Estimation Efficiency Under Privacy Constraints

We investigate the problem of estimating a random variable <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula> under a privacy constraint dictated by another correlated random variable <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula>. When <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula> are discrete, we express the underlying privacy-utility tradeoff in terms of the privacy-constrained guessing probability <inline-formula> <tex-math notation="LaTeX">${\mathcal {h}}(P_{XY}, \varepsilon)$ </tex-math></inline-formula>, and the maximum probability <inline-formula> <tex-math notation="LaTeX">$\mathsf {P}_{\mathsf {c}}(Y|Z)$ </tex-math></inline-formula> of correctly guessing <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula> given an auxiliary random variable <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula>, where the maximization is taken over all <inline-formula> <tex-math notation="LaTeX">$P_{Z|Y}$ </tex-math></inline-formula> ensuring that <inline-formula> <tex-math notation="LaTeX">$\mathsf {P}_{\mathsf {c}}(X|Z)\leq \varepsilon $ </tex-math></inline-formula> for a given privacy threshold <inline-formula> <tex-math notation="LaTeX">$\varepsilon \geq 0$ </tex-math></inline-formula>. We prove that <inline-formula> <tex-math notation="LaTeX">${\mathcal {h}}(P_{XY}, \cdot)$ </tex-math></inline-formula> is concave and piecewise linear, which allows us to derive its expression in closed form for any <inline-formula> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula> when <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula> are binary. In the non-binary case, we derive <inline-formula> <tex-math notation="LaTeX">${\mathcal {h}}(P_{XY}, \varepsilon)$ </tex-math></inline-formula> in the high-utility regime (i.e., for sufficiently large, but nontrivial, values of <inline-formula> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula>) under the assumption that <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula> have the same alphabets. We also analyze the privacy-constrained guessing probability for two scenarios in which <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula> are binary vectors. When <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula> are continuous random variables, we formulate the corresponding privacy-utility tradeoff in terms of <inline-formula> <tex-math notation="LaTeX">${\mathsf {sENSR}}(P_{XY}, \varepsilon)$ </tex-math></inline-formula>, the smallest normalized minimum mean squared-error (mmse) incurred in estimating <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula> from a Gaussian perturbation <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula>. Here, the minimization is taken over a family of Gaussian perturbations <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula> for which the mmse of <inline-formula> <tex-math notation="LaTeX">$f(X)$ </tex-math></inline-formula> given <inline-formula> <tex-math notation="LaTeX">$Z$ </tex-math></inline-formula> is within a factor <inline-formula> <tex-math notation="LaTeX">$1- \varepsilon $ </tex-math></inline-formula> from the variance of <inline-formula> <tex-math notation="LaTeX">$f(X)$ </tex-math></inline-formula> for any non-constant real-valued function <inline-formula> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula>. We derive tight upper and lower bounds for <inline-formula> <tex-math notation="LaTeX">${\mathsf {sENSR}}$ </tex-math></inline-formula> when <inline-formula> <tex-math notation="LaTeX">$Y$ </tex-math></inline-formula> is Gaussian. For general absolutely continuous random variables, we obtain a tight lower bound for <inline-formula> <tex-math notation="LaTeX">${\mathsf {sENSR}}(P_{XY}, \varepsilon)$ </tex-math></inline-formula> in the high privacy regime, i.e., for small <inline-formula> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula>.