1. On the credal structure of consistent probabilities Department of Computing School of Technology, Oxford Brookes University 19/6/2008 Fabio Cuzzolin

2. Background Masters thesis on gesture recognition at the University of Padova Visiting student, ESSRL, Washington University in St. Louis, and at the University of California at Los Angeles Ph.D. thesis on random sets and uncertainty theory Researcher at Politecnico di Milano with the Image and Sound Processing group Post-doc at the University of California at Los Angeles, UCLA Vision Lab Marie Curie fellow at INRIA Rhone-Alpes Lecturer, Oxford Brookes University

3. My background research Discrete math linear independence on lattices and matroids Uncertainty theory geometric approach algebraic analysis generalized total probability Machine learning Manifold learning for dynamical models Computer vision gesture and action recognition 3D shape analysis and matching Gait ID pose estimation

4. assumption: not enough evidence to determine the actual probability describing the problem second-order distributions (Dirichlet), interval probabilities credal sets Uncertainty measures: Intervals, credal sets Belief functions [Shafer 76]: special case of credal sets a number of formalisms have been proposed to extend or replace classical probability

5. Belief functions as random sets if m is a mass function s.t. A B belief function b:2 s.t. probabilities are additive: if A B= then p(A B)=p(A)+p(B) probability on a finite set: function p: 2 Θ -> [0,1] with p(A)= x m(x), where m: Θ -> [0,1] is a mass function

6. Examples of belief functions Some example of belief functions Finite domain of size 4 b 2 ({a 1, a 3 })=0; b 1 ({a 1, a 3 })=m 1 ({a 1 }); b 2 ({a 2,a 3,a 4 })=m 2 ({a 2,a 3,a 4 }); b 1 ({a 2,a 3,a 4 })=0. b 1 : m({a 1 })=0.7, m({a 1, a 2 })=0.3 a1a1 a2a2 a3a3 a4a4 b 2 : m( )=0.1, m({a 2, a 3, a 4 })=0.9

7. it has the shape of a simplex IEEE Tr. SMC-C '08, Ann. Combinatorics '06, FSS '06, IS '06, IJUFKS'06 Geometric approach to uncertainty belief functions can be seen as points of a Cartesian space of dimension 2 n -2 belief space: the space of all the belief functions on a given frame Each subset is a coordinate in this space

8. how to transform a measure of a certain family into a different uncertainty measure can be done geometrically Approximation problem Probabilities, fuzzy sets, possibilities are special cases of b.f.s

9. Belief functions as credal sets consistent probabilities: the probabilities that can be obtained from b by redistributing the mass of each event for each event A with mass m(A) we assign a fraction x m(A) to each x A turns out to be the probabilities which dominates b

10. Binary example Belief functions on a domain of size 2 are points of R 2 credal set of probabilities consistent with b

11. Example of probability distributions consistent with a belief function Half of {x,y} to x, half to y All of {y,z} to y

12. Classical result the credal set of consistent probs is known to have a quite complex structure it is a polytope, the convex closure of a number of points (distributions) what are those distributions?? if E i ={x 1,...,x m } i=1..n are the focal elements of b, the extremal probs are:

13. Ternary example graphical illustration, domain of size 3

14. Our result the vertices of the credal set of consistent probabilities are associated with permutations of the elements {x 1,...,x n } of the domain consider all possible permutations (x (1),...,x (n) ) the actual vertices of the polytope of consistent probs are: to each element x (i) it assigns the mass of all events A containing it but not its predecessors in the permutation

15. Example again consider a ternary frame {x,y,z} and a belief function with mass: m(x)=0.2, m(y)=0.1, m(z)=0.3, m(x,y)=0.1, m(y,z)=0.2, m(x,y,z)=0.1

16. Conclusions Belief functions as polytopes of probabilities Not all such polytopes (credal sets) are belief functions For b.f.s: vertices are each associated with a permutation of the elements