On the properties of relative plausibilities

Computer Science Department

UCLA

Fabio Cuzzolin

SMC’05, Hawaii, October 10-12 2005

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1

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

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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

BmAs )(

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 s(A) in an Euclidean space

vertices: b.f. assigning 1 to a single set A

),( APClS 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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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)

s

s tt

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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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AB

cs BmAsAP )()(1*

Plausibility space

plausibility function associated with s

the space of plausibility functions is also a simplex

P =(0,0)

=(1,1)

P =(1,0)x

P =(0,1)y

s

Ps*

Ps*~

m

m

3Relative plausibility and the approximation problem

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),(minarg spdpPp

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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Probabilistic approximations

Geometry of the probabilistic region

Several probability functions related to a given belief function s

(submitted to SMC-B 2005)P =(0,0)

=(1,1)

P =(0,1)=y

ps]=s]=Pign(s)

y

P =(1,0)=x x

s

Ps

S

P

m

m

1-m

1-mx y

x

y

p~

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xs

Axs

s xP

xPAP

)(

)()(

~*

*

* 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

pPps s *~

Fundamental property: the relative plausibility perfectly represents s when combined with another probability using Dempster’s rule

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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

dttptsdistsStPp

),(minarg

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

All the b.f. on the line s Ps* are perfect representatives

P =(0,0)

=(1,1)

P =(1,0)x

P =(0,1)y

s

Ps*~

c

t

s'

s cs s'

s t++

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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