Bayesian perspectives on multiple comparisons

We discuss Bayesian attitudes towards adjusting inferences for multiplicities. In the simplest Bayesian view, there is no need for adjustments and the Bayesian perspective is similar to that of the frequentist who makes inferences on a per-comparison basis. However, as we explain, Bayesian thinking can lead to making adjustments that are in the same spirit as those made by frequentists who subscribe to preserving the familywise error rate. We describe the differences between assuming independent prior distributions and hierarchical prior distributions. As an example of the latter, we illustrate the use of a Dirichlet process prior distribution in the context of multiplicities. We also discuss some quasi-Bayesian procedures which combine Bayesian and frequentist ideas. This shows the potential of Bayesian methodology to yield procedures that can be evaluated using "objective" criteria. Finally, we comment on the role of subjectivity in Bayesian approaches to the complex realm of multiple comparisons problems, and on robust vs. informative priors, (C) 1999 Elsevier Science B.V. All rights reserved. MSG: 62J15; 62A15; 62F15.