Distributed Hypothesis Testing Against Dependence

Abstract

We study distributed hypothesis testing and establish the exact error exponent in single-letter form for new testing problems. In distributed hypothesis testing, a receiver decides between $\mathcal{H}_0:P_{XY}$ and $\mathcal{H}_1:Q_{XY}$ based on $Y^n$ and a rate-limited description of $X^n$. So far, such single-letter forms are known only for testing against independence, studied by Ahlswede and Csiszár, and testing against conditional independence, studied by Rahman and Wagner. In this paper, we study testing against dependence, where $P_{XY}=P_XP_Y$, and show that its error exponent is given by Han's exponent, which is established by single-letterizing a multi-letter version of Han's exponent. Our result disproves a previous conjecture by Han claiming that the error exponent is given by the lautum information. We then consider the Cartesian product of testing against dependence and testing against independence, and derive a single-letter characterization of its error exponent. Finally, we study testing against conditional dependence, which is the dependence-testing counterpart of the setting studied by Rahman and Wagner. We derive a single-letter converse bound for the error exponent by introducing and solving a related setting where the side information is also available to the transmitter. We show that our converse bound is tight in some cases by using a conditional-coding version of the quantization scheme, which improves upon existing achievability schemes.