A Mathematical Analysis of Information-Preserving Transformations Between Probabilistic and Possibilistic Formulations of Uncertainty

Geer, James; Klir, George J.

doi:10.1080/03081079208945024

Cited by 102 publications

(24 citation statements)

References 20 publications

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“…There are two basic approaches to possibility/probability transformations. We presented one in the previous section 4.1, the other one is due to Klir [74,58]. Klir's approach relies on a principle of information invariance, while the other one, described above, is based on optimizing information content [41].…”

Section: Alternative Approaches To Probability-possibility Transformsmentioning

confidence: 99%

Possibility theory and statistical reasoning

Dubois¹

2006

Computational Statistics & Data Analysis

385

250

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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.

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Section: Alternative Approaches To Probability-possibility Transformsmentioning

confidence: 99%

Possibility theory and statistical reasoning

Dubois¹

2006

Computational Statistics & Data Analysis

385

250

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“…These alternative views are discussed in [15], [32]. The most prominent one, due to Klir [18], [24], [25] is based on a principle of information invariance. In Klir's view, the transformation should be based on three assumptions:…”

Section: And Only If P(x) > P(x′) (Order Preservation)mentioning

confidence: 99%

Probability-Possibility Transformations, Triangular Fuzzy Sets, and Probabilistic Inequalities

Dubois

Foulloy²,

Mauris³

et al. 2004

Reliable Computing

498

339

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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 well-known probability inequalities of Bienaymé-Chebychev and Camp-Meidel. 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.

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“…The principle postulates that the amount of uncertainty should be preserved in each transformation of uncertainty from one mathematical framework to another. The principle was first studied in the context of probabilitypossibility transformations [29,30,32,9,42,68]. For an alternative approach see also [24,25].…”

Section: Applications Of Measures Of Uncertaintymentioning

confidence: 99%

Measure of Uncertainty and Information

Klir

2005

Uncertainty and Information

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A Mathematical Analysis of Information-Preserving Transformations Between Probabilistic and Possibilistic Formulations of Uncertainty

Cited by 102 publications

References 20 publications

Possibility theory and statistical reasoning

Possibility theory and statistical reasoning

Probability-Possibility Transformations, Triangular Fuzzy Sets, and Probabilistic Inequalities

Measure of Uncertainty and Information

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