# The geometry of of relative plausibilities

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The geometry of of relative plausibilities
3/28/2017 The geometry of of relative plausibilities Fabio Cuzzolin Computer Science Department University of California at Los Angeles IPMU’06, Paris, July

1 2 3 today we’ll be… …introducing our research
3/28/2017 today we’ll be… 1 …introducing our research 2 …the geometric approach to the ToE... 3 …presenting the paper

…the author PhD student, University of Padova, Italy, Department of
3/28/2017 …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

… the research research Computer vision Discrete mathematics
3/28/2017 … the research Computer vision object and body tracking data association gesture and action recognition Discrete mathematics linear independence on lattices research Belief functions and imprecise probabilities geometric approach algebraic analysis total belief problem

Geometry of belief functions
3/28/2017 2 Geometry of belief functions

3/28/2017 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 this induces a belief function, i.e the total probability function:

3/28/2017 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 the space of all the belief functions on a given frame is a simplex (ISIPTA’01, submitted to IEEE SMC-C, 2005) Vertices: b.f. assigning 1 to a single set A Coordinates of b in B: m(A)

Geometry of Dempster’s rule
3/28/2017 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 b  b’ b’ b foci of a conditional subspace (IEEE Trans. SMC-B 2004)

Relative plausibility
3/28/2017 Relative plausibility plausibility function plb associated with b using the plausibility function one can build a probability by computing the plausibility of singletons relative plausibility of singletons it is a probability, i.e. it sums to 1

Duality principle plausibilities basic plausibility assignment
3/28/2017 Duality principle plausibilities basic plausibility assignment convex geometry of plausibility space belief functions basic probability assignment convex geometry of belief space

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)

Geometry of the relative plausibility of singletons
3/28/2017 3 Geometry of the relative plausibility of singletons

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

A three-plane geometry
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

Non-Bayesianity flag Special conditions on b can be expressed in terms of the “Non-Bayesianity flag”, i.e. the probability f1 =p/2 iff R(x)=1/n; f2 = 0 iff R || plQ iff n  2 f3 = 0 iff R =

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

Perspective: the approximation problem
3/28/2017 4 Perspective: the approximation problem

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

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 Dempster-based approximation: finding the probability which behaves as the original b.f. when combined using Dempster’s rule

All the b.f. on the line (b, ) are perfect representatives
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 Dempster’s rule All the b.f. on the line (b, ) are perfect representatives

Conclusions 1 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 2 3 4

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