Hypothesis Testing of Mixture Distributions using Compressed Data

TL;DR

This paper investigates hypothesis testing for mixture distributions with compressed data, deriving optimal error exponents and extending results to related problems, under a Neyman-Pearson framework.

Contribution

It introduces new bounds on error exponents for mixture distribution testing with compression, including conditions for maximum achievable error exponents and connections to the Wyner-Ahlswede-Körner problem.

Findings

Derived optimal error exponents for mixture distributions under compression.

Established conditions for maximum epsilon-achievable error exponents.

Extended results to the Wyner-Ahlswede-Körner problem with mixture sources.

Abstract

In this paper we revisit the binary hypothesis testing problem with one-sided compression. Specifically we assume that the distribution in the null hypothesis is a mixture distribution of iid components. The distribution under the alternative hypothesis is a mixture of products of either iid distributions or finite order Markov distributions with stationary transition kernels. The problem is studied under the Neyman-Pearson framework in which our main interest is the maximum error exponent of the second type of error. We derive the optimal achievable error exponent and under a further sufficient condition establish the maximum $\epsilon$-achievable error exponent. It is shown that to obtain the latter, the study of the exponentially strong converse is needed. Using a simple code transfer argument we also establish new results for the Wyner-Ahlswede-K{\"o}rner problem in which the source distribution is a mixture of iid components.