Research Theme: The Geometry of Uncertainty








Belief Space: The space of belief functions
Geometry of Dempster's rule
Fiber bundle structure of the belief space
My major contribution in the field of uncertainty measures is a geometric approach to uncertainty theory and generalized probabilities, which uncertainty measures of different nature (probabilities, possibilities, belief functions, random sets) are represented as points of a Cartesian space and there analyzed. In more detail, I studied the geometry of the region where different classes of measures live in terms of the notion of structured collections of simplices or 'simplicial complexes'. In the simplest case, probability distribution live in a higher-dimensional triangle, a 'simplex'. Many fundamental problems can be solved in this framework by geometric means: how to approximate a belief function or a random set with a probability, how to compute distances between different uncertainty measures, etcetera.
Dempster's rule of combination, Shafer's original proposal for updating in the theory of belief functions, can also be studied from a geometric point of view.
The study of the orthogonal sums of affine subspaces allows us to unveil a convex decomposition of Dempster's rule of combination in terms of Bayes' rule of conditioning and prove that under specific conditions orthogonal sum and affine closure commute. A direct consequence of these results is the simplicial shape of the conditional subspaces, i.e., the sets of all the possible combinations of a given belief function.
Dempster's rule exhibits a rather elegant behavior when applied to belief functions assigning the same mass to a fixed subset. The resulting affine spaces have a common intersection that is characteristic of the conditional subspace, that we call focus. The affine geometry of these foci eventually suggests an interesting geometric construction of the orthogonal sum of two belief functions.
The study of finite non-additive measures or belief functions has been recently posed in connection with combinatorics and convex geometry. As a matter of fact, as belief functions are completely specified by the associated belief values on the events of the frame on which they are defined, they can be represented as points of a Cartesian space. The space of all belief functions B or "belief space" is a simplex whose vertices are BF focused on single events.
An alternative description of the space of belief functions in terms of differential geometry can also be proposed. The belief space possesses indeed a recursive bundle structure inherently related to the mass assignment mechanism, in which basic probability is recursively assigned to events of increasing size. A formal proof of the decomposition of B together with a characterization of bases and fibers as simplices are provided.