Estimation Efficiency Under Privacy Constraints

We investigate the problem of estimating a random variable Y ∈ Y under a privacy constraint dictated by another correlated random variable X ∈ X , where estimation efficiency and privacy are assessed in terms of two different loss functions. In the discrete case, we use the Hamming loss function and express the corresponding utility-privacy tradeoff in terms of the privacy-constrained guessing probability h(PXY , ε), the maximum probability Pc(Y |Z) of correctly guessing Y given an auxiliary random variable Z ∈ Z , where the maximization is taken over all PZ|Y ensuring that Pc(X |Z) ≤ ε for a given privacy threshold ε ≥ 0. We prove that h(PXY , ·) is concave and piecewise linear, which allows us to derive its expression in closed for any ε when X and Y are binary. In the non-binary case, we derive h(PXY , ε) in the high utility regime (i.e., for sufficiently large, but nontrivial, values of ε) under the assumption that Z takes values in Y . We also analyze the privacy-constrained guessing probability for two binary vector scenarios. When X and Y are continuous random variables, we use the squared-error loss function and express the corresponding utility-privacy tradeoff in terms of sENSR(PXY , ε), which is the smallest normalized minimum mean squared-error (mmse) incurred in estimating Y from its Gaussian perturbation Z , such that the mmse of f(X) given Z is within ε of the variance of f(X) for any non-constant real-valued function f . We derive tight upper and lower bounds for sENSR when Y is Gaussian. We also obtain a tight lower bound for sENSR(PXY , ε) for general absolutely continuous random variables when ε is sufficiently small.