1 / 17

On the properties of relative plausibilities

On the properties of relative plausibilities. Fabio Cuzzolin. Computer Science Department UCLA. SMC’05, Hawaii, October 10-12 2005. 3. …presenting the paper. 2. …the geometric approach to the ToE. today we’ll be…. …introducing our research. 1.

Download Presentation

On the properties of relative plausibilities

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related