The Restricted Risk Bayes Approach and its Application to Linear State Estimation

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