Interfaces between Bayesian and Frequentist Multiplte Testing

This thesis investigates frequentist properties of Bayesian multiple testing procedures in a variety of scenarios and depicts the asymptotic behaviors of Bayesian methods. Both Bayesian and frequentist approaches to multiplicity control are studied and compared, with special focus on understanding the multiplicity control behavior in situations of dependence between test statistics.

Chapter 2 examines a problem of testing mutually exclusive hypotheses with dependent data. The Bayesian approach is shown to have excellent frequentist properties and is argued to be the most effective way of obtaining frequentist multiplicity control without sacrificing power. Chapter 3 further generalizes the model such that multiple signals are acceptable, and depicts the asymptotic behavior of false positives rates and the expected number of false positives. Chapter 4 considers the problem of dealing with a sequence of different trials concerning some medical or scientific issue, and discusses the possibilities for multiplicity control of the sequence. Chapter 5 addresses issues and efforts in reconciling frequentist and Bayesian approaches in sequential endpoint testing. We consider the conditional frequentist approach in sequential endpoint testing and show several examples in which Bayesian and frequentist methodologies cannot be made to match.