  This paper deals with the problem of simultaneously making many (M) binary
decisions based on one realization of a random data matrix X. M is typically
large and X will usually have M rows associated with each of the M decisions to
make, but for each row the data may be low dimensional. A Bayesian
decision-theoretic approach for this problem is implemented with the overall
loss function being a cost-weighted linear combination of Type I and Type II
loss functions. The class of loss functions considered allows for the use of
the false discovery rate (FDR), false nondiscovery rate (FNR), and missed
discovery rate (MDR) in assessing the decision. Through this Bayesian paradigm,
the Bayes multiple decision function (BMDF) is derived and an efficient
algorithm to obtain the optimal Bayes action is described. In contrast to many
works in the literature where the rows of the matrix X are assumed to be
stochastically independent, we allow in this paper a dependent data structure
with the associations obtained through a class of frailty-induced Archimedean
copulas. In particular, non-Gaussian dependent data structure, which is the
norm rather than the exception when dealing with failure-time data, can be
entertained. The numerical implementation of the determination of the Bayes
optimal action is facilitated through sequential Monte Carlo techniques. The
main theory developed could also be extended to the problem of multiple
hypotheses testing, multiple classification and prediction, and
high-dimensional variable selection. The proposed procedure is illustrated for
the simple versus simple and for the composite hypotheses setting via
simulation studies. The procedure is also applied to a subset of a real
microarray data set from a colon cancer study.
