Geometry of Dempster’s rule

1. Geometry of Dempster’s rule Fabio Cuzzolin NAVLAB - Autonomous Navigation and Computer Vision Lab Department of Information Engineering University of Padova, Italy FSKD’02, Singapore, November 19 2002

2. 2 • presenting the geometric approach: the belief space 3 • analyzing the local geometry of Dempster’s rule 4 • perspectives of geometric approach The talk 1 • introducing the theory of evidence

3. 1 The theory of evidence

4. generalize classical finite probabilities A B1 B2 • focal elements • normalization Belief functions

5. are combined by means of Dempster’s rule AiÇBj=A Ai • intersection of focal elements Bj Dempster’s rule

6. 2 Geometry of belief functions

7. it has the shape of a simplex Belief space • the space of all the belief functions on a frame  • each subset A  A-th coordinate s(A)

8. example: binary frame ={x,y} Global geometry of  • Dempster’s rule and convex closure commute • conditional subspace: “future” of s

9. 3 Local geometry of Dempster’s rule

10. Dempster’s sum of convex combinations • decomposition in terms of Bayes’ rule Convex form of 

11. Local geometry in S2

12. set of belief functions with equal mass k assigned to a subset A • expression as convex closure Constant mass loci

13. intersection of all the subspaces • it is an affine subspace • generators: focal points Foci of conditional subspaces

14. …conclusions 4 • a new approach to the theory of evidence: the belief space • geometric behavior of Dempster’s rule • applications: approximation, decomposition, fuzzy measures