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<inf>XY</inf>,ε) as the maximum probability of correctly guessing Y given an auxiliary random variable Z, where the maximization is taken over all P<inf>Z</inf>|<inf>Y</inf> ensuring that the probability of correctly guessing X given Z does not exceed ε. We show that the map ε → h(P<inf>XY</inf>,ε) is strictly increasing, concave, and piecewise linear, which allows us to derive a closed form expression for h(P<inf>xY</inf>,ε) when X and Y are connected via a binary-input binary-output channel. For {(X<inf>i</inf>, Y<inf>i</inf>)}<sup>n</sup><inf>i=1</inf> being pairs of independent and identically distributed binary random vectors, we similarly define ẖ<inf>n</inf>(P<inf>X n Y n, ε</inf>) under the assumption that Z<sup>n</sup> is also a binary vector. Then we obtain a closed form expression for ẖ<inf>n</inf>(P<inf>X n Y n, ε</inf>) for sufficiently large, but nontrivial values of ε.

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