Credal semantics of Bayesian transformations Fabio Cuzzolin, Department of Computing, Oxford Brookes University 6 th INTERNATIONAL SYMPOSIUM ON IMPRECISE PROBABILITIES AND THEIR APPLICATIONS, ISIPTA’09 University of Durham, UK, July Epistemic transformations Probability transformation Credal interpretation of BetP Lower, upper, interval constraints TBM-like frameworks? A unified description the relation between belief functions and probabilities is the relation between belief functions and probabilities is well studied in the ToE well studied in the ToE different goals: efficient combination, decision making different goals: efficient combination, decision making central in Transferable Belief Model: central in Transferable Belief Model: credal representation of evidence credal representation of evidence pignistic probability transform pignistic probability transform some probability transformations seek commutativity with respect some probability transformations seek commutativity with respect to important operators to important operators affine family: transformations which commute with affine affine family: transformations which commute with affine combination in the Cartesian space combination in the Cartesian space epistemic family: transfs commuting with aggregation operators epistemic family: transfs commuting with aggregation operators (Dempster's rule) (Dempster's rule) Their credal representation belief functions have a credal interpretation, as they belief functions have a credal interpretation, as they determine a convex set of (consistent) probability measures determine a convex set of (consistent) probability measures pignistic transform: maps a belief function pignistic transform: maps a belief function to the center of mass of this credal set to the center of mass of this credal set epistemic transformations are in fact transformations of epistemic transformations are in fact transformations of lower, upper, or interval probability systems respectively lower, upper, or interval probability systems respectively lower probability  b lower probability  b upper probability  pl upper probability  pl interval probability  p[b  interval probability  p[b  do they possess a credal interpretation? do they possess a credal interpretation? what about their transformations? what about their transformations?   the lower prob constraint determines a lower simplex T 1 [b] in the lower prob constraint determines a lower simplex T 1 [b] in the simplex of all prob measures the simplex of all prob measures the upper prob constraint determines an upper simplex T n-1 [b] the upper prob constraint determines an upper simplex T n-1 [b] the interval prob system determines the intersection of lower the interval prob system determines the intersection of lower and upper simplices and upper simplices this is in general different from the polytope of probs consistent this is in general different from the polytope of probs consistent with a BF b with a BF b the relative belief transform lies at the intersection of the lines the relative belief transform lies at the intersection of the lines joining corresponding vertices of P and T 1 [b] joining corresponding vertices of P and T 1 [b] the relative plausibility transform lies at the intersection of the the relative plausibility transform lies at the intersection of the lines joining corresponding vertices of P and T n-1 [b] lines joining corresponding vertices of P and T n-1 [b] the intersection probability lies at the intersection of the lines the intersection probability lies at the intersection of the lines joining corresponding vertices of T 1 [b] and T n-1 [b] joining corresponding vertices of T 1 [b] and T n-1 [b] upper, lower, and interval probability systems are all associated upper, lower, and interval probability systems are all associated with pairs of simplices with pairs of simplices the corresponding probability transformations have a similar the corresponding probability transformations have a similar behavior in the probability simplex  focus of a pair of simplices behavior in the probability simplex  focus of a pair of simplices focus: point of the Cartesian plane with the same affine focus: point of the Cartesian plane with the same affine coordinates in the two simplices coordinates in the two simplices t2t2 t1t1 t3t3 s1s1 s2s2 s3s3 T S relative belief = focus of lower simplex and P relative belief = focus of lower simplex and P its affine coordinate is the inverse of the total mass of its affine coordinate is the inverse of the total mass of singletons singletons relative plausibility = focus of upper simplex and P relative plausibility = focus of upper simplex and P its affine coordinate is the inverse of the total plausibility its affine coordinate is the inverse of the total plausibility of singletons of singletons intersection probability = focus of lower and upper simplices intersection probability = focus of lower and upper simplices its affine coordinate is the relative width of the associate its affine coordinate is the relative width of the associate probability interval probability interval pignistic transform is based on a rationality principle AND has a strong pignistic transform is based on a rationality principle AND has a strong credal interpretation credal interpretation same can be said for the epistemic transforms same can be said for the epistemic transforms rationale: the transform has to behave homogeneously rationale: the transform has to behave homogeneously in the two sets of constraints in the two sets of constraints formulation of TBM-like frameworks for lower, upper, or interval formulation of TBM-like frameworks for lower, upper, or interval probability constraints? probability constraints? relative plausibility of singletons relative belief of singletons relative belief of singletons intersection probability intersection probability Epistemic transforms as foci B P b p[b] f(S,T)