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The geometry of of relative plausibilities Computer Science Department University of California at Los Angeles Fabio Cuzzolin IPMU06, Paris, July 2-7 2006

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

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…the author PhD student, University of Padova, Italy, Department of Information Engineering (NAVLAB laboratory) 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 INRIA – Rhone-Alpes, Grenoble

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

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2 Geometry of belief functions

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6 A Belief functions B2B2 B1B1 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 this induces a belief function, i.e. the total probability function:

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7 Belief space Belief functions can be seen as points of an Euclidean space each subset A A-th coordinate b(A) in an Euclidean space Vertices: b.f. assigning 1 to a single set A Coordinates of b in B: m(A) the space of all the belief functions on a given frame is a simplex (ISIPTA01, submitted to IEEE SMC-C, 2005)

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8 Geometry of Dempsters rule two belief functions can be combined using Dempsters rule Dempsters sum as intersection of linear spaces conditional subspace foci of a conditional subspace (IEEE Trans. SMC-B 2004) b b b

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9 Relative plausibility plausibility function pl b associated with b relative plausibility of singletons it is a probability, i.e. it sums to 1 using the plausibility function one can build a probability by computing the plausibility of singletons

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10 Duality principle belief functions basic probability assignment convex geometry of belief space plausibilities basic plausibility assignment convex geometry of plausibility space

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11 Bayesian relatives of b Belief and plausibility spaces are both simplices Several probability functions related to a given belief function b (submitted to IEEE SMC-B 2005)

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3 Geometry of the relative plausibility of singletons

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13 Location Plausibility and belief of singletons ( ) is the intersection of the line joining the vacuous belief function b and the plausibility (belief) of singletons with the Bayesian subspace plausibility and belief of singletons intersect P in

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14 Plausibility and belief of singletons have corresponding dual quantities the geometry of relative plausibility and associated quantities is a geometry of three planes three angles relate to belief values of b A three-plane geometry

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15 Non-Bayesianity flag Special conditions on b can be expressed in terms of theNon-Bayesianity flag, i.e. the probability = /2 iff R(x)=1/n; = 0 iff R || pl iff n 2 = 0 iff R =

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16 Two families of Bayesian rel. Pignistic function i.e. center of mass of consistent probabilities orthogonal projection of b onto P Intersection sigma of the line (b,pl b ) with P Relative plausibility of singletons Relative belief of singletons Relative non-Bayesian contribution R[b] of singletons 2-additivity Equi-distribution

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4 Perspective: the approximation problem

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18 Approximation problem Probabilistic approximation: finding the probability p which is the closest to a given belief function b Not unique: choice of a criterion Several proposals: pignistic function, orthogonal projection, relative plausibility of singletons

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

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20 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 Fundamental property: the relative plausibility perfectly represents b when combined with another probability using Dempsters rule All the b.f. on the line (b, ) are perfect representatives

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21 4 1 2 3 Conclusions Belief functions as representation of uncertain evidence Geometric approach to the ToE Geometry of the relative plausibility of singletons Relative plausibility as solution of the Dempster-based approximation problem

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