Distributed Hypothesis Testing with Variable-Length Coding

TL;DR

This paper determines the best possible decay rate of the type-II error in distributed hypothesis testing with an expected rate constraint, showing it depends on the allowed type-I error level and differs from maximum rate constraint scenarios.

Contribution

It introduces a characterization of the optimal error exponent under expected rate constraints, extending classical results to more flexible coding rate conditions.

Findings

Optimal type-II error exponent depends on type-I error limit

Expected rate constraint leads to different exponents than maximum rate constraint

Results generalize classical hypothesis testing bounds

Abstract

This paper characterizes the optimal type-II error exponent for a distributed hypothesis testing-against-independence problem when the \emph{expected} rate of the sensor-detector link is constrained. Unlike for the well-known Ahlswede-Csiszar result that holds under a \emph{maximum} rate constraint and where a strong converse holds, here the optimal exponent depends on the allowed type-I error exponent. Specifically, if the type-I error probability is limited by $\epsilon$, then the optimal type-II error exponent under an \emph{expected} rate constraint $R$ coincides with the optimal type-II error exponent under a \emph{maximum} rate constraint of $(1-\epsilon)R$.