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Starting with the vector observation model , robust Bayesian estimates of the vector are constructed for the following two distinct situations: 1) the state is Gaussian and the observation error 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 . Monte Carlo results demonstrate the robustness of the proposed estimation procedure, which might be termed a robustified Kalman filter.