On the properties of relative plausibilities

1. On the properties of relative plausibilities Fabio Cuzzolin Computer Science Department UCLA SMC’05, Hawaii, October 10-12 2005

2. 3 …presenting the paper 2 …the geometric approach to the ToE... today we’ll be… …introducing our research 1

3. PhD student,University of Padova, Italy, Department of Information Engineering (NAVLABlaboratory) • Visiting student, Washington University in St. Louis • Post-doc in Padova, Control and Systems Theory group • Research assistant, Image and Sound Processing Group (ISPG), Politecnico di Milano, Italy • Post-doc, Vision Lab, UCLA, Los Angeles …the author

4. Computer vision Discrete mathematics • object and body tracking • data association • gesture and action recognition • linear independence on lattices Belief functions and imprecise probabilities • geometric approach • algebraic analysis • total belief problem … the research research

5. 2 Geometry of belief functions

6. this induces a belief function, i.e. the total probability function: Belief functions • belief functions are the natural generalization of finite probabilities • Probabilities assign a number (mass) between 0 and 1 to elements of a set  • consider instead a function m assigning masses to the subsets of  A B1 B2

7. the space of all the belief functions on a given set has the form of a simplex (submitted to SMC-C, 2005) Belief space • Belief functions can be seen as points of an Euclidean space • each subset A  A-th coordinate s(A) in an Euclidean space • vertices: b.f. assigning 1 to a single set A

8. Geometry of Dempster’s rule • two belief functions can be combined using Dempster’s rule  • Dempster’s sum as intersection of linear spaces • conditional subspace s  t t s • foci of a conditional subspace • (IEEE Trans. SMC-B 2004)

9. plausibilities • basic plausibility assignment • convex geometry of plausibility space Duality principle • belief functions • basic probability assignment • convex geometry of belief space

10. the space of plausibility functions isalso a simplex Plausibility space • plausibility function associated with s

11. 3 Relative plausibility and the approximation problem

12. Approximation problem • Probabilistic approximation: finding the probability p which is the “closest” to a given belief function s • Not unique: choice of a criterion • Several proposals: pignistic function, orthogonal projection, relative plausibility of singletons

13. Probabilistic approximations • Geometry of the probabilistic region • Several probability functions related to a given belief function s • (submitted to SMC-B 2005)

14. relative plausibility of singletons • it is a probability, i.e. it sums to 1 • Fundamental property: the relative plausibility perfectly represents s when combined with another probability using Dempster’s rule Relative plausibility • using the plausibility function one can build a probability by computing the plausibility of singletons

15. Dempster-based approximation: finding the probability which behaves as the original b.f. when combined using Dempster’s rule Dempster-based criterion • the theory of evidence has two pillars: representing evidence as belief functions, and fusing evidence using Dempster’s rule of combination • Any approximation criterion must encompass both

16. Towards a formal proof • Conjecture: the relative plausibility function is the solution of the Dempster – based approximation problem • This can be proved through geometrical methods • All the b.f. on the line s Ps*are perfect representatives

17. Conclusions 1 • Belief functions as representation of uncertain evidence • Geometric approach to the ToE • Probabilistic approximation problem • Relative plausibility of singletons • Relative plausibility as solution of the approximation problem 2 3