  Optimal state estimation for linear discrete-time systems is considered.
Motivated by the literature on differential privacy, the measurements are
assumed to be corrupted by Laplace noise. The optimal least mean square error
estimate of the state is approximated using a randomized method. The method
relies on that the Laplace noise can be rewritten as Gaussian noise scaled by
Rayleigh random variable. The probability of the event that the distance
between the approximation and the best estimate is smaller than a constant is
determined as function of the number of parallel Kalman filters that is used in
the randomized method. This estimator is then compared with the optimal linear
estimator, the maximum a posteriori (MAP) estimate of the state, and the
particle filter.
