A Powerful Bayesian Test for Equality of Means in High Dimensions

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 <i>p</i> × <i>p</i> covariance matrix, whereas Bayes factors based on Jeffrey’s prior suffer the same impediment as the classical Hotelling <i>T</i>² 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.

本文针对高维场景下的两总体均值比较问题，构建了基于贝叶斯因子（Bayes factor）的检验流程。在“大p小n（large-p-small-n）”场景中，基于恰当先验的贝叶斯因子需要设定一个规模庞大且结构复杂的p×p阶协方差矩阵；而基于杰弗里先验（Jeffrey's prior）的贝叶斯因子则与经典霍特林T²（Hotelling T²）检验统计量面临相同的困境，即均需对病态样本协方差矩阵求逆。为规避这一局限，本文提出将贝叶斯因子构建于高维数据向量的低维随机投影之上。我们在备择假设框架下选取先验分布，以在固定阈值水平下最大化检验功效，由此得到受限最优势贝叶斯检验（Restricted Most Powerful Bayesian Test, RMPBT）。最终的检验统计量基于多组随机投影数据所对应的贝叶斯因子集成。本文证明该检验具备无偏性，且在温和正则条件下同时具有局部相合性。我们通过模拟数据集与真实数据集示例验证了该方法的有效性。本文的补充材料可在线获取。