# Testing Mixtures of Discrete Distributions

## TL;DR

This paper introduces a new noise model for distribution testing, where the noisy distribution is a known mixture of the original and noise, and demonstrates that testing in this setting can be as sample-efficient as in the noise-free case.

## Contribution

The authors propose a tractable mixture noise model for distribution testing and show that testing complexity remains unchanged compared to classical methods.

## Key findings

- Sample complexity matches classical non-mixture testing
- Mixture testing is more tractable under the proposed noise model
- Results apply to identity and closeness testing problems

## Abstract

There has been significant study on the sample complexity of testing properties of distributions over large domains. For many properties, it is known that the sample complexity can be substantially smaller than the domain size. For example, over a domain of size $n$, distinguishing the uniform distribution from distributions that are far from uniform in $\ell_1$-distance uses only $O(\sqrt{n})$ samples. However, the picture is very different in the presence of arbitrary noise, even when the amount of noise is quite small. In this case, one must distinguish if samples are coming from a distribution that is $\epsilon$-close to uniform from the case where the distribution is $(1-\epsilon)$-far from uniform. The latter task requires nearly linear in $n$ samples [Valiant 2008, Valian and Valiant 2011]. In this work, we present a noise model that on one hand is more tractable for the testing problem, and on the other hand represents a rich class of noise families. In our model, the noisy distribution is a mixture of the original distribution and noise, where the latter is known to the tester either explicitly or via sample access; the form of the noise is also known a priori. Focusing on the identity and closeness testing problems leads to the following mixture testing question: Given samples of distributions $p, q_1,q_2$, can we test if $p$ is a mixture of $q_1$ and $q_2$? We consider this general question in various scenarios that differ in terms of how the tester can access the distributions, and show that indeed this problem is more tractable. Our results show that the sample complexity of our testers are exactly the same as for the classical non-mixture case.