Geometry of Dempster's rule of combination


Fabio Cuzzolin 
IEEE Transactions on Systems, Man, and Cybernetics  part B, 2004 

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 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 , 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. 
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BibTeX Entry 
@article{cuzzolin04smcb,
AUTHOR = "Fabio Cuzzolin",
TITLE = "Geometry of Dempster's rule of combination",
JOURNAL = "IEEE Transactions on Systems, Man, and Cybernetics  part B",
VOLUME = "34",
NUMBER = "2",
PAGES = "961977",
YEAR = "2004"
}

