Department of Engineering Math, University of Bristol A geometric approach to uncertainty Oxford Brookes Vision Group Oxford Brookes University 12/03/2009 Fabio Cuzzolin

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

Examples of belief functions two examples of belief functions on domain of size 4 b 2 ({x 1, x 3 }) = 0; b 1 ({x 1, x 3 })=m 1 ({x 1 }); b 2 ({x 2,x 3,x 4 }) = m 2 ({x 2,x 3,x 4 }); b 1 ({x 2,x 3,x 4 })=0. x1x1 x2x2 x3x3 x4x4 b 1 : m({x 1 })=0.7, m({x 1, x 2 })=0.3 b 2 : m( )=0.1, m({x 2, x 3, x 4 })=0.9

Consistent probabilities interpretation: 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 2 n -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: Dempsters rule

Example of Dempster's sum b 1 : m({x 1 })=0.7, m({x 1, x 2 })=0.3 x1x1 b 1 b 2 : m({x 1 }) = 0.7*0.1/0.37 = 0.19 m({x 2 }) = 0.3*0.9/0.37 = 0.73 m({x 1, x 2 }) = 0.3*0.1/0.37 = 0.08 b 2 : m( )=0.1, m({x 2, x 3, x 4 })=0.9 x2x2 x3x3 x4x4

18 Convex form of Dempster's sum commutes with affine combination can be decomposed in terms of Bayes' rule

Geometry of Dempsters rule Dempsters sum intersection of linear spaces! [IEEE SMC-B04] Possible futures of b conditional subspace b b b other operators have been proposed by Smets, Denoeux etc their 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 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 P intersection 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 sets each 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,T 1,T n-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,T 1 ) relative plausibility = focus of (P,T n-1 ) intersection probability = focus of (T 1,T n- 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 complex consonant 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 L p norms in geometric approach

Projection onto a complex idea: belief function has a partial approximation on all simplicial components of CS global solution = best such approximation b CS x CS y CS z

the binary case again consistent/cons onant approximation are the same in the binary case not so in the general case

Partial L p approximations L 1 = L 2 approximations have a simple interpretation in terms of belief [IEEE TFS07] left: a belief functionright: its consistent approx focused on x m'(A x) = m(A) A

Outer consonant approximations each component of the consonant complex maximal chain of events A 1 A i A n 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 inversion Belief function are sum functions analogous of integral in calculus derivative = Moebius inversion belief function b.b.a. plausibility function commonality function ? ?

Congruent simplices plausibility functions pl(A) live in a simplex too same is true for commonality functions Q(A) geometrically, they form congruent simplices

Equivalent theories they all have a Moebius inverse alternative formulations of the theory can be given in terms of such assignments [IJUFKS07] belief functionb.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