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Department of Engineering Math, University of Bristol A geometric approach to uncertainty Oxford Brookes Vision Group Oxford Brookes University 12/03/2009 Fabio Cuzzolin Slide 2 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 Slide 3 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 Slide 4 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 Slide 5 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 Slide 6 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 Slide 7 Belief functions as sum functions if m: 2 -> [0,1] is a mass function s.t. the belief value of an event A is m: -> [0,1] is a mass function s.t. probability function p: 2 -> [0,1] the probability value of an event A is Slide 8 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 Slide 9 18 Convex form of Dempster's sum commutes with affine combination can be decomposed in terms of Bayes' rule Slide 19 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? Slide 20 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 Slide 21 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 Slide 22 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 Slide 23 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] Slide 24 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) Slide 25 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 Slide 26 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 Slide 27 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 Slide 28 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 Slide 29 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] Slide 30 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 Slide 31 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 Slide 32 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 Slide 33 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 Slide 34 Consonant complex consonant belief functions form a complex [FSS08] examples: the binary and ternary cases Slide 35 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 Slide 36 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 Slide 37 the binary case again consistent/cons onant approximation are the same in the binary case not so in the general case Slide 38 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 Slide 39 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 Slide 40 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 Slide 41 Moebius inversion Belief function are sum functions analogous of integral in calculus derivative = Moebius inversion belief function b.b.a. plausibility function commonality function ? ? Slide 42 Congruent simplices plausibility functions pl(A) live in a simplex too same is true for commonality functions Q(A) geometrically, they form congruent simplices Slide 43 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. Slide 44 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