# Kernel-Based Tests for Likelihood-Free Hypothesis Testing

arXiv: https://arxiv.org/abs/2308.09043 · HTML: https://mlsanity.com/paper/2308.09043/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 51 references that are in the map.
- First posted on arXiv 17 Aug 2023.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Revisiting Classifier Two-Sample Tests](https://mlsanity.com/paper/1610.06545/) (2016)
- [Sequential Neural Likelihood: Fast Likelihood-free Inference with Autoregressive Flows](https://mlsanity.com/paper/1805.07226/) (2018)
- [Learning Deep Kernels for Non-Parametric Two-Sample Tests](https://mlsanity.com/paper/2002.09116/) (2020)
- [Generative Models and Model Criticism via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/1611.04488/) (2016)
- [Likelihood-free inference by ratio estimation](https://mlsanity.com/paper/1611.10242/) (2016)
- [Mining gold from implicit models to improve likelihood-free inference](https://mlsanity.com/paper/1805.12244/) (2018)
- [Learning Summary Statistic for Approximate Bayesian Computation via Deep Neural Network](https://mlsanity.com/paper/1510.02175/) (2015)
- [Classification Logit Two-sample Testing by Neural Networks](https://mlsanity.com/paper/1909.11298/) (2019)
- [A Test of Relative Similarity For Model Selection in Generative Models](https://mlsanity.com/paper/1511.04581/) (2015)
- [On the Optimality of Gaussian Kernel Based Nonparametric Tests against Smooth Alternatives](https://mlsanity.com/paper/1909.03302/) (2019)
- [High-Dimensional Density Ratio Estimation with Extensions to Approximate Likelihood Computation](https://mlsanity.com/paper/1404.7063/) (2014)
- [Confidence Sets and Hypothesis Testing in a Likelihood-Free Inference Setting](https://mlsanity.com/paper/2002.10399/) (2020)
- [Optimal Testing of Discrete Distributions with High Probability](https://mlsanity.com/paper/2009.06540/) (2020)
- [Informative Features for Model Comparison](https://mlsanity.com/paper/1810.11630/) (2018)

## Built on by (most cited first, 2 of 2)

- [Anchor-based Maximum Discrepancy for Relative Similarity Testing](https://mlsanity.com/paper/2510.10477/) (2025)
- [Sparse, self-organizing ensembles of local kernels detect rare statistical anomalies](https://mlsanity.com/paper/2511.03095/) (2025)

## Abstract (arXiv)

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.