Recursive nonlinear estimation: A geometric approach

Abstract

The paper deals with approximation of nonlinear estimation within the Bayesian framework. The underlying idea is to project the true posterior density orthogonally onto a prespecified approximation family. The problem to be coped with is that the true posterior density is not typically at disposal when estimation is implemented recursively. It is shown that there exists a Bayes-closed description of the posterior density which is recursively computable without complete knowledge of the true posterior. We study a mutual relationship between the equivalence class, composed of densities matching the current description, and a prespecified parametric family. It is proved that if the approximation family is of the mixture type, the equivalence classes can be made orthogonal to this family. Then the approximating density done by the orthogonal projection of the true posterior density minimizes the Kullback-Leibler distance between both densities. On the contrary, if the approximation family is of the exponential type, the analogous result holds at most locally. To be able to give a sensible definition of the orthogonal projection, we have been forced to introduce a Riemannian geometry on the family of probability distributions. Being aware that the differential-geometric concepts and tools do not belong to common knowledge of control engineers, we include necessary preliminary information. © 1990.