The geometry of of relative plausibilities

Computer Science Department

University of California at Los Angeles

Fabio Cuzzolin

IPMU’06, Paris, July 2-7 2006

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today we’ll be…

…introducing our research

3…presenting the paper

2…the geometric approach to the ToE...

…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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… 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

2Geometry of belief functions

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A

Belief functions

B2

B1

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

AB

BmAb )(

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

Vertices: b.f. assigning 1 to a single set A Coordinates of b in B: m(A)

),( AbClB A

the space of all the belief functions on a given frame is a simplex (ISIPTA’01, submitted to IEEE SMC-C, 2005)

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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

foci of a conditional subspace

(IEEE Trans. SMC-B 2004)

b

b b’b’

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AB

cb BmAbApl )()(1

Relative plausibility

plausibility function plb associated with b

xb

Axb

b xpl

xplAlp

)(

)()(

~ 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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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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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)

3Geometry of the relative plausibility of singletons

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Location

x

xbb bxpllp )(

Plausibility and belief of singletons

x

xbxmb )(

( ) is the intersection of the line joining the vacuous belief function b and the plausibility (belief) of singletons with the Bayesian subspace

b~

blp~

)()(][)()( xmxplbxmx b

1||

1||

||)(

)(

][

A

A

AAm

Am

b

plausibility and belief of singletons intersect P in

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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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Non-Bayesianity flag

yb

b

ymypl

xmxplxbR

)()(

)()()]([

Special conditions on b can be expressed in terms of the “Non-Bayesianity flag”, i.e. the probability

=/2 iff R(x)=1/n;

= 0 iff R || pliff n 2

= 0 iff R = b~

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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,plb) with P

Relative plausibility of singletons

Relative belief of singletons

Relative non-Bayesian contribution R[b] of singletons

2-additivity Equi-distribution

4Perspective: the approximation problem

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),(minarg bpdpPp

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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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

')','(minarg'

dbbpbbdistbBbPp

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

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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

plppb b ~

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 representativesblp~

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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