A label efficient two-sample test

Two-sample tests evaluate whether two samples are realizations of the same distribution (the null hypothesis) or two different distributions (the alternative hypothesis). In the traditional formulation of this problem, the statistician has access to both the measurements (feature variables) and the group variable (label variable). However, in several important applications, feature variables can be easily measured but the binary label variable is unknown and costly to obtain. In this paper, we consider this important variation on the classical two-sample test problem and pose it as a problem of obtaining the labels of only a small number of samples in service of performing a two-sample test. We devise a label efficient three-stage framework: firstly, a classifier is trained with samples uniformly labeled to model the posterior probabilities of the labels; secondly, a novel query scheme dubbed \emph{bimodal query} is used to query labels of samples from both classes with maximum posterior probabilities, and lastly, the classical Friedman-Rafsky (FR) two-sample test is performed on the queried samples. Our theoretical analysis shows that bimodal query is optimal for two-sample testing using the FR statistic under reasonable conditions and that the three-stage framework controls the Type I error. Extensive experiments performed on synthetic, benchmark, and application-specific datasets demonstrate that the three-stage framework has decreased Type II error over uniform querying and certainty-based querying with same number of labels while controlling the Type I error. Source code for our algorithms and experimental results is available at https://github.com/wayne0908/Label-Efficient-Two-Sample.