Privacy-Aware Guessing Efficiency

TL;DR

This paper analyzes the optimal guessing efficiency of a discrete variable under privacy constraints, deriving closed-form solutions for specific binary scenarios and revealing the properties of the guessing function.

Contribution

It introduces a novel framework for quantifying guessing efficiency under privacy constraints and provides explicit formulas for binary cases, advancing understanding of privacy-utility trade-offs.

Findings

The guessing function is strictly increasing, concave, and piecewise linear.

Closed-form expressions are derived for binary variables connected via a binary-input binary-output channel.

Explicit solutions are obtained for large but nontrivial privacy levels in i.i.d. binary vectors.

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 $(X^n, Y^n)$ being pairs of independent and identically distributed binary random vectors, we similarly define $\underline{h}_n(P_{X^nY^n}, \epsilon)$ under the assumption that $Z^n$ is also a binary vector. Then we obtain a closed form expression for $\underline{h}_n(P_{X^nY^n}, \epsilon)$ for sufficiently large, but nontrivial values of $\epsilon$.