Optimal Bayesian Kalman Filtering With Prior Update

In many practical filter design problems, the exact statistical information of the underlying random processes is not available. One robust filtering approach in these situations is to design an intrinsically Bayesian robust filter that provides optimal solution relative to the prior distribution governing the uncertainty class of all possible joint random process models. In this context, the intrinsically Bayesian robust Kalman filter has been recently introduced for the case that the second-order statistics of the observation and process noise in the state-space model are unknown. However, such a filter does not utilize the additional information embedded in the data being observed. In this paper, we derive the optimal Bayesian Kalman filter, which is optimal over posterior distribution obtained from incorporating data into the prior distribution. This filter has the same recursive structure as that of the classical Kalman filter, except that it is designed relative to the posterior effective noise statistics , which are found by employing the method of factor graphs through formulating the problem of computing the likelihood function as a message passing algorithm.