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On the properties of relative plausibilities Computer Science Department UCLA Fabio Cuzzolin SMC05, Hawaii, October 10-12 2005

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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

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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 s(A) in an Euclidean space vertices: b.f. assigning 1 to a single set A the space of all the belief functions on a given set has the form of a simplex (submitted to 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) s s t t

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9 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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10 Plausibility space plausibility function associated with s the space of plausibility functions is also a simplex

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3 Relative plausibility and the approximation problem

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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

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

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

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15 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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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 P s * are perfect representatives

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17 1 2 3 Conclusions 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

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01/18 Lab meeting Fabio Cuzzolin

01/18 Lab meeting Fabio Cuzzolin

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