Asymptotic Theory of a Bayesian Non-Marginal Multiple Testing Procedure and Comparison With Existing Methods

Recently, Chandra and Bhattacharya (2017) proposed a novel and general Bayesian multiple comparison method such that the decision on any hypothesis depends upon the joint posterior probability of the hypotheses on which the current hypothesis is strongly dependent. Here we investigate the asymptotic properties of their methodology, establishing in particular rates of convergence to zero of several versions of Bayesian false discovery rate and Bayesian false non-discovery rate associated with the non-marginal approach, as the sample size tends to infinity. We also establish convergence properties of several other established multiple testing methods and their Bayesian and randomized modifications that we propose, each representing a class of methodologies, and show that the non-marginal method is least as good as the existing ones in that the associated versions of Bayesian false non-discovery rate converge to zero at rates at least as fast as the other methods, when versions of Bayesian false discovery rate are asymptotically controlled.