Collaborative Distributed Hypothesis Testing

TL;DR

This paper investigates collaborative distributed hypothesis testing with limited multi-round communication, establishing feasibility results for error exponents and analyzing the impact of interaction and zero-rate constraints.

Contribution

It extends distributed hypothesis testing to multi-round interactions and provides new feasibility results, including the case of zero-rate communication where interaction offers no benefit.

Findings

Feasibility of error exponents for general hypotheses established.

Interaction does not improve asymptotic performance under zero-rate communication.

Special cases like testing against independence are analyzed with both feasibility and unfeasibility results.

Abstract

A collaborative distributed binary decision problem is considered. Two statisticians are required to declare the correct probability measure of two jointly distributed memoryless process, denoted by $X^n=(X_1,\dots,X_n)$ and $Y^n=(Y_1,\dots,Y_n)$, out of two possible probability measures on finite alphabets, namely $P_{XY}$ and $P_{\bar{X}\bar{Y}}$. The marginal samples given by $X^n$ and $Y^n$ are assumed to be available at different locations. The statisticians are allowed to exchange limited amount of data over multiple rounds of interactions, which differs from previous work that deals mainly with unidirectional communication. A single round of interaction is considered before the result is generalized to any finite number of communication rounds. A feasibility result is shown, guaranteeing the feasibility of an error exponent for general hypotheses, through information-theoretic methods. The special case of testing against independence is revisited as being an instance of this result for which also an unfeasibility result is proven. A second special case is studied where zero-rate communication is imposed (data exchanges grow sub-exponentially with $n$) for which it is shown that interaction does not improve asymptotic performance.