non-exchangeable hypotheses. Finally, we illustrate that simple numerical recipes motivated by our theoretical results can be effective in examining some key model assumptions commonly assumed in both Bayesian and frequentist procedures (e.g., zero assumption).