  In this article, we consider the problem of simultaneous testing of
hypotheses when the individual test statistics are not necessarily independent.
Specifically, we consider the problem of simultaneous testing of point null
hypotheses against two-sided alternatives for the mean parameters of normally
distributed random variables. We assume that conditionally given the vector of
means, these random variables jointly follow a multivariate normal distribution
with a known but arbitrary covariance matrix. We consider a Bayesian framework
where each unknown mean parameter is modeled through a two-component "spike and
slab" mixture prior. This way, unconditionally the test statistics jointly have
a mixture of multivariate normal distributions. A new testing procedure is
developed that uses the dependence among the test statistics and works in a
"step-down" manner. This procedure is general enough to be applied for
non-normal data. A decision theoretic justification in favor of the proposed
testing procedure has been provided by showing that unlike many traditional
p-value based stepwise procedures, this new method possesses a certain
"convexity property" which makes it admissible with respect to a vector risk
function that captures the risks for the individual testing problems. An
alternative representation of the proposed test statistics has also been
established resulting in great simplification in the computational complexity.
It is demonstrated through extensive simulations that for various forms of
dependence and a wide range of sparsity levels, the proposed testing procedure
compares quite favorably with several existing multiple testing procedures
available in the literature in terms of overall misclassification probability.
