Asymptotic Theory of Dependent Bayesian Multiple Testing Procedures Under Possible Model Misspecification

The effect of dependence among multiple hypothesis testing have recently attracted attention of the statistical community. Statisticians have studied the effect of dependence in multiple testing procedures under different setups. In this article, we study asymptotic properties of Bayesian multiple testing procedures. Specifically, we provide sufficient conditions for strong consistency under general dependence structure. We also consider a novel Bayesian non-marginal multiple testing procedure and associated error measures that coherently account for the dependence structure present in the model and the prior. We advocate posterior versions of FDR and FNR as appropriate error rates in Bayesian multiple testing and show that the asymptotic convergence rates of the error rates are directly associated with the Kullback-Leibler divergence from the true model. Indeed, all our results hold very generally even under the setup where the class of postulated models is misspecified. We illustrate our general asymptotic theory in a time-varying covariate selection problem with autoregressive response variables, viewed from a multiple testing perspective. We show that for any proper prior distribution on the parameters, consistency of certain Bayesian multiple testing procedures hold. We compare the Bayesian non-marginal procedure with some existing Bayesian multiple testing methods through an extensive simulation study in the above time-varying covariate selection problem. Superior performance of the new procedure compared to the others vindicate that proper exploitation of the dependence structure by the multiple testing methods is indeed important.