On the role of the prior in multiplicity adjustment. (2nd April 2016) - Li, Dandan
ABSTRACT: Multiplicity adjustment in Bayesian analysis is achieved through the use of a prior distribution, for the probability that a variable is in the (unknown) model in the context of model selection, or for the probability that a null hypothesis is true in the context of multiple testing. However, it is not obvious how the prior distribution brings about multiplicity adjustment. In 2010 Scott and Berger stated there is an "air of paradox" in how multiplicity adjustment is achieved in the fully Bayesian approach. They gave useful insight into the role of the prior distribution in multiplicity adjustment by using the prior odds ratio ( POR ), the ratio of prior probabilities of a smaller model to a larger model, and used a uniform distribution for the prior in their illustration. In this article, we identify certain characteristics of POR based on the uniform prior that help explain the role of the prior in multiplicity adjustment, and provide generalizations of these properties to more general priors. We use these results to develop a summary measure to quantify the degree of multiplicity adjustment inherent in a prior distribution, and discuss the relative roles of the hyperparameters in a Beta prior or mixtures of them. We use simulation results to illustrate these findings.