Kernel-Based Tests for Likelihood-Free Hypothesis Testing

Given n observations from two balanced classes, consider the task of labeling an additional m inputs that are known to all belong to \emph{one} of the two classes. Special cases of this problem are well-known: with completeknowledge of class distributions ( n \infty) theproblem is solved optimally by the likelihood-ratio test; when m 1 it corresponds to binary classification; and when m\approx n it is equivalent to two-sample testing. The intermediate settings occur in the field of likelihood-free inference, where labeled samples are obtained by running forward simulations and the unlabeled sample is collected experimentally. In recent work it was discovered that there is a fundamental trade-offbetween m and n: increasing the data sample m reduces the amount n of training/simulationdata needed. In this work we (a) introduce a generalization where unlabeled samples come from a mixture of the two classes -- a case often encountered in practice; (b) study the minimax sample complexity for non-parametric classes of densities under \textit{maximum meandiscrepancy} (MMD) separation; and (c) investigate the empirical performance of kernels parameterized by neural networks on two tasks: detectionof the Higgs boson and detection of planted DDPM generated images amidstCIFAR-10 images.