  A fast Bayesian method that seamlessly fuses classification and hypothesis
testing via discriminant analysis is developed. Building upon the original
discriminant analysis classifier, modelling components are added to identify
discriminative variables. A combination of cake priors and a novel form of
variational Bayes we call reverse collapsed variational Bayes gives rise to
variable selection that can be directly posed as a multiple hypothesis testing
approach using likelihood ratio statistics. Some theoretical arguments are
presented showing that Chernoff-consistency (asymptotically zero type I and
type II error) is maintained across all hypotheses. We apply our method on some
publicly available genomics datasets and show that our method performs well in
practice for its computational cost. An R package VaDA has also been made
available on Github.
