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Abstract:
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We develop a Bayes factor based testing procedure for comparing two population means in high dimensional setting. In `large-p-small-n' setting, traditional Bayes factor based procedures suffer the same impediment as the classical Hotelling $T^{2}$ test statistic since they both involve inversion of ill-formed 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 resulting test is unbiased and, under mild conditions, is also consistent. We demonstrate the efficacy of the approach through simulated and real data examples.
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