High-dimensional Asymptotic Theory of Bayesian Multiple Testing Procedures Under General Dependent Setup and Possible Misspecification

Abstract: In this article, we investigate the asymptotic properties of Bayesian multiple testing procedures under general dependent setup, when the sample size and the number of hypotheses both tend to infinity. Specifically, we investigate strong consistency of the procedures and asymptotic properties of different versions of false discovery and false non-discovery rates under the high dimensional setup. We particularly focus on a novel Bayesian non-marginal multiple testing procedure and its associated error rates in this regard. Our results show that the asymptotic convergence rates of the error rates are directly associated with the Kullback-Leibler divergence from the true model, and the results hold even when the postulated class of models is misspecified. For illustration of our high-dimensional asymptotic theory, we consider a Bayesian variable selection problem in a time-varying covariate selection framework, with autoregressive response variables. We particularly focus on the setup where the number of hypotheses increases at a faster rate compared to the sample size, which is the so-called ultra-high dimensional situation.