Privacy-Aware MMSE Estimation

TL;DR

This paper explores how to optimally estimate a variable while preserving privacy by controlling the predictability of correlated variables under MMSE constraints, using binary and continuous models.

Contribution

It derives the optimal random mapping for MMSE minimization under privacy constraints in binary and continuous settings.

Findings

Optimal random mapping $P_{Z|Y}$ minimizes MMSE of $Y$ while maintaining privacy of $X$.

Explicit solutions for binary-input symmetric-output channels.

Analysis of additive noise channels for continuous variables.

Abstract

We investigate the problem of the predictability of random variable $Y$ under a privacy constraint dictated by random variable $X$, correlated with $Y$, where both predictability and privacy are assessed in terms of the minimum mean-squared error (MMSE). Given that $X$ and $Y$ are connected via a binary-input symmetric-output (BISO) channel, we derive the \emph{optimal} random mapping $P_{Z|Y}$ such that the MMSE of $Y$ given $Z$ is minimized while the MMSE of $X$ given $Z$ is greater than $(1-\epsilon)\mathsf{var}(X)$ for a given $\epsilon\geq 0$. We also consider the case where $(X,Y)$ are continuous and $P_{Z|Y}$ is restricted to be an additive noise channel.