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∣Y ensuring 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 (Xn,Yn) being pairs of independent
and identically distributed binary random vectors, we similarly define
hn(PXnYn,ϵ) under the assumption that Zn is also
a binary vector. Then we obtain a closed form expression for
hn(PXnYn,ϵ) for sufficiently large, but nontrivial
values of ϵ.Comment: ISIT 201