  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.
