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
A geometric approach to uncertainty theory
Uncertainty measures
A geometric approach
Geometry of combination rules
The approximation problem
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
A geometric approach to uncertainty theory
Uncertainty measures
A geometric approach
Geometry of combination rules
The approximation problem
Credal semantics of Bayesian transformations
Complex of consonant and consistent belief functions
Moebius inverses of plaus and common
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?
A geometric approach to uncertainty theory
Uncertainty measures
A geometric approach
Geometry of combination rules
The Bayesian approximation problem
Credal semantics of Bayesian transformations
Complex of consonant and consistent belief functions
Moebius inverses of plaus and common
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
A geometric approach to uncertainty theory
Uncertainty measures
A geometric approach
Geometry of combination rules
The approximation problem
Credal semantics of Bayesian transformations
Complex of consonant and consistent belief functions
Moebius inverses of plaus and common
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
A geometric approach to uncertainty theory
Uncertainty measures
A geometric approach
Geometry of combination rules
The approximation problem
Credal semantics of Bayesian transformations
Complex of consonant and consistent belief functions
Moebius inverses of plaus and common
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
A geometric approach to uncertainty theory
Uncertainty measures
A geometric approach
Geometry of combination rules
The approximation problem
Credal semantics of Bayesian transformations
Complex of consonant and consistent belief functions
Moebius inverses of plaus and common
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