Non-marginal Decisions: New Bayesian Multiple Testing Procedures

In the multiple testing literature, either Bayesian or non-Bayesian, the decision rules are usually functions of the marginal probabilities of the corresponding individual hypotheses. In the Bayesian paradigm,when the hypotheses are dependent, then it is desirable that the decision on each hypothesis depends on decisions on every other hypotheses through the joint posterior probabilities of all the hypotheses. In this paper we develop novel procedures that coherently take this requirement into consideration. Our multiple testing procedures are based on new notions of error and non-error terms associated with breaking up the total number of hypotheses. With respect to the new notions, we maximize the posteror expectation of the number of true positives, while controlling the posterior expectations of several error terms. Although the resulting function to be maximized is additive in nature, it is a non-linear function of the decisions and so does not compromise in any way with our requirement of non-marginalized decisions. We propose two different criteria; in one criterion each term involves decisions on all the hypotheses, and in another, the terms involve decisions on only those hypotheses that have ranks less than the indices of the respective terms in the summation, with respect to some ranking procedure, such as Bayes factors. The optimal decisions are not available in closed forms and we resort to block-relaxation algorithms, maximizing sequentially with respect to a decision, holding the remaining decisions fixed. We draw connection of our block relaxation algorithms with specialized Gibbs samplers, and obtain, using absorbing Markov chain theory, the expected number of iterations required for our algorithms to converge.