  This paper proposes an optimal stochastic sampling approach to privacy, in
which a sensor observes a process which is correlated to private information,
and a sampler decides to keep or discard the sensor's observations. The kept
samples are shared with an adversary who might attempt to infer the private
process. The privacy leakages are captured with the mutual information between
the private process and sampler's output. We cast the optimal sampling design
as an optimization problem that (i) minimizes the reconstruction error of the
observed process using the sampler's output, (ii) and reduces privacy leakages.
We first show the optimal reconstruction is obtained by solving a one-step
optimization problem at each time step. We derive the optimality equations of
the sampler for a general processes via the dynamic decomposition method, and
show the sampler controls adversary's belief about the private input. Also, we
propose a simplified design for linear Gaussian processes by restricting the
sampling policy to a special collection. We show that the optimal
reconstruction of states, the belief state and the optimization objective can
be analytically expressed based on a conditional mean and covariance matrix. We
develop an numerical algorithm to optimize the sampling and reconstruction
policies based on the implicit function theorem. Finally, we verify our design
and show its capabilities in state reconstruction, privacy protection and data
size reduction via simulations.
