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In recent years, multiple hypothesis testing has come to the forefront ofstatistical research, ostensibly in relation to applications in genomics andsome other emerging fields. The false discovery rate (FDR) and its variantsprovide very important notions of errors in this context comparable to the roleof error probabilities in classical testing problems. Accurate estimation ofpositive FDR (pFDR), a variant of the FDR, is essential in assessing andcontrolling this measure. In a recent paper, the authors proposed a model-basednonparametric Bayesian method of estimation of the pFDR function. Inparticular, the density of p-values was modeled as a mixture of decreasing betadensities and an appropriate Dirichlet process was considered as a prior on themixing measure. The resulting procedure was shown to work well in simulations.In this paper, we provide some theoretical results in support of the betamixture model for the density of p-values, and show that, under appropriateconditions, the resulting posterior is consistent as the number of hypothesesgrows to infinity.

Posterior consistency of Dirichlet mixtures of beta densities in estimating positive false discovery rates