Distributed Hypothesis Testing over a Noisy Channel: Error-exponents Trade-off

TL;DR

This paper investigates the fundamental limits of distributed binary hypothesis testing over noisy channels, deriving bounds on error exponents trade-offs and proposing schemes that outperform traditional separation methods.

Contribution

It introduces two inner bounds for the error-exponents trade-off, including a joint scheme that outperforms separation-based approaches in certain scenarios.

Findings

Separation-based scheme recovers known bounds for noiseless channels.

Joint scheme achieves tighter bounds than separation in some cases.

Provides a comprehensive analysis of error-exponent trade-offs in noisy distributed testing.

Abstract

A two-terminal distributed binary hypothesis testing problem over a noisy channel is studied. The two terminals, called the observer and the decision maker, each has access to $n$ independent and identically distributed samples, denoted by $\mathbf{U}$ and $\mathbf{V}$, respectively. The observer communicates to the decision maker over a discrete memoryless channel, and the decision maker performs a binary hypothesis test on the joint probability distribution of $(\mathbf{U},\mathbf{V})$ based on $\mathbf{V}$ and the noisy information received from the observer. The trade-off between the exponents of the type I and type II error probabilities is investigated. Two inner bounds are obtained, one using a separation-based scheme that involves type-based compression and unequal error-protection channel coding, and the other using a joint scheme that incorporates type-based hybrid coding. The separation-based scheme is shown to recover the inner bound obtained by Han and Kobayashi for the special case of a rate-limited noiseless channel, and also the one obtained by the authors previously for a corner point of the trade-off. Finally, we show via an example that the joint scheme achieves a strictly tighter bound than the separation-based scheme for some points of the error-exponents trade-off.