Abstract-A deterministic procedure for optimal approximation of arbitrary probability density functions by means of Dirac mixtures with equal weights is proposed. The optimality of this approximation is guaranteed by minimizing the distance of the approximation from the true density. For this purpose a distance measure is required, which is in general not well defined for Dirac mixtures. Hence, a key contribution is to compare the corresponding cumulative distribution functions.This paper concentrates on the simple and intuitive integral quadratic distance measure. For the special case of a Dirac mixture with equally weighted components, closedform solutions for special types of densities like uniform and Gaussian densities are obtained. Closed-form solution of the given optimization problem is not possible in general. Hence, another key contribution is an efficient solution procedure for arbitrary true densities based on a homotopy continuation approach.In contrast to standard Monte Carlo techniques like particle filters that are based on random sampling, the proposed approach is deterministic and ensures an optimal approximation with respect to a given distance measure. In addition, the number of required components (particles) can easily be deduced by application of the proposed distance measure. The resulting approximations can be used as basis for recursive nonlinear filtering mechanism alternative to Monte Carlo methods.

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Abstract-Intention recognition is an important topic in human-robot cooperation that can be tackled using probabilistic model-based methods. A popular instance of such methods are Bayesian networks where the dependencies between random variables are modeled by means of a directed graph. Bayesian networks are very efficient for treating networks with conditionally independent parts. Unfortunately, such independence sometimes has to be constructed by introducing so called hidden variables with an intractably large state space. An example are human actions which depend on human intentions and on other human actions. Our goal in this paper is to find models for intention-action mapping with a reduced state space in order to allow for tractable on-line evaluation. We present a systematic derivation of the reduced model and experimental results of recognizing the intention of a real human in a virtual environment.

Abstract-This paper introduces a new approach to the recursive propagation of probability density functions through discrete-time stochastic nonlinear dynamic systems. An efficient recursive procedure is proposed that is based on the optimal approximation of the posterior densities after each prediction step by means of Dirac mixtures. The parameters of the individual components are selected by systematically minimizing a suitable distance measure in such a way that the future evolution of the approximate densities is as close to the exact densities as possible.

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