Kernel-Based Tests for Likelihood-Free Hypothesis Testing

TL;DR

This paper develops kernel-based methods for likelihood-free hypothesis testing, exploring the trade-off between labeled and unlabeled data, and demonstrates their effectiveness on complex real-world tasks like Higgs boson detection.

Contribution

It introduces a generalized setting with mixture unlabeled samples, analyzes the minimax sample complexity under MMD separation, and empirically tests neural network kernels on challenging detection tasks.

Findings

Confirmed the asymmetric m vs n data trade-off in practice.

Established theoretical bounds for sample complexity under MMD separation.

Demonstrated neural network kernels' effectiveness on Higgs and CIFAR-10 detection tasks.

Abstract

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 complete knowledge of class distributions ($n=\infty$) the problem 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-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 \textit{maximum 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.