  We propose optimal Bayesian two-sample tests for testing equality of
high-dimensional mean vectors and covariance matrices between two populations.
In many applications including genomics and medical imaging, it is natural to
assume that only a few entries of two mean vectors or covariance matrices are
different. Many existing tests that rely on aggregating the difference between
empirical means or covariance matrices are not optimal or yield low power under
such setups. Motivated by this, we develop Bayesian two-sample tests employing
a divide-and-conquer idea, which is powerful especially when the difference
between two populations is sparse but large. The proposed two-sample tests
manifest closed forms of Bayes factors and allow scalable computations even in
high-dimensions. We prove that the proposed tests are consistent under
relatively mild conditions compared to existing tests in the literature.
Furthermore, the testable regions from the proposed tests turn out to be
optimal in terms of rates. Simulation studies show clear advantages of the
proposed tests over other state-of-the-art methods in various scenarios. Our
tests are also applied to the analysis of the gene expression data of two
cancer data sets.
