Belief fuctions: theory and algorithms

The subject of this thesis is belief function theory and its application in different contexts. Belief function theory (also known as Dempster-Shafer theory) is a mathematical framework for describing quantified beliefs held by an agent. It can be interpreted as a generalization of Bayesian probability theory and makes it possible to distinguish between different types of uncertainty. In particular, belief functions can make uncertainty resulting from a lack of evidence explicit. In this thesis, applications of belief function theory are explored both on a theoretical and on an algorithmic level. One of the major criticisms raised against the use of belief functions is the exponential complexity associated with their representation and combination. This criticism is addressed in this thesis by showing how efficient algorithms can be developed based on Monte-Carlo approximations and exploitation of independence. First, the context of temporal processes subject to uncertainty is considered where uncertainty can be modeled by belief functions. Here, evidential temporal update equations are derived that generalize Bayesian filtering and allow the state of a dynamical system to be estimated recursively over time. In order to reduce the exponential complexity of solving these equations, a Monte-Carlo approximation resulting in an evidential particle filter algorithm is presented. This evidential particle filter algorithm constitutes a generalization of probabilistic particle filters in discrete domains and reduces the exponential time and space complexity of the analytical filtering solution to a complexity that is linear in the size of the state space. The second context considered in this thesis is spatial uncertainty; specifically, the problem of simultaneous localization and mapping (SLAM). For SLAM, a mobile robot exploring an unknown environment is tasked with constructing a map of the environment while, at the same time, localizing itself using this map. It is proved in this thesis that the joint distribution of the robot’s path and the map can be factorized into a probabilistic path component and an evidential map component. This factorized joint distribution is then approximated using a Rao-Blackwellized particle filter, resulting in an evidential SLAM algorithm that generalizes the popular probabilistic FastSLAM algorithm. The grid maps produced by the algorithm are described by belief functions and thus provide the robot with additional information about the uncertainty in the map. The time complexity of incorporating a new measurement is linear in the number of grid cells and is therefore identical to the complexity