Nonlinear filtering. A weighted mean squares approach and a Bayesian one via the maximum entropy principle

Abstract This paper suggests two approaches to nonlinear filtering, which make abdication of the differential equation of the conditional probability density given the observation. In the first one, which refers to the statistical point of view, one derives the trajectory of the state estimation by minimizing a weighted cost function which can be thought of as an extended mean-squares criterion, and the result is compared with the Kalman filter (in the linear case!). The second approach is the Bayesian one, and the basic problem is to determine the probability density of the state as it is defined by the Fokker-Planck equation. To this end one suggests two approaches via the maximum entropy principle. The first one refers to the state moments of the system, while the second one involves a slight extension of this principle via the time integral of the entropy.