Post on 28-Mar-2015
transcript
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
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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