Geometry of Dempster's rule of combination

IEEE Transactions on Systems, Man, and Cybernetics. Part B, Cybernetics : a Publication of the IEEE Systems, Man, and Cybernetics Society
Fabio Cuzzolin

Abstract

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 Stheta of all the admissible BFs defined over a given finite domain 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 <s>, 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.

Citations

Aug 19, 2007·IEEE Transactions on Systems, Man, and Cybernetics. Part B, Cybernetics : a Publication of the IEEE Systems, Man, and Cybernetics Society·Fabio Cuzzolin
Feb 13, 2015·IEEE Transactions on Cybernetics·Mehena LoudahiOlivier Colot

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