The Maximum Mean Discrepancy (MMD) has been the state-of-the-art nonparametric test for tackling the two-sample problem. Its statistic is given by the difference in expectations of the witness function, a real-valued function defined as a weighted sum of kernel evaluations on a set of basis points. Typically the kernel is optimized on a training set, and hypothesis testing is performed on a separate test set to avoid overfitting (i.e., control type-I error). That is, the test set is used to simultaneously estimate the expectations and define the basis points, while the training set only serves to select the kernel and is discarded. In this work, we propose to use the training data to also define the weights and the basis points for better data efficiency. We show that 1) the new test is consistent and has a well-controlled type-I error; 2) the optimal witness function is given by a precision-weighted mean in the reproducing kernel Hilbert space associated with the kernel; and 3) the test power of the proposed test is comparable or exceeds that of the MMD and other modern tests, as verified empirically on challenging synthetic and real problems (e.g., Higgs data).

翻译：最大平均值差异(MMD)是处理两个抽样问题的最先进的非参数性测试,其统计数据来自对证人功能的预期差异,即根据一组基点对内核评价的加权总和进行实际估价的功能,即一组基点的证人功能的差异。一般而言,内核优化在一组培训中,假设测试在一套单独的测试中进行,以避免过度匹配(即控制型号I错误)。这就是,测试组用来同时估计期望和界定基点,而培训组只用来选择内核并被丢弃。在这项工作中,我们提议使用培训数据来界定加权数和基点,以提高数据效率。我们表明:(1)新的测试是一致的,并且有严格控制的型号I错误;(2)最佳的证人功能是由与内核相关的再生产内核空间的精密加权平均值来设定的;(3)拟议测试的测试力可比较或超过MMD和其他现代数据具有挑战性的测试。(经核实的、模拟的、模拟和模拟的)