  This paper studies the design of an optimal privacyaware estimator of a
public random variable based on noisy measurements which contain private
information. The public random variable carries non-private information,
however, its estimate will be correlated with the private information due to
the estimation process. It is assumed that the estimate of the public random
variable is revealed to an untrusted party. The objective is to design an
estimator for the public random variable such that the leakage of the private
information, via the estimation process, is kept below a certain level. The
privacy metric is defined as the discrete conditional entropy of the private
random variable, which carries the private information, given the output of the
estimator. A binary loss function is considered for the estimation of the
public random variable. It is shown that the optimal privacy-aware estimator is
the solution of a (possibly infinite-dimensional) convex optimization problem.
We next study the optimal perfect privacy estimator which ensures that the
estimate of the public random variable is independent of the private
information. A necessary and sufficient condition is derived which guarantees
that an estimator satisfies the perfect privacy requirement. It is shown that
the optimal perfect privacy estimator is the solution of a linear optimization
problem. A sufficient condition for the existence of the optimal perfect
privacy estimator is derived when the estimator has access to discretized
measurements of sensors.
