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12
Towards a unified theory of imprecise probability
 Int. J. Approx. Reasoning
, 2000
"... Belief functions, possibility measures and Choquet capacities of order 2, which are special kinds of coherent upper or lower probability, are amongst the most popular mathematical models for uncertainty and partial ignorance. I give examples to show that these models are not sufficiently general to ..."
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Cited by 40 (0 self)
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Belief functions, possibility measures and Choquet capacities of order 2, which are special kinds of coherent upper or lower probability, are amongst the most popular mathematical models for uncertainty and partial ignorance. I give examples to show that these models are not sufficiently general to represent some common types of uncertainty. Coherent lower previsions and sets of probability measures are considerably more general but they may not be sufficiently informative for some purposes. I discuss two other models for uncertainty, involving sets of desirable gambles and partial preference orderings. These are more informative and more general than the previous models, and they may provide a suitable mathematical setting for a unified theory of imprecise probability.
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 38 (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...
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 26 (2 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
A Random Set Description of a Possibility Measure and Its Natural Extension
 IEEE Transactions on Systems, Man and Cybernetics
, 1997
"...  The relationship is studied between possibility and necessity measures dened on arbitrary spaces, the theory of imprecise probabilities, and elementary random set theory. It is shown how special random sets can be used to generate normal possibility and necessity measures, as well as their natural ..."
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Cited by 17 (7 self)
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 The relationship is studied between possibility and necessity measures dened on arbitrary spaces, the theory of imprecise probabilities, and elementary random set theory. It is shown how special random sets can be used to generate normal possibility and necessity measures, as well as their natural extensions. This leads to interesting alternative formulas for the calculation of these natural extensions. KeywordsUpper probability, upper prevision, coherence, natural extension, possibility measure, random sets. I. Introduction P OSSIBILITY measures were introduced by Zadeh [1] in 1978. In his view, these supremum preserving set functions are a mathematical representation of the information conveyed by typical armative statements in natural language. For recent discussions of this interpretation within the behavioural framework of the theory of imprecise probabilities, we refer to [2], [3], [4]. Supremum preserving set functions can also be found in the literature under a number o...
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 8 (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.
Possibilistic Previsions
, 1998
"... The paper deals with a possibilistic imprecise secondorder probability model. It is argued that such models appear naturally in a number of situations. They lead to the introduction of a new type of previsions, called possibilistic previsions, which formally generalise coherent upper and lower prev ..."
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Cited by 4 (2 self)
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The paper deals with a possibilistic imprecise secondorder probability model. It is argued that such models appear naturally in a number of situations. They lead to the introduction of a new type of previsions, called possibilistic previsions, which formally generalise coherent upper and lower previsions. The converse problem is also looked at: given a possibilistic prevision, under what conditions can it be generated by a secondorder possibility distribution? This leads to the definition of normality, representability and natural extension of possibilistic previsions. Finally, some attention is paid to the special class of full possibilistic previsions, which can be formally related to Zadeh's fuzzy probabilities. The results have immediate applicability in decision making and statistical reasoning. 1 Introduction Consider an unstable radioactive nucleus. Its probability of decay in a given time interval t is given by 1 \Gamma e \Gammat , where the parameter is the decay rate o...
Practical Implementation of Possibilistic Probability Mass Functions
 In Proceedings of Fifth Workshop on Uncertainty Processing (WUPES 2000) (Jindvrichouv Hradec, Czech Republic
, 2000
"... Probability assessments of events are often linguistic in nature. We model them by means of possibilistic probabilities (a version of Zadeh's fuzzy probabilities with a behavioural interpretation) with a suitable shape for practical implementation (on a computer). Employing the tools of interval ..."
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Cited by 4 (2 self)
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Probability assessments of events are often linguistic in nature. We model them by means of possibilistic probabilities (a version of Zadeh's fuzzy probabilities with a behavioural interpretation) with a suitable shape for practical implementation (on a computer). Employing the tools of interval analysis and the theory of imprecise probabilities we argue that the verication of coherence for these possibilistic probabilities, the corrections of noncoherent to coherent possibilistic probabilities and their extension to other events and gambles can be performed by nite and exact algorithms. The model can furthermore be transformed into an imprecise rstorder model, useful for decision making and statistical inference. 1 Introduction Consider a football match in which the three possible outcomes are win (w), draw (d) and loss (l) for the home team. Suppose we have the following probability judgements for a specic match: win is likely to occur, draw and loss both have a chan...
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.
Coherence Of Rules For Defining Conditional Possibility
, 1998
"... . Possibility measures and conditional possibility measures are given a behavioural interpretation as marginal betting rates against events. Under this interpretation, possibility measures should satisfy two consistency criteria, known as `avoiding sure loss' and `coherence'. We survey the rules tha ..."
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Cited by 1 (0 self)
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. Possibility measures and conditional possibility measures are given a behavioural interpretation as marginal betting rates against events. Under this interpretation, possibility measures should satisfy two consistency criteria, known as `avoiding sure loss' and `coherence'. We survey the rules that have been proposed for defining conditional possibilities and investigate which of them satisfy our consistency criteria in two situations of practical interest. Only two of these rules satisfy the criteria in both cases studied, and the conditional possibilities produced by these rules are highly uninformative. We introduce a new rule that is more informative and is also coherent in both cases. 1. Introduction When the term `possibility measure' was coined by Zadeh [29] for what is essentially a supremum preserving set function, the significance of this type of set function had already been recognised in a number of other contexts [21, 22, 23]. These set functions are especially simple t...
Some Views on Information Fusion and Logic Based Approaches in Decision Making under Uncertainty
"... Abstract: Decision making under uncertainty is a key issue in information fusion and logic based reasoning approaches. The aim of this paper is to show noteworthy theoretical and applicational issues in the area of decision making under uncertainty that have been already done and raise new open rese ..."
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Abstract: Decision making under uncertainty is a key issue in information fusion and logic based reasoning approaches. The aim of this paper is to show noteworthy theoretical and applicational issues in the area of decision making under uncertainty that have been already done and raise new open research related to these topics pointing out promising and challenging research gaps that should be addressed in the coming future in order to improve the resolution of decision making problems under uncertainty.