Privacy-aware guessing efficiency
Shahab Asoodeh, Mario Díaz, Fady Alajaji, Tamás Linder · 2017
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(PXY,ε) as the maximum probability of correctly guessing Y given an auxiliary random variable Z, where the maximization is taken over all PZ|Yensuring that the probability of correctly guessing X given Z does not exceed ε. We show that the map ε → h(PXY,ε) is strictly increasing, concave, and piecewise linear, which allows us to derive a closed form expression for h(PxY,ε) when X and Y are connected via a binary-input binary-output channel. For {(Xi, Yi)}ni=1being pairs of independent and identically distributed binary random vectors, we similarly define h_n(PX n Y n, ε) under the assumption that Znis also a binary vector. Then we obtain a closed form expression for h_n(PX n Y n, ε) for sufficiently large, but nontrivial values of ε.