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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 40 (15 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.
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. Moreover it s ..."
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Cited by 39 (5 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...
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
Possibilistic Semantics and Measurement Methods in Complex Systems
 in: Proc. 2nd Int. Symposium on Uncertainty Modeling and Analysis, ed. Bilal Ayyub
, 1993
"... Possibility theory is a new mathematical theory for the representation of uncertainty. It is related to, but distinct from, probability theory, DempsterShafer evidence theory, and fuzzy set theory. It has been applied almost exclusively to knowledgebased engineering systems, with measurements take ..."
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Cited by 6 (6 self)
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Possibility theory is a new mathematical theory for the representation of uncertainty. It is related to, but distinct from, probability theory, DempsterShafer evidence theory, and fuzzy set theory. It has been applied almost exclusively to knowledgebased engineering systems, with measurements taken from subjective evaluations. Toward the end of developing a strictly possibilistic semantics of natural systems, the following will be considered: the semantics of possibility statements in relation to modal logic, natural language, and mathematical possibility theory; a strong consistency relation for probability and possibility; the basis for the application of possibility theory to complex systems; and physical measurement procedures for possibility. 1 Introduction Possibility theory, as a new method for the representation of uncertainty, is similar to, and yet distinct from, probability theory. Together with other new mathematical theories of uncertainty (fuzzy sets, fuzzy measures, D...
An ObjectOriented Architecture for Possibilistic Models
 in: Proc. 1994 Conf. ComputerAided Systems Technology
, 1994
"... . An architecture for the implementation of possibilistic models in an objectoriented programming environment (C++ in particular) is described. Fundamental classes for special and general random sets, their associated fuzzy measures, special and general distributions and fuzzy sets, and possibilist ..."
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Cited by 4 (4 self)
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. An architecture for the implementation of possibilistic models in an objectoriented programming environment (C++ in particular) is described. Fundamental classes for special and general random sets, their associated fuzzy measures, special and general distributions and fuzzy sets, and possibilistic processes are specified. Supplementary methodsincluding the fast Mobius transform, the maximum entropy and Bayesian approximations of random sets, distribution operators, compatibility measures, consonant approximations, frequency conversions, and possibilistic normalization and measurement methodsare also introduced. Empirical results to be investigated are also described. 1 Introduction Possibility theory [4] is an alternative information theory to that based on probability. Although possibility theory is logically independent of probability theory, they are related: both arise in DempsterShafer evidence theory as fuzzy measures defined on random sets; and their distributions a...
Review of uncertainty reasoning approaches as guidance for maritime and offshore safetybased assessment
 Journal of UK Safety and Reliability Society
, 2003
"... Many different formal techniques have been developed over the past two decades for dealing with uncertain information for decision making. In this paper we review some of the most important ones, i.e., Bayesian theory of probability, DempsterShafer theory of evidence, and fuzzy set theory, describe ..."
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Cited by 4 (4 self)
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Many different formal techniques have been developed over the past two decades for dealing with uncertain information for decision making. In this paper we review some of the most important ones, i.e., Bayesian theory of probability, DempsterShafer theory of evidence, and fuzzy set theory, describe how they work and in what ways they differ from one another, and show their strength and weakness respectively as well as their connection. We also consider hybrid approaches which combine two or more approximate reasoning techniques within a single reasoning framework. These have been proposed to address limitations in the use of individual techniques. The study is intended to provide guidance in the process of developing frameworks for safetybased decision analysis using different methods for reasoning under uncertainty. 1. Uncertainty in Decision Making In conventional information processing techniques it is often assumed that problems are well structured, complete information is always available and information processing procedures can be clearly defined. However, in many realworld decision making problems, this is not always the case and decisions making is often associated with uncertainty. “Uncertainty ” is a context dependent concept. There does not exist a comprehensive and unique definition of uncertainty. One definition of uncertainty is given as follows [Zimmermann 2000]: “Uncertainty is a situation in which a person does not have the quantitatively and qualitatively appropriate
On Possibilistic Automata
 in: Lecture Notes in Computer Science
, 1993
"... . General automata are considered with respect to normalization over semirings. Possibilistic automata are defined as normal pessimistic fuzzy automata. Possibilistic automata are analogous to stochastic automata where stochastic (+=\Theta) semirings are replaced by possibilistic (=) semirings; but ..."
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Cited by 3 (3 self)
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. General automata are considered with respect to normalization over semirings. Possibilistic automata are defined as normal pessimistic fuzzy automata. Possibilistic automata are analogous to stochastic automata where stochastic (+=\Theta) semirings are replaced by possibilistic (=) semirings; but where stochastic automata must be normal, fuzzy automata may be (resulting in possibilistic automata) or may not be (resulting in fuzzy automata proper). While possibilistic automata are direct generalizations of nondeterministic automata, stochastic automata are not. Some properties of possibilistic automata are considered, and the identity of possibilistic automata strongly consistent with stochastic automata is shown. 1 Introduction Fuzzy automata were introduced and examined in the late 1960's in a series of papers by Santos [20], Santos and Wee [22], and Wee and Fu [25]. Their most commonly considered form is the pessimistic maxmin fuzzy automata, a system e A : =\Omega Q; A; F; ¯ ...
Probabilities, Possibilities, and Fuzzy Sets
 Fuzzy Sets and Systems
, 1994
"... A formal analysis of probabilities, possibilities, and fuzzy sets is presented in this paper. A number of theorems proved show that probabilities carry more information per bit than both possibilities and fuzzy sets. The cost of this higher capacity is increased computational complexity and reduced ..."
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Cited by 3 (0 self)
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A formal analysis of probabilities, possibilities, and fuzzy sets is presented in this paper. A number of theorems proved show that probabilities carry more information per bit than both possibilities and fuzzy sets. The cost of this higher capacity is increased computational complexity and reduced computational efficiency. The resulting tradeoff of high complexity and information capacity versus computational efficiency is discussed under the spectrum of experimental systems and applications. 1 Introduction Probabilities, possibilities, and fuzzy sets are all measures used to formalize and quantify uncertainty. There is an ongoing debate regarding the appropriateness of each measure in formalizing uncertainty. Arguments in favour of probabilities can be found in [17, 2] while arguments more in favour of possibilities and fuzzy sets are presented in [16, 13]. A brief presentation and qualitative comparison of the above measures as well as MYCIN's certainty factors ([26]) and Dempster...
Measure of uncertainty and information
 Imprecise Probability Project,1999 (http://ippserv.rug.ac.be/home/ipp.html
, 1999
"... Abstract. This contribution overviews the approaches, results and history of attempts at measuring uncertainty and information in the various theories of imprecise probabilities. The main focus, however, is on the theory of belief functions (or the DempsterShafer theory) [62] and the possibility th ..."
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Abstract. This contribution overviews the approaches, results and history of attempts at measuring uncertainty and information in the various theories of imprecise probabilities. The main focus, however, is on the theory of belief functions (or the DempsterShafer theory) [62] and the possibility theory [7] as most of the development so far has happened there. Due to the limited space I am focusing on the main ideas and point to references for details. There are
On Modeling Possibilistic Uncertainty in TwoState Reliability Theory
"... : In this paper, we show how a possibilistic uncertainty model can be used to represent and manipulate uncertainty about the states of a system and of its components. At the same time, we present a thorough study of the incorporation of this possibilistic uncertainty model in classical, twostate re ..."
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Cited by 2 (0 self)
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: In this paper, we show how a possibilistic uncertainty model can be used to represent and manipulate uncertainty about the states of a system and of its components. At the same time, we present a thorough study of the incorporation of this possibilistic uncertainty model in classical, twostate reliability theory. The possibilistic reliability of a component or system is introduced and studied. Furthermore, we introduce the important notion of a possibilistic structure function, based upon the concept of a classical, twovalued structure function. Under certain conditions of possibilistic independence, it allows us to calculate the possibilistic reliability of a system in terms of the possibilistic reliabilities of its components. Finally, we give straightforward methods for determining a possibilistic structure function from its classical, twovalued counterpart. In this way, we intend to show that a possibilistic uncertainty model in twostate reliability theory is formally analogo...