Bayesian Frequentist Multiple Testing

We introduce a Bayesian approach to multiple testing. The method is an extension of the false discovery rate (FDR) method due to Benjamini and Hochberg (1995). We also examine the empirical Bayes approach to simultaneous inference proposed by Efron, Tibshirani, Storey and Tusher (2001). We show that, in contrast to the single hypothesis case - where Bayes and frequentist tests do not agree even asymptotically - in the multiple testing case we do have asymptotic agreement.