Bayesian Nonparametric Approach to Multiple Testing

Motivated by the problems in genomics, astronomy and some other emerging fields, multiple hypothesis testing has come to the forefront of statistical research in the recent years. In the context of multiple testing, new error measures such as the false discovery rate (FDR) occupy important roles comparable to the role of type I error in classical hypothesis testing. Assuming that a random mechanism decides the truth of a hypothesis, substantial gain in power is possible by estimating error measures from the data. Nonparametric Bayesian approaches are proven to be particularly suitable for estimation of error measure in multiple testing situation. A Bayesian approach based on a nonparametric mixture model for p-values can utilize special features of the distribution of p-values that significantly improves the quality of estimation. In this paper we describe the nonparametric Bayesian modeling exercise of the distribution of the p-values. We begin with a brief review of Bayesian nonparametric concepts of Dirichlet process and Dirichlet mixtures and classical multiple hypothesis testing. We then review recently proposed nonparametric Bayesian methods for estimating errors based on a Dirichlet mixture of prior for the p-value density. When the test statistics are independent, a mixture of beta kernels can adequately model the p-value density, whereas in the dependent case one can consider a Dirichlet mixture of multivariate skew-normal kernel prior for probit transforms of