Maximal Guessing Coupling and Its Applications

Abstract

A coupling of two distributions $P_{X}$ and $P_{\mathrm{Y}}$ is a joint distribution $P_{XY}$ with marginal distributions equal to $P_{X}$ and $P_{\mathrm{Y}}$ . Given marginals $P_{X}$ and $P_{Y}$ and the maximal guessing probability function $g(P_{XY}):=\max_{f:\mathcal{X}\rightarrow \mathcal{Y}}\mathbb{P}_{P}\{Y=f(X)\}$ of the joint distribution $P_{XY}$ , what is its maximum over all couplings $P_{XY}$ of $P_{X}$ and $P_{Y}$ ? This is the maximal guessing coupling problem. We study this problem and show that it is equivalent to the probability distribution approximation problem. Therefore, some existing results on the latter problem can be used to derive the asymptotics of the maximal guessing coupling problem. We apply these results to two new information-theoretic problems: channel capacity with input distribution constraint, and perfect stealth-secrecy communication.