  In this paper we consider the problem of multiple testing when the hypotheses
are dependent. In most of the existing literature, either Bayesian or
non-Bayesian, the decision rules mainly focus on the validity of the test
procedure rather than actually utilizing the dependency to increase efficiency.
Moreover, the decisions regarding different hypotheses are marginal in the
sense that they do not depend upon each other directly. However, in realistic
situations, the hypotheses are usually dependent, and hence it is desirable
that the decisions regarding the dependent hypotheses are taken jointly.
  In this article we develop a novel Bayesian multiple testing procedure that
coherently takes this requirement into consideration. Our method, which is
based on new notions of error and non-error terms, substantially enhances
efficiency by judicious exploitation of the dependence structure among the
hypotheses. We prove that our method minimizes the posterior expected loss
associated with a an additive "0-1" loss function, we also prove theoretical
results on the relevant error probabilities, establishing the coherence and
usefulness of our method. The optimal decision configuration is not available
in closed form and we propose a novel and efficient simulated annealing
algorithm for the purpose of optimization, which is also generically applicable
to binary optimization problems.
  Numerical studies demonstrate that in dependent situations, our method
performs significantly better than some existing popular conventional multiple
testing methods, in terms of accuracy and power control. Moreover, application
of our ideas to a real, spatial data set associated with radionuclide
concentration in Rongelap islands yielded insightful results.
