1. 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

2. 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

3. 1 The theory of evidence

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

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

6. 2 Geometry of belief functions

7. 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)

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

9. 3 Local geometry of Dempsters rule

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

11. 11 Local geometry in S 2

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

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

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