Privacy-Aware Guessing Efficiency

Abstract: We investigate the problem of guessing a discrete random variable $Y$ under a privacy constraint dictated by another correlated discrete random variable $X$, where both guessing efficiency and privacy are assessed in terms of the probability of correct guessing. We define $h(P_{XY}, \epsilon)$ as the maximum probability of correctly guessing $Y$ given an auxiliary random variable $Z$, where the maximization is taken over all $P_{Z|Y}$ ensuring that the probability of correctly guessing $X$ given $Z$ does not exceed $\epsilon$. We show that the map $\epsilon\mapsto h(P_{XY}, \epsilon)$ is strictly increasing, concave, and piecewise linear, which allows us to derive a closed form expression for $h(P_{XY}, \epsilon)$ when $X$ and $Y$ are connected via a binary-input binary-output channel. For $(X^n, Y<sup>n)$ being pairs of independent and identically distributed binary random vectors, we similarly define $$\underline{h}<em>n(P</em>{X^{nY^n},} \epsilon)$$ under the assumption that $Z<sup>n$ is also a binary vector. Then we obtain a closed form expression for $$\underline{h}<em>n(P</em>{X^{nY^n},} \epsilon)$$ for sufficiently large, but nontrivial values of $\epsilon$.