  In this paper, we develop an information-theoretic framework for the optimal
privacy-aware estimation of the states of a (linear or nonlinear) system. In
our setup, a private process, modeled as a first-order Markov chain, derives
the states of the system, and the state estimates are shared with an untrusted
party who might attempt to infer the private process based on the state
estimates. As the privacy metric, we use the mutual information between the
private process and the state estimates. We first show that the privacy-aware
estimation is a closed-loop control problem wherein the estimator controls the
belief of the adversary about the private process. We also derive the Bellman
optimality principle for the optimal privacy-aware estimation problem, which is
used to study the structural properties of the optimal estimator. We next
develop a policy gradient algorithm, for computing an optimal estimation
policy, based on a novel variational formulation of the mutual information. We
finally study the performance of the optimal estimator in a building automation
application.
