Open Access
2023 Bayesian Optimal Two-Sample Tests for High-Dimensional Gaussian Populations
Kyoungjae Lee, Kisung You, Lizhen Lin
Author Affiliations +
Bayesian Anal. Advance Publication 1-25 (2023). DOI: 10.1214/23-BA1373


We propose minimax 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 differences between two populations are rare 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 minimax 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.

Funding Statement

This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A4A1018207). LL would like to acknowledge the generous support of NSF grants DMS CAREER 1654579 and DMS 2113642.


We are very grateful to the Editor, Associate Editor and two reviewers for their valuable comments which have led to significant improvement in the earlier version of our paper.


Download Citation

Kyoungjae Lee. Kisung You. Lizhen Lin. "Bayesian Optimal Two-Sample Tests for High-Dimensional Gaussian Populations." Bayesian Anal. Advance Publication 1 - 25, 2023.


Published: 2023
First available in Project Euclid: 27 February 2023

Digital Object Identifier: 10.1214/23-BA1373

Primary: 62F03 , 62F15
Secondary: 62H15

Keywords: Bayes factor consistency , Bayesian hypothesis test , high-dimensional covariance matrix , optimal high-dimensional tests

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