Restricted Risk Bayes Linear State Estimation

The problem of state estimation with stochastic uncertainties in the initial state, model noise, and measurement noise is considered using the restricted risk Bayes approach. It is assumed that the a priori distributions of these quantities are not perfectly known, but that some information about them may be available. While offering robustness, the restricted risk Bayes approach incorporates the available a priori information to give less conservative state estimators than the Gamma-minimax approach popular in the literature. When attention is restricted to linear estimators based on a quadratic loss function, a systematic method to derive restricted risk Bayes estimators is proposed. Applying to the filtering problem, the restricted risk Bayes approach provides us with a robust method to calibrate the Kalman filter (KF), considering the presence of stochastic uncertainties. This method is illustrated with a target tracking example and a wireless channel tracking example for which the Bayes, minimax, and restricted risk Bayes estimators are derived and their performance is compared.