department of engineering math, university of bristol a geometric approach to uncertainty oxford...

Department of Engineering Math, University of Bristol

A geometric approach to uncertainty

Oxford Brookes Vision Group

Oxford Brookes University12/03/2009

Fabio Cuzzolin

My path

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

Visiting student, ESSRL, Washington University in St. Louis

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

A geometric approach to uncertainty theory

Uncertainty measures

A geometric approach

Geometry of combination rules

Simplex of probability measures

Complex of possibility measures

The approximation problem

Generalized total probability

Vision applications and developments

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

Multi-valued maps and belief functions

suppose you have two different but related problems ...

... that we have a probability distribution for the first one

... and that the two are linked by a map one to many

[Dempster'68, Shafer'76]

the probability P on S induces a belief function

on T

Belief functions as sum functions

1)( B

Bm• if m: 2Θ -> [0,1] is a mass function s.t.

• the belief value of an event A is

AB

BmAb )(

• m: Θ -> [0,1] is a mass function s.t.• probability function p: 2Θ -> [0,1]• the probability value of an event A is

1)( x

xm

Ax

xmAp )(

Examples of belief functions

two examples of belief functions on domain of size 4

b2({x1,x3}) = 0; b1({x1,x3})=m1({x1});

b2({x2,x3,x4}) = m2({x2,x3,x4}); b1({x2,x3,x4})=0.

x1 x

2

x3

x4

• b1:

m({x1})=0.7, m({x1 ,x2})=0.3

• b2:

m()=0.1, m({x2 ,x3 ,x4})=0.9

Consistent probabilitiesinterpretation: mass m(A) can “float” inside A

Ex of probabilities consistent with a belief function

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

all of {y,z} to y

10

Two equivalent formulations

belief function b(A)

is the lower bound to the probability of A for a probability consistent with b

plausibility function pl(A)

is the upper bound to the probability of A for a consistent probability






Credal semantics of Bayesian transformations

Complex of consonant and consistent belief functions

Moebius inverses of plaus and common

Vision applications

Belief functions as points

if n=||=2, a belief function b is specified by b(x) and b(y) as for all bfs, b()=0 and b()=1 belief functions can be seen as points of a Cartesian space of dimension 2n-2

Belief functions as credal sets

a belief function can be seen as the lower bound to a convex set of “consistent” probabilities

it has the shape of a simplex

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

A geometric approach to uncertainty

belief space: the space of all the belief functions on a given frame









Vision applications

Dempster's sum

comes from the original definition in terms of multi-valued maps

assumes conditional independence of the bodies of evidence

several aggregation or elicitation operators proposed

• original proposal: Dempster’s rule

Example of Dempster's sum

• b1:

m({x1})=0.7, m({x1 ,x2})=0.3

x1

• b1 b2 :

m({x1}) = 0.7*0.1/0.37 = 0.19

m({x2}) = 0.3*0.9/0.37 = 0.73

m({x1 ,x2}) = 0.3*0.1/0.37 = 0.08

• b2:

m()=0.1, m({x2 ,x3 ,x4})=0.9

x2

x3

x4

18

Convex form of

i

iii

ii bbbb

jjj

iii

AA

A

Bbb

bb bbBplBm

AplAmbb

, '

'

)()(

)()('

• Dempster's sum commutes with affine combination

can be decomposed in terms of Bayes' rule

Geometry of Dempster’s rule

Dempster’s sum <-> intersection of linear spaces!

[IEEE SMC-B04]

Possible “futures” of b conditional subspace

b

b b’b’

other operators have been proposed by Smets, Denoeux etctheir geometric behavior? commutativity with respect to affine operator?





The Bayesian approximation problem




Vision applications

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 all special cases of b.f.s

IEEE Tr. SMC-B '07, IEEE Tr. Fuzzy Systems '07, AMAI '08, AI '08, IEEE Tr. Fuzzy Systems '08

22

),(minarg bpdpPp

Probability transformations

finding the probability which is the “closest” to a given belief function

different criteria can be chosen

pignistic function

• relative plausibility of singletons

23

Geometric approximations

the approximation problem can be posed in the geometric approach [IEEE SMC-B07]

pignistic function BetP as barycenter of P[b]

• orthogonal projection [b]

intersection probability p[b]

24

Intersection probability

it is derived from geometric arguments [IEEE SMC-B07]

but is inherently associated with probability intervals

it is the unique probability such that [AIJ08]

p(x) = b(x) + (pl(x) - b(x))

b(x) pl(x) b(y) pl(y) b(z) pl(z)

25

Two families of probability transformations (or three..)

• Pignistic function i.e. center of mass of consistent probabilities

• orthogonal projection of b onto Pintersection probability

• Relative plausibility of singletons

• Relative belief of singletons [IEEE TFS08]

• Relative uncertainty of singletons [AMAI08]

commute with affine

combination

commute with Dempster's combination









Vision applications

Interval probs as credal setseach belief function “is” a credal set

each belief function is also associated with an interval probability

interval probabilities correspond to credal sets too

“lower” and “upper” simplices

Focus of a pair of simplices

different Bayesian transformations can be seen as foci of a pair of simplices among (P,T1,Tn-1)

• focus = point with the same simplicial coordinates in the two simplices

when interior is the intersection of the lines joining corresponding vertices

Bayesian transformations as foci

relative belief = focus of (P,T1)

relative plausibility = focus of (P,Tn-1)

intersection probability = focus of (T1,Tn-1)

[IEEE SMC-B08]

TBM-like frameworks

Transferable Belief Model: belief are represented as credal sets, decisions made after pignistic transformation [Smets]

reasoning frameworks similar to the TBM can be imagined ...

... in which upper, lower, and interval constraints are repr. as credal sets ...

... while decisions are made after appropriate transformation









Vision applications

Consonant belief functions

focal elements = non-zero mass events

consonant belief function = belief function whose focal elements are nested

they correspond to possibility

measures

they correspond to fuzzy sets

Simplicial complexes

simplicial complex = structured collection of simplices

All faces of a simplex belong to the complex

Pairs of simplices intersect in their faces only

Consonant complexconsonant belief functions form a complex [FSS08]

examples: the binary and ternary cases

Consistent belief functions

consistent belief function = belief function whose focal elements have non-empty intersection

consonant bfs <-> possibility/necessity measures

consistent bfs <-> possibility distributions

they possess the same geometry in terms of complexes

consistent approximation → allows to preserve consistency of the body of evidence [IEEE TFS07]

can be done using Lp norms in geometric approach

Projection onto a complexidea: belief function has a partial approximation on all simplicial components of CS

global solution = best such approximation

b CSxCSy

CSz

the binary case again

consistent/consonant approximation are the same in the binary case

not so in the general case

Partial Lp

approximationsL1 = L2 approximations have a simple interpretation in terms of belief [IEEE TFS07]

left: a belief function right: its consistent approx

focused on x

m'(Ax) = m(A) A

Outer consonant approximations

each component of the consonant complex → maximal chain of events A1 Ai An

in each component of the complex consonant approxs form a simplex

[FSS08]

each “vertex” is obtained by re-assigning the mass of each event to an element of the chain

Moebius inversionBelief function are sum functions

analogous of integral in calculus

derivative = Moebius inversion

belief function

b.b.a.

plausibility function

commonality function

?

?

Congruent simplicesplausibility functions pl(A) live in a simplex too

same is true for commonality functions Q(A)

geometrically, they form congruent

simplices

Equivalent theoriesthey all have a Moebius inverse

alternative formulations of the theory can be given in terms of such assignments [IJUFKS07]

belief function

b.b.a.

plausibility function

commonality function

b.pl.a.

b.comm.a.

Conclusions

uncertainty measures of different classes can be represented as points of a Cartesian space

evidence aggregation/revision operators can be analyzed

the crucial approximation problem can be posed in a geometric setup

geometric quantities and loci have an epistemic interpretation!

extension to continuous case: random sets, gambles

behavior of aggregation operators: conjunctive/disjunctive rule, t-norms, natural extension

department of engineering math, university of bristol a geometric approach to uncertainty oxford...

Documents

classical probability

t slide

probability function

probability intervals

probability value

estimation slide

developments slide

uncertainty theory uncertainty