  Hypothesis testing of structure in correlation and covariance matrices is of
broad interest in many application areas. In high dimensions and/or small to
moderate sample sizes, high error rates in testing is a substantial concern.
This article focuses on increasing power through a frequentist assisted by
Bayes (FAB) procedure. This FAB approach boosts power by including prior
information on the correlation parameters. In particular, we suppose there is
one of two sources of prior information: (i) a prior dataset that is distinct
from the current data but related enough that it may contain valuable
information about the correlation structure in the current data; and (ii)
knowledge about a tendency for the correlations in different parameters to be
similar so that it is appropriate to consider a hierarchical model. When the
prior information is relevant, the proposed FAB approach can have significant
gains in power. A divide-and-conquer algorithm is developed to reduce
computational complexity in massive testing dimensions. We show improvements in
power for detecting correlated gene pairs in genomic studies while maintaining
control of Type I error or false discover rate (FDR).
