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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.
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.
Supremum Preserving Upper Probabilities
, 1998
"... We study the relation between possibility measures and the theory of imprecise probabilities, and argue that possibility measures have an important part in this theory. It is shown that a possibility measure is a coherent upper probability if and only if it is normal. A detailed comparison is giv ..."
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Cited by 39 (12 self)
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We study the relation between possibility measures and the theory of imprecise probabilities, and argue that possibility measures have an important part in this theory. It is shown that a possibility measure is a coherent upper probability if and only if it is normal. A detailed comparison is given between the possibilistic and natural extension of an upper probability, both in the general case and for upper probabilities dened on a class of nested sets. We prove in particular that a possibility measure is the restriction to events of the natural extension of a special kind of upper probability, dened on a class of nested sets. We show that possibilistic extension can be interpreted in terms of natural extension. We also prove that when either the upper or the lower cumulative distribution function of a random quantity is specied, possibility measures very naturally emerge as the corresponding natural extensions. Next, we go from upper probabilities to upper previsions...
A behavioural model for vague probability assessments
, 2003
"... I present an hierarchical uncertainty model that is able to represent vague probability assessments, and to make inferences based on them. This model can be given an interpretation in terms of the behaviour of a modeller in the face of uncertainty, and is based on Walley’s theory of imprecise proba ..."
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Cited by 16 (1 self)
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I present an hierarchical uncertainty model that is able to represent vague probability assessments, and to make inferences based on them. This model can be given an interpretation in terms of the behaviour of a modeller in the face of uncertainty, and is based on Walley’s theory of imprecise probabilities. It is formally closely related to Zadeh’s fuzzy probabilities, but it has a different interpretation, and a different calculus. Through rationality (coherence) arguments, the hierarchical model is shown to lead to an imprecise firstorder uncertainty model that can be used in decision making, and as a prior in statistical reasoning.
nmonotone exact functionals
, 2006
"... We study nmonotone functionals, which constitute a generalisation of nmonotone set functions. We investigate their relation to the concepts of exactness and natural extension, which generalise the notions of coherence and natural extension in the behavioural theory of imprecise probabilities. We ..."
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Cited by 13 (11 self)
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We study nmonotone functionals, which constitute a generalisation of nmonotone set functions. We investigate their relation to the concepts of exactness and natural extension, which generalise the notions of coherence and natural extension in the behavioural theory of imprecise probabilities. We improve upon a number of results in the literature, and prove among other things a representation result for exact nmonotone functionals in terms of Choquet integrals.
Generalized Information Theory for Engineering Modeling and Simulation
, 2003
"... this paper, we survey some of the most prominent of the GIT mathematical formalisms in the context of the classical approaches, including probability theory itself. Our emphasis will be primarily on introducing the formal specifications of a range of theories, although we will also take some time to ..."
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Cited by 8 (1 self)
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this paper, we survey some of the most prominent of the GIT mathematical formalisms in the context of the classical approaches, including probability theory itself. Our emphasis will be primarily on introducing the formal specifications of a range of theories, although we will also take some time to discuss semantics, applications, and implementations
Approximations of upper and lower probabilities by measurable selections
"... A random set can be regarded as the result of the imprecise observation of a random variable. Following this interpretation, we study to which extent the upper and lower probabilities induced by the random set keep all the information about the values of the probability distribution of the random va ..."
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Cited by 3 (2 self)
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A random set can be regarded as the result of the imprecise observation of a random variable. Following this interpretation, we study to which extent the upper and lower probabilities induced by the random set keep all the information about the values of the probability distribution of the random variable. We link this problem to the existence of selectors of a multivalued mapping and with the inner approximations of the upper probability, and prove that under fairly general conditions (although not in all cases), the upper and lower probabilities are an adequate tool for modelling the available information. In doing this, we generalise a number of results from the literature. Finally, we study the particular case of consonant random sets and we also derive a relationship between Aumann and Choquet integrals.
Integration in Possibility Theory
 INTERNATIONAL JOURNAL OF GENERAL SYSTEMS
, 1997
"... The paper discusses integration in possibility theory, both in an ordinal and in a numerical (behavioral) context. It is shown that in an ordinal context, the fuzzy integral has an important part in at least three areas: the extension of possibility measures to larger domains, the construction of p ..."
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Cited by 2 (0 self)
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The paper discusses integration in possibility theory, both in an ordinal and in a numerical (behavioral) context. It is shown that in an ordinal context, the fuzzy integral has an important part in at least three areas: the extension of possibility measures to larger domains, the construction of product measures from marginals and the denition of conditional possibilities. In a numerical (behavioral) context, integration can be used to extend upper probabilities to upper previsions. It is argued that the role of the fuzzy integral in this context is limited, as it can only be used to dene a coherent upper prevision if the associated upper probability is 01valued, in which case it moreover coincides with the Choquet integral. These results are valid for arbitrary coherent upper probabilities, and therefore also relevant for possibility theory. It follows from the discussion that in a numerical context, the Choquet integral is better suited than the fuzzy integral for producing coherent upper previsions starting from possibility measures. At the same time, alternative expressions for the Choquet integral associated with a possibility measure are derived.