Estimation Efficiency Under Privacy Constraints

We investigate the problem of estimating a random variable Y∈ Y under a privacy constraint dictated by another 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 mathsfP_mathsfc(Y|Z) of correctly guessing Y given an auxiliary random variable Z∈ Z, where the maximization is taken over all PZ|Y ensuring that mathsfP_mathsfc(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 form 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 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 mathsfsENSR(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 mathsfsENSR when Y is Gaussian. We also obtain a tight lower bound for mathsfsENSR(PXY, ε) for general absolutely continuous random variables when ε is sufficiently small.