  With the routine collection of massive-dimensional predictors in many
application areas, screening methods that rapidly identify a small subset of
promising predictors have become commonplace. We propose a new MOdular Bayes
Screening (MOBS) approach, which involves several novel characteristics that
can potentially lead to improved performance. MOBS first applies a Bayesian
mixture model to the marginal distribution of the response, obtaining posterior
samples of mixture weights, cluster-specific parameters, and cluster
allocations for each subject. Hypothesis tests are then introduced,
corresponding to whether or not to include a given predictor, with posterior
probabilities for each hypothesis available analytically conditionally on
unknowns sampled in the first stage and tuning parameters controlling borrowing
of information across tests. By marginalizing over the first stage posterior
samples, we avoid under-estimation of uncertainty typical of two-stage methods.
We greatly simplify the model specification and reduce computational complexity
by using {\em modularization}. We provide basic theoretical support for this
approach, and illustrate excellent performance relative to competitors in
simulation studies and the ability to capture complex shifts beyond simple
differences in means. The method is illustrated with applications to genomics
by using a very high-dimensional cis-eQTL dataset with roughly 38 million SNPs.
