Privacy-aware MMSE estimation

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 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-ε)var(X) for a given ε ≥ 0. We also consider the case where (X, Y ) are continuous and P
_Z|Y
is restricted to be an additive-noise channel.