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115
The Transferable Belief Model
 ARTIFICIAL INTELLIGENCE
, 1994
"... We describe the transferable belief model, a model for representing quantified beliefs based on belief functions. Beliefs can be held at two levels: (1) a credal level where beliefs are entertained and quantified by belief functions, (2) a pignistic level where beliefs can be used to make decisions ..."
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Cited by 486 (15 self)
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We describe the transferable belief model, a model for representing quantified beliefs based on belief functions. Beliefs can be held at two levels: (1) a credal level where beliefs are entertained and quantified by belief functions, (2) a pignistic level where beliefs can be used to make decisions and are quantified by probability functions. The relation between the belief function and the probability function when decisions must be made is derived and justified. Four paradigms are analyzed in order to compare Bayesian, upper and lower probability, and the transferable belief approaches.
Possibility Theory as a Basis for Qualitative Decision Theory
, 1995
"... A counterpart to von Neumann and Morgenstern' expected utility theory is proposed in the framework of possibility theory. The existence of a utility function, representing a preference ordering among possibility distributions (on the consequences of decisionmaker's actions) that satisfies ..."
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Cited by 127 (29 self)
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A counterpart to von Neumann and Morgenstern' expected utility theory is proposed in the framework of possibility theory. The existence of a utility function, representing a preference ordering among possibility distributions (on the consequences of decisionmaker's actions) that satisfies a series of axioms pertaining to decisionmaker's behavior, is established. The obtained utility is a generalization of Wald's criterion, which is recovered in case of total ignorance; when ignorance is only partial, the utility takes into account the fact that some situations are more plausible than others. Mathematically, the qualitative utility is nothing but the necessity measure of a fuzzy event in the sense of possibility theory (a socalled Sugeno integral). The possibilistic representation of uncertainty, which only requires a linearly ordered scale, is qualitative in nature. Only max, min and orderreversing operations are used on the scale. The axioms express a riskaverse behavior of the d...
Toward normative expert systems: part I. The Pathfinder project
 12 The Scientific World Journal Methods of Information in Medicine
, 1992
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Decision Making in the TBM: the Necessity of the Pignistic Transformation
, 2004
"... In the transferable belief model(TBM), pignistic probabilities are used for decision making. The nature of the pignistic transformation is justified by a linearity requirement. We justify the origin of this requirement showing it is not ad hoc but unavoidable provides one accepts expected utility th ..."
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Cited by 93 (1 self)
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In the transferable belief model(TBM), pignistic probabilities are used for decision making. The nature of the pignistic transformation is justified by a linearity requirement. We justify the origin of this requirement showing it is not ad hoc but unavoidable provides one accepts expected utility theory.
Probabilitypossibility transformations, triangular fuzzy sets and probabilistic inequalities
 Reliable Computing
, 2004
"... Abstract. A possibility measure can encode a family of probability measures. This fact is the basis for a transformation of a probability distribution into a possibility distribution that generalises the notion of best interval substitute to a probability distribution with prescribed confidence. Thi ..."
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Cited by 73 (25 self)
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Abstract. A possibility measure can encode a family of probability measures. This fact is the basis for a transformation of a probability distribution into a possibility distribution that generalises the notion of best interval substitute to a probability distribution with prescribed confidence. This paper describes new properties of this transformation, by relating it with the wellknown probability inequalities of BienayméChebychev and CampMeidel. The paper also provides a justification of symmetric triangular fuzzy numbers in the spirit of such inequalities. It shows that the cuts of such a triangular fuzzy number contains the “confidence intervals ” of any symmetric probability distribution with the same mode and support. This result is also the basis of a fuzzy approach to the representation of uncertainty in measurement. It consists in representing measurements by a family of nested intervals with various confidence levels. From the operational point of view, the proposed representation is compatible with the recommendations of the ISO Guide for the expression of uncertainty in physical measurement. 1.
On Possibility/Probability Transformations
 Proceedings of Fourth IFSA Conference
, 1993
"... this paper that the probabilistic representations and the possibilistic ones are not just two equivalent representations of uncertainty. Hence there should be no symmetry between the two mutual conversion procedures. The possibilistic representation is weaker because it explicitly handles imprecisio ..."
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Cited by 65 (11 self)
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this paper that the probabilistic representations and the possibilistic ones are not just two equivalent representations of uncertainty. Hence there should be no symmetry between the two mutual conversion procedures. The possibilistic representation is weaker because it explicitly handles imprecision (e.g. incomplete knowledge) and because possibility measures are based on an ordering structure rather than an additive one. Turning a probability measure into a possibility measure may be useful in the presence of other weak sources of information, or when computing with possibilities is simpler than computing with probabilities. Turning a possibility measure into a probability measure might be of interest in the scope of decisionmaking (Smets, 1990). The next section suggests that the transformations should be guided by two different information principles : the principle of insufficient reason from possibility to probability, and the principle of maximum specificity from probability to possibility. The first principle aims at finding a probability measure which preserves the uncertainty of choice between outcomes, while the second principle aims at finding the most informative possibility distribution, under the constraints dictated by the possibility/probability consistency principle. The paper then proposes two transformations that obey these principles. In the discrete case they are already known. But here, results in the continuous case are given. It is pointed out that these transformations are not related to each other, and the converse transformations are shown to be inadequate. In the last section we discuss the relationship between our approach and other works pertaining to the same topic. Some lines of research are considered.
Possibility theory and statistical reasoning
 Computational Statistics & Data Analysis Vol
, 2006
"... Numerical possibility distributions can encode special convex families of probability measures. The connection between possibility theory and probability theory is potentially fruitful in the scope of statistical reasoning when uncertainty due to variability of observations should be distinguished f ..."
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Cited by 59 (4 self)
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Numerical possibility distributions can encode special convex families of probability measures. The connection between possibility theory and probability theory is potentially fruitful in the scope of statistical reasoning when uncertainty due to variability of observations should be distinguished from uncertainty due to incomplete information. This paper proposes an overview of numerical possibility theory. Its aim is to show that some notions in statistics are naturally interpreted in the language of this theory. First, probabilistic inequalites (like Chebychev’s) offer a natural setting for devising possibility distributions from poor probabilistic information. Moreover, likelihood functions obey the laws of possibility theory when no prior probability is available. Possibility distributions also generalize the notion of confidence or prediction intervals, shedding some light on the role of the mode of asymmetric probability densities in the derivation of maximally informative interval substitutes of probabilistic information. Finally, the simulation of fuzzy sets comes down to selecting a probabilistic representation of a possibility distribution, which coincides with the Shapley value of the corresponding consonant capacity. This selection process is in agreement with Laplace indifference principle and is closely connected with the mean interval of a fuzzy interval. It sheds light on the “defuzzification ” process in fuzzy set theory and provides a natural definition of a subjective possibility distribution that sticks to the Bayesian framework of exchangeable bets. Potential applications to risk assessment are pointed out. 1
Fuzzy sets and probability : Misunderstandings, bridges and gaps
 In Proceedings of the Second IEEE Conference on Fuzzy Systems
, 1993
"... This paper is meant to survey the literature pertaining to this debate, and to try to overcome misunderstandings and to supply access to many basic references that have addressed the "probability versus fuzzy set" challenge. This problem has not a single facet, as will be claimed here. Mor ..."
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Cited by 59 (6 self)
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This paper is meant to survey the literature pertaining to this debate, and to try to overcome misunderstandings and to supply access to many basic references that have addressed the "probability versus fuzzy set" challenge. This problem has not a single facet, as will be claimed here. Moreover it seems that a lot of controversies might have been avoided if protagonists had been patient enough to build a common language and to share their scientific backgrounds. The main points made here are as follows. i) Fuzzy set theory is a consistent body of mathematical tools. ii) Although fuzzy sets and probability measures are distinct, several bridges relating them have been proposed that should reconcile opposite points of view ; especially possibility theory stands at the crossroads between fuzzy sets and probability theory. iii) Mathematical objects that behave like fuzzy sets exist in probability theory. It does not mean that fuzziness is reducible to randomness. Indeed iv) there are ways of approaching fuzzy sets and possibility theory that owe nothing to probability theory. Interpretations of probability theory are multiple especially frequentist versus subjectivist views (Fine [31]) ; several interpretations of fuzzy sets also exist. Some interpretations of fuzzy sets are in agreement with probability calculus and some are not. The paper is structured as follows : first we address some classical misunderstandings between fuzzy sets and probabilities. They must be solved before any discussion can take place. Then we consider probabilistic interpretations of membership functions, that may help in membership function assessment. We also point out nonprobabilistic interpretations of fuzzy sets. The next section examines the literature on possibilityprobability transformati...
ON THE PLAUSIBILITY TRANSFORMATION METHOD FOR TRANSLATING BELIEF FUNCTION MODELS TO PROBABILITY MODELS
, 2006
"... In this paper, we propose the plausibility transformation method for translating DempsterShafer (DS) belief function models to probability models, and describe some of its properties. There are many other transformation methods used in the literature for translating belief function models to proba ..."
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Cited by 47 (1 self)
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In this paper, we propose the plausibility transformation method for translating DempsterShafer (DS) belief function models to probability models, and describe some of its properties. There are many other transformation methods used in the literature for translating belief function models to probability models. We argue that the plausibility transformation method produces probability models that are consistent with DS semantics of belief function models, and that, in some examples, the pignistic transformation method produces results that appear to be inconsistent with Dempster’s rule of combination.
New Semantics For Quantitative Possibility Theory
 2ND INTERNATIONAL SYMPOSIUM ON IMPRECISE PROBABILITIES AND THEIR APPLICATIONS, ITHACA, NEW YORK
, 2001
"... New semantics for numerical values given to possibility measures are provided. For epistemic possibilities, the new approach is based on the semantics of the transferable belief model, itself based on betting odds interpreted in a less drastic way than what subjective probabilities presupposes. It i ..."
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Cited by 40 (5 self)
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New semantics for numerical values given to possibility measures are provided. For epistemic possibilities, the new approach is based on the semantics of the transferable belief model, itself based on betting odds interpreted in a less drastic way than what subjective probabilities presupposes. It is shown that the least informative among the belief structures that are compatible with prescribed betting rates is nested, i.e. corresponds to a possibility measure. It is also proved that the idempotent conjunctive combination of two possibility measures corresponds to the hypercautious conjunctive combination of the belief functions induced by the possibility measures. This view di#ers from the subjective semantics first proposed by Giles and relying on upper and lower probability induced by nonexchangeable bets. For objective possibility degrees, the semantics is based on the most informative possibilistic approximation of a probability measure derived from a histogram. The motivation for this semantics is its capability to extend a wellknown kind of confidence intervals around the mode of a distribution to a fuzzy confidence interval. We show how the idempotent disjunctive combination of possibility functions is related to the convex mixture of probability distributions.