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

Starting with the vector observation model
y = Hx + v
, robust Bayesian estimates
\hat{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_{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.