  Hypothesis testing of structure in covariance matrices is of significant
importance, but faces great challenges in high-dimensional settings. Although
consistent frequentist one-sample covariance tests have been proposed, there is
a lack of simple, computationally scalable, and theoretically sound Bayesian
testing methods for large covariance matrices. Motivated by this gap and by the
need for tests that are powerful against sparse alternatives, we propose a
novel testing framework based on the maximum pairwise Bayes factor. Our initial
focus is on one-sample covariance testing; the proposed test can {\it
optimally} distinguish null and alternative hypotheses in a frequentist
asymptotic sense. We then propose diagonal tests and a scalable covariance
graph selection procedure that are shown to be consistent. A simulation study
evaluates the proposed approach relative to competitors. We illustrate
advantages of our graph selection method on a gene expression data set.
