Privacy-Aware Guessing Efficiency

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 vertical bar Y ensuring that the probability of correctly guessing X given Z does not exceed epsilon. We show that the map epsilon -> 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-i, Y-i)}(n)(i-1) being pairs of independent and identically distributed binary random vectors, we similarly define (h) under bar (n) (P-XnY n, epsilon) under the assumption that Z(n) is also a binary vector. Then we obtain a closed form expression for (h) under bar (n) (P-XnY n, epsilon) for sufficiently large, but nontrivial values of epsilon.