1. The geometry of of relative plausibilities
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The geometry of of relative plausibilities
Fabio Cuzzolin
Computer Science Department
University of California at Los Angeles
IPMU’06, Paris, July

2. 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

3. …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

4. … 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

5. Geometry of belief functions
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Geometry of belief functions

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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:

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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)

8. Geometry of Dempster’s rule
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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)

9. Relative plausibility
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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

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

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)

12. Geometry of the relative plausibility of singletons
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Geometry of the relative plausibility of singletons

13. 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

14. 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

15. 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 =

16. 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

17. Perspective: the approximation problem
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Perspective: the approximation problem

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

19. 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

20. 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

21. 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