An application of the information theory to filtering problems

The purpose of this paper is to study the filtering problems from the viewpoint of the information theory. For a linear system it is proved that the necessary and sufficient condition for maximizing the mutual information between a state and the estimate is to minimize the entropy of the estimation error. Then we derive the Kalman-Bucy filter for both the discrete-time and the continuous-time systems by an application of the information theory. Furthermore, the approach is extended to the nonlinear dynamical systems with noisy observations and then the information structures of the optimal filter for a continuous-time nonlinear system are made clear, which has been presented as the interesting open problems by Bucy.