Nonlinear Bayesian estimation with convex sets of probability densities

This paper presents a theoretical framework for Bayesian estimation in the case of imprecisely known probability density functions. The lack of knowledge about the true density functions is represented by sets of densities. A formal Bayesian estimator for these sets is introduced, which is intractable for infinite sets. To obtain a tractable filter, properties of convex sets in form of convex polytopes of densities are investigated. It is shown that pathwise connected sets and their convex hulls describe the same ignorance. Thus, an exact algorithm is derived, which only needs to process the hull, delivering tractable results in the case of a proper parametrization. Since the estimator delivers a convex hull of densities as output, the theoretical grounds are laid for deriving efficient Bayesian estimators for sets of densities. The derived filter is illustrated by means of an example.