Use of the Kalman filter for inference in state-space models with unknown noise distributions

The Kalman filter is frequently used for state estimation in state-space models when the standard Gaussian noise assumption does not apply. A problem arises, however, in that inference based on the incorrect Gaussian assumption can lead to misleading or erroneous conclusions about the relationship of the Kalman filter estimate to the true (unknown) state. This note shows how inequalities from probability theory associated with the probabilities of convex sets have potential for characterizing the estimation error of a Kalman filter in such a non-Gaussian (distribution-free) setting.