A Powerful Bayesian Test for Equality of Means in High Dimensions
针对高维数据中比较两个总体均值的难题，提出基于随机投影和贝叶斯因子的检验方法，通过最大化检验功效得到受限最有效贝叶斯检验，适用于大p小n场景。
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.