From Bayes to Kalman

Abstract

The Kalman filter [10] was introduced in the preceding chapter as a generalization of a simple recursive processor. This was a “bottom up” point of view and was presented without proof. In this chapter it will be shown that the Kalman filter is a special case of a more general processing structure called a Bayesian filter (BF). That is, it is presented as a “top down” description. None of the material in this book will deal in depth with the Bayesian filter, but it is introduced for completeness and also to demonstrate that the Kalman filter logically derives from it and is therefore a Bayesian processor. It then follows that for the case where the measurement model and the system model are both linear, and the measurement noise and system noise are both Gaussian, the Kalman filter is optimum and therefore an optimum realization of a Bayesian filter.