Estimation Efficiency Under Privacy Constraints

We investigate the problem of estimating a random variable Y under a privacy constraint dictated by another correlated random variable X. When X and Y are discrete, we express the underlying privacy-utility tradeoff in terms of the privacy-constrained guessing probability h(P-XY, epsilon), and the maximum probability P-c(Y vertical bar Z) of correctly guessing Y given an auxiliary random variable Z, where the maximization is taken over all P-Z vertical bar Y ensuring that P-c(X vertical bar Z) <= epsilon for a given privacy thresholds epsilon >= 0. We prove that h(P-XY, .) is concave and piecewise linear, which allows us to derive its expression in closed form for any epsilon when X and Y are binary. In the non-binary case, we derive h(P-XY, epsilon) in the high-utility regime (i.e., for sufficiently large, but nontrivial, values of epsilon) under the assumption that Y and Z have the same alphabets. We also analyze the privacy-constrained guessing probability for two scenarios in which X, Y, and Z are binary vectors. When X and Y are continuous random variables, we formulate the corresponding privacy-utility tradeoff in terms of sENSR(P-XY, epsilon), the smallest normalized minimum mean squared-error (mmse) incurred in estimating Y from a Gaussian perturbation Z. Here, the minimization is taken over a family of Gaussian perturbations Z for which the mmse of f (X) given Z is within a factor 1 - epsilon from 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. For general absolutely continuous random variables, we obtain a tight lower bound for sENSR(P-XY, epsilon) in the high privacy regime, i.e., for small epsilon.