Bayesian MAP multiple testing procedures

We consider a Bayesian approach to multiple hypothesis testing. A hierarchical prior model is based on imposing a prior distribution $\pi(k)$ on the number of hypotheses arising from alternatives (false nulls). We then apply the maximum a posteriori (MAP) rule to find the most likely configuration of null and alternative hypotheses. The resulting MAP procedure and its closely related step-up and step-down versions compare ordered Bayes factors of individual hypotheses with a sequence of critical values depending on the prior. We discuss the relations between the proposed MAP procedure and the existing frequentist and Bayesian counterparts. A more detailed analysis is given for the normal data, where we show, in particular, that choosing a specific $\pi(k)$, the MAP procedure can mimic several known familywise error (FWE) and false discovery rate (FDR) controlling procedures. The performance of MAP procedures is illustrated on a simulated example.