ABSTRACT We develop a Bayes factor-based testing procedure for comparing two population means in high-dimensional settings. In ‘large-p-small-n” settings, Bayes factors based on proper priors require eliciting a large and complex p × p covariance matrix, whereas Bayes factors based on Jeffrey’s prior suffer the same impediment as the classical Hotelling T2 test statistic as they involve inversion of ill-formed sample covariance matrices. To circumvent this limitation, we propose that the Bayes factor be based on lower dimensional random projections of the high-dimensional data vectors. We choose the prior under the alternative to maximize the power of the test for a fixed threshold level, yielding a restricted most powerful Bayesian test (RMPBT). The final test statistic is based on the ensemble of Bayes factors corresponding to multiple replications of randomly projected data. We show that the test is unbiased and, under mild conditions, is also locally consistent. We demonstrate the efficacy of the approach through simulated and real data examples. Supplementary materials for this article are available online.

中文翻译：

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高维均值相等的强大贝叶斯检验


摘要 我们开发了一种基于贝叶斯因子的测试程序，用于比较高维设置中的两个总体平均值。在“大-p-小-n”设置中，基于正确先验的贝叶斯因子需要引出一个大而复杂的 p × p 协方差矩阵，而基于 Jeffrey 的先验的贝叶斯因子与经典的 Hotelling T2 检验统计量遭受相同的障碍，因为它们涉及格式错误的样本协方差矩阵的求逆。为了规避这一限制，我们建议贝叶斯因子基于高维数据向量的低维随机投影。我们在替代方案中选择先验，以最大化固定阈值水平的测试功效，从而产生受限制的最强大贝叶斯测试（RMPBT）。最终的检验统计量基于与随机投影数据的多次重复相对应的贝叶斯因子的集合。我们证明该测试是公正的，并且在温和条件下也具有局部一致性。我们通过模拟和真实数据示例证明了该方法的有效性。本文的补充材料可在线获取。