Geometry of Dempster´s rule of combination

In this paper, we analyze Shafer´s belief functions (BFs) as geometric entities, focusing in particular on the geometric behavior of Dempster´s rule of combination in the belief space, i.e., the set S/sub /spl Theta// of all the admissible BFs defined over a given finite domain /spl Theta/. 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 BF s. We show how Dempster´s rule exhibits a rather elegant behavior when applied to BFs assigning the same mass to a fixed subset (constant mass loci). The resulting affine spaces have a common intersection that is characteristic of the conditional subspace, called focus. The affine geometry of these foci eventually suggests an interesting geometric construction of the orthogonal sum of two BFs.