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
,ε) 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 ε. We show that the map ε → h(P
_XY
,ε) is strictly increasing, concave, and piecewise linear, which allows us to derive a closed form expression for h(P
_xY
,ε) 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_
_n
(P
_{X n Y n, ε}
) under the assumption that Z
ⁿ
is also a binary vector. Then we obtain a closed form expression for h_
_n
(P
_{X n Y n, ε}
) for sufficiently large, but nontrivial values of ε.