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Consistent approximations of belief functions
"... Consistent belief functions represent collections of coherent or noncontradictory pieces of evidence. As most operators used to update or elicit evidence do not preserve consistency, the use of consistent transformations cs[·] in a reasoning process to guarantee coherence can be desirable. Such tra ..."
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Consistent belief functions represent collections of coherent or noncontradictory pieces of evidence. As most operators used to update or elicit evidence do not preserve consistency, the use of consistent transformations cs[·] in a reasoning process to guarantee coherence can be desirable. Such transformations are turn linked to the problem of approximating an arbitrary belief function with a consistent one. We study here the consistent approximation problem in the case in which distances are measured using classical Lp norms. We show that, for each choice of the element we want them to focus on, the partial approximations determined by the L1 and L2 norms coincide, and can be interpreted as classical focused consistent transformations. Global L1 and L2 solutions do not in general coincide, however, nor are they associated with the highest plausibility element.
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"... Metric consistent approximations of belief functions We study here the problem of approximating an arbitrary belief function with a consistent one, in a geometric framework in which b.f.s are represented as points of a linear space. We show that the approximations determined by both L1 and L2 norms ..."
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Metric consistent approximations of belief functions We study here the problem of approximating an arbitrary belief function with a consistent one, in a geometric framework in which b.f.s are represented as points of a linear space. We show that the approximations determined by both L1 and L2 norms are unique and coincide, besides having an elegant interpretation in terms of degrees of belief. The L ∞ norms determines an entire polytope of solutions whose barycenter lies on the L1/L2 approximation.
Consistent transformations of belief functions
, 2011
"... Consistent belief functions represent collections of coherent or noncontradictory pieces of evidence, but most of all they are the counterparts of consistent knowledge bases in belief calculus. The use of consistent transformations cs[·] in a reasoning process to guarantee coherence can therefore b ..."
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Consistent belief functions represent collections of coherent or noncontradictory pieces of evidence, but most of all they are the counterparts of consistent knowledge bases in belief calculus. The use of consistent transformations cs[·] in a reasoning process to guarantee coherence can therefore be desirable, and generalizes similar techniques in classical logics. Transformations can be obtained by minimizing an appropriate distance measure between the original belief function and the collection of consistent ones. We focus here on the case in which distances are measured using classical Lp norms, in both the “mass space ” and the “belief space ” representation of belief functions. While mass consistent approximations reassign the mass not focussed on a chosen element of the frame either to the whole frame or to all supersets of the element on an equal basis, approximations in the belief space do distinguish these focal elements according to the “focussed consistent transformation” principle. The different approximations are interpreted and compared, with the help of examples.