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Consistent approximations of belief functions
"... Consistent belief functions represent collections of coherent or non-contradictory 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 non-contradictory 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.
An interpretation of consistent belief functions in terms of simplicial complexes
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Belief functions combination without the assumption of independence of the information sources
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A Generalization of the Pignistic Transform for Partial Bet
"... Abstract. The Transferable Belief Model is a powerful interpretation of belief function theory where decision making is based on the pignistic transform. Smets has proposed a generalization of the pignistic transform which appears to be equivalent to the Shapley value in the transferable utility mod ..."
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Abstract. The Transferable Belief Model is a powerful interpretation of belief function theory where decision making is based on the pignistic transform. Smets has proposed a generalization of the pignistic transform which appears to be equivalent to the Shapley value in the transferable utility model. It corresponds to the situation where the decision maker bets on several hypotheses by associating a subjective probability to non-singleton subsets of hypotheses. Naturally, the larger the set of hypotheses is, the higher the Shapley value is. As a consequence, it is impossible to make a decision based on the comparison of two sets of hypotheses of different size, because the larger set would be promoted. This behaviour is natural in a game theory approach of decision making, but, in the TBM framework, it could be useful to model other kinds of decision processes. Hence, in this article, we propose another generalization of the pignistic transform where the belief in too large focal elements is normalized in a different manner prior to its redistribution. 1
Two k-additive generalizations of the pignistic transform for imprecise decision making
"... The Transferable Belief approach to the Theory of Evidence is based on the pignistic transform which, mapping belief functions to probability distributions, allows to make “precise ” decisions on a set of disjoint hypotheses via classical utility theory. In certain scenarios, however, such as medica ..."
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The Transferable Belief approach to the Theory of Evidence is based on the pignistic transform which, mapping belief functions to probability distributions, allows to make “precise ” decisions on a set of disjoint hypotheses via classical utility theory. In certain scenarios, however, such as medical diagnosis, the need for an “imprecise ” approach to decision making arises, in which sets of possible outcomes are compared. We propose here a framework for imprecise decision derived from the TBM, in which belief functions are mapped to k-additive belief functions (i.e., belief functions whose focal elements have maximal cardinality equal to k) rather than Bayesian ones. We do so by introducing two alternative generalizations of the pignistic transform to the case of k-additive belief functions. The latter has several interesting properties: depending on which properties are deemed the most important, the two distinct generalizations arise. The proposed generalized transforms are empirically validated by applying them to imprecise decision in concrete pattern recognition problems.
