  Over the last decade, large-scale multiple testing has found itself at the
forefront of modern data analysis. In many applications data are correlated, so
that the observed test statistic used for detecting a non-null case, or signal,
at each location in a dataset carries some information about the chances of a
true signal at other locations. Brown, Lazar, Datta, Jang, and McDowell (2014)
proposed in the neuroimaging context a Bayesian multiple testing model that
accounts for the dependence of each volume element on the behavior of its
neighbors through a conditional autoregressive (CAR) model. Here, we propose a
generalized CAR model that allows for inclusion of points with no neighbors at
all, something that is not possible under conventional CAR models. We consider
also neighborhoods based on criteria other than physical location, such as
genetic pathways in microarray determined from existing biological knowledge.
This generalization provides a unified framework for the simultaneous modeling
of dependent and independent cases, resulting in stronger Bayesian learning in
the posterior and increased precision in the estimates of interesting signals.
We justify the selected prior distribution and prove that the resulting
posterior distribution is proper. We illustrate the effectiveness and
applicability of our proposed model by using it to analyze both simulated and
real microarray data in which the genes exhibit dependence that is determined
by physical adjacency on a chromosome or predefined gene pathways.
