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 one of the two classes. Special cases of this problem are well-known: with complete knowledge of class distributions (n=∞) the problem is solved optimally by the likelihood-ratio test; when m=1 it corresponds to binary classification; and when m≈ 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-off between m and n: increasing the data sample m reduces the amount n of training/simulation data 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 textitmaximum mean discrepancy (MMD) separation; and (c) investigate the empirical performance of kernels parameterized by neural networks on two tasks: detection of the Higgs boson and detection of planted DDPM generated images amidst CIFAR-10 images. For both problems we confirm the existence of the theoretically predicted asymmetric m vs n trade-off.