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Geometry of Dempsters rule NAVLAB - Autonomous Navigation and Computer Vision Lab Department of Information Engineering University of Padova, Italy Fabio Cuzzolin FSKD02, Singapore, November 19 2002

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2 1 The talk zintroducing the theory of evidence 2 zpresenting the geometric approach: the belief space 3 zanalyzing the local geometry of Dempsters rule 4 zperspectives of geometric approach

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1 The theory of evidence

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4 zgeneralize classical finite probabilities A Belief functions znormalization B2B2 B1B1 zfocal elements

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5 Dempsters rule zare combined by means of Dempsters rule AiAi BjBj A i B j =A zintersection of focal elements

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2 Geometry of belief functions

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7 zit has the shape of a simplex Belief space zthe space of all the belief functions on a frame zeach subset A A-th coordinate s(A)

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8 Global geometry of zDempsters rule and convex closure commute zconditional subspace: future of s zexample: binary frame ={x,y}

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3 Local geometry of Dempsters rule

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10 Convex form of zDempsters sum of convex combinations zdecomposition in terms of Bayes rule

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11 Local geometry in S 2

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12 Constant mass loci zset of belief functions with equal mass k assigned to a subset A zexpression as convex closure

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13 zintersection of all the subspaces Foci of conditional subspaces zit is an affine subspace zgenerators: focal points

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14 4 …conclusions za new approach to the theory of evidence: the belief space zgeometric behavior of Dempsters rule zapplications: approximation, decomposition, fuzzy measures

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