Maximum pairwise bayes factors for covariance structure testing

Hypothesis testing of structure in covariance matrices is of sig-nificant importance, but faces great challenges in high-dimensional settings. Although consistent frequentist one-sample covariance tests have been pro-posed, 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 al-ternatives, 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 optimally distinguish null and alternative hypothe-ses in a frequentist asymptotic sense. We then propose diagonal tests and a scalable covariance graph selection procedure that are shown to be con-sistent. A simulation study evaluates the proposed approach relative to competitors. We illustrate advantages of our graph selection method on a gene expression data set.