Non-Marginal Decisions: New Bayesian Multiple Testing Procedures

• Activity Number: 491
• Type: Contributed
• Date/Time: Wednesday, August 12, 2015 : 8:30 AM to 10:20 AM
• Sponsor: International Indian Statistical Association
• Author(s): Noirrit Kiran Chandra* and Sourabh Bhattacharya
• Companies: Indian Statistical Institute, Kolkata and Indian Statistical Institute, Kolkata
• Keywords: Multiple testing ; Non-marginal Decisions ; Simulated Annealing ; False Discovery Rate

Abstract

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 posterior 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. The optimal decisions are not available in closed forms and we resort to Simulated Annealing algorithm, maximizing with respect to the decision configurations.