1. Consistent approximations of belief functions Fabio Cuzzolin, Department of Computing, Oxford Brookes University 6 th INTERNATIONAL SYMPOSIUM ON IMPRECISE PROBABILITIES AND THEIR APPLICATIONS, ISIPTA’09 Durham University, UK, July 14-18 2009 Propositional logic Consistent belief functions Geometry of consistency Approximation on a complex Results: binary case Interpretation Perspectives and future research Results: general case in the binary case, the consistent in the binary case, the consistent simplex has 2 components simplex has 2 components using L p norms produces using L p norms produces interesting results for probabilities interesting results for probabilities for probabilities: for probabilities: - L 1 norm → probabilities compatible with b - L 1 norm → probabilities compatible with b - L 2, L  norms → pignistic function - L 2, L  norms → pignistic function for consistent Bfs: for consistent Bfs: - L  norm → convex set of approxs - L  norm → convex set of approxs - L 1, L 2 norms → pointwise approx - L 1, L 2 norms → pointwise approx bottom line: to project a point onto a complex, we need to project it onto all its simplicial components (partial solutions)‏ bottom line: to project a point onto a complex, we need to project it onto all its simplicial components (partial solutions)‏ all partial solutions have later to be compared to find the global optimal solution all partial solutions have later to be compared to find the global optimal solution pointwise approxs have intepretation in terms of degrees of pointwise approxs have intepretation in terms of degrees of belief! belief! focused consistent transformation: mass of A and A  {x} is focused consistent transformation: mass of A and A  {x} is assigned to {x} assigned to {x} corresponds to L 1,2 partial approximation on the component CS x corresponds to L 1,2 partial approximation on the component CS x the partial L 1,2 approxs concide on each component of the the partial L 1,2 approxs concide on each component of the consistent complex consistent complex their b.p.a. is, for all x , their b.p.a. is, for all x , m(A) = m b (A) + m b (A \ {x}) m(A) = m b (A) + m b (A \ {x}) the global L 1 approx is the one associated with the element the global L 1 approx is the one associated with the element x such that: x such that: the global L 2 approx is the one associated with the element the global L 2 approx is the one associated with the element x such that: x such that: description and interpretation ofL  approximations description and interpretation of L  approximations show similarity with credal set of consistent probabilities show similarity with credal set of consistent probabilities consistent probabilities are all probabilities “greater” than b, consistent probabilities are all probabilities “greater” than b, given the usual order relation: p(A)  b(A)  A  given the usual order relation: p(A)  b(A)  A  are the L  credal approximations of b associated with are the L  credal approximations of b associated with some order relation? some order relation? consonant BFs also form a simplicial complex consonant BFs also form a simplicial complex similar treatment of L p consonant approximations? similar treatment of L p consonant approximations? consistent belief functions: counterparts of consistent consistent belief functions: counterparts of consistent knowledge bases of propositional logic knowledge bases of propositional logic turn out to be the class of BFs with non-empty intersection of turn out to be the class of BFs with non-empty intersection of all focal elements all focal elements internal conflict is null iff a BF is consistent internal conflict is null iff a BF is consistent no two complementary propositions receive non-zero no two complementary propositions receive non-zero support under a consistent BF support under a consistent BF Belief logic sets of propositions or formulas are assumed as sets of propositions or formulas are assumed as knowledge bases, and inference is made by knowledge bases, and inference is made by using the operators of the language using the operators of the language consistent knowledge base: no two consistent knowledge base: no two complementary propositions are both supported complementary propositions are both supported each proposition is represented by its set of interpretations each proposition is represented by its set of interpretations we can collect all possible interpretations in a frame we can collect all possible interpretations in a frame... and assign a belief value to each such set... and assign a belief value to each such set → we can assign belief values to propositions → we can assign belief values to propositions consistent belief functions live on a simplicial complex, a consistent belief functions live on a simplicial complex, a structured collection of simplices‏ structured collection of simplices‏ in the binary case, it is formed by two segments in the binary case, it is formed by two segments Why a consistent transform? generic elief functions incorporate generic belief functions incorporate conflicting evidence conflicting evidence consistent Bfs do not support consistent Bfs do not support contradictory propositions contradictory propositions get rid of internal conflict, prior to making get rid of internal conflict, prior to making a decision or taking an action a decision or taking an action