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

Abstract

Starting with the vector observation modely = Hx + v, robust Bayesian estimates\hat{x}of the vectorxare constructed for the following two distinct situations: 1) the statexis Gaussian and the observation errorvis (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 setupy_{k}= H_{k}x_{k}+ v_{k}, x_{k+1}=\Phi_{k}x_{k}+w_{k}. Monte Carlo results demonstrate the robustness of the proposed estimation procedure, which might be termed a robustified Kalman filter.