On the credal structure of consistent probabilities

Department of Computing

School of Technology, Oxford Brookes University

19/6/2008

Fabio Cuzzolin

Background

Master’s thesis on gesture recognition at the University of Padova

Visiting student, ESSRL, Washington University in St. Louis, and at the University of California at Los Angeles

Ph.D. thesis on random sets and uncertainty theory

Researcher at Politecnico di Milano with the Image and Sound Processing group

Post-doc at the University of California at Los Angeles, UCLA Vision Lab

Marie Curie fellow at INRIA Rhone-Alpes

Lecturer, Oxford Brookes University

My background

research

Discrete math

linear independence on lattices and matroids

Uncertainty theory

geometric approach

algebraic analysis

generalized total probability

Machine learning

Manifold learning for dynamical models

Computer vision gesture and action recognition

3D shape analysis and matching

Gait ID

pose estimation

assumption: not enough evidence to determine the actual probability describing the problem

second-order distributions (Dirichlet), interval probabilities

credal sets

Uncertainty measures: Intervals, credal sets

Belief functions [Shafer 76]: special case of

credal sets

a number of formalisms have been proposed to extend or replace classical probability

Belief functions as random sets

1)( B

Bmif m is a mass function s.t.

AB

BmAb )(

A

B• belief function b:2

s.t.

• probabilities are additive: if AB= then p(AB)=p(A)+p(B)

• probability on a finite set: function p: 2Θ -> [0,1] with

p(A)=x m(x), where m: Θ -> [0,1] is a mass function

Examples of belief functions

Some example of belief functions

Finite domain of size 4

b2({a1,a3})=0; b1({a1,a3})=m1({a1});

b2({a2,a3,a4})=m2({a2,a3,a4}); b1({a2,a3,a4})=0.

• b1:

m({a1})=0.7, m({a1 ,a2})=0.3

a1

a2

a3

a4

• b2:

m()=0.1, m({a2 ,a3 ,a4})=0.9

it has the shape of a simplex

IEEE Tr. SMC-C '08, Ann. Combinatorics '06, FSS '06, IS '06, IJUFKS'06

Geometric approach to uncertainty

belief functions can be seen as points of a Cartesian space of dimension 2n-2 belief space: the space of all the belief functions on a given frame

Each subset is a coordinate in this space

how to transform a measure of a certain family into a different uncertainty measure → can be done geometrically

Approximation problem

Probabilities, fuzzy sets, possibilities are special cases of b.f.s

Belief functions as credal sets

consistent probabilities: the probabilities that can be obtained from b by redistributing the mass of each event

for each event A with mass m(A)

we assign a fraction xm(A) to each xA

turns out to be the probabilities which dominates b

Binary example

Belief functions on a domain of size 2 are points of R2

credal set of probabilities consistent with b

Example

of probability distributions consistent with a belief function

Half of {x,y} to x, half to y

All of {y,z} to y

Classical result

the credal set of consistent probs is known to have a quite complex structure

it is a polytope, the convex closure of a number of points (distributions)

what are those distributions??

if Ei={x1,...,xm} i=1..n are the focal elements of b, the extremal probs are:

Ternary example

graphical illustration, domain of size 3

Our result

the vertices of the credal set of consistent probabilities are associated with permutations of the elements {x1,...,xn} of the domain

consider all possible permutations (x(1),...,x(n))

the actual vertices of the polytope of consistent probs are:

to each element x(i) it assigns the mass of all events A containing it but not its predecessors in the permutation

Example again

consider a ternary frame {x,y,z}

and a belief function with mass:

m(x)=0.2, m(y)=0.1, m(z)=0.3,

m(x,y)=0.1, m(y,z)=0.2, m(x,y,z)=0.1

Conclusions

Belief functions as polytopes of probabilities

Not all such polytopes (“credal sets”) are belief functions

For b.f.s: vertices are each associated with a permutation of the elements