Robust bayesian estimation for the linear model and robustifying the Kalman filter

Starting with the vector observation model y = Hx + v, robust Bayesian estimates x of the vector x are constructed for the following two distinct situations: 1) the state x is Gaussian and the observation error v is (heavy-tailed) non-Gaussian and 2) the state is heavy-tailed non-Gaussian and the observation error is Gaussian. Bounds with respect to broad symmetric non-Gaussian families are derived for the error covariance matrix of these estimates. These "one-step" robust estimates are then used to obtain robust estimates for the Kalman filter setup y₊= H₊x₊+ v₊, x₊+₁=₊x₊+w₊. Monte Carlo results demonstrate the robustness of the proposed estimation procedure, which might be termed a robustified Kalman filter.