Conditioning in possibility theory with strict order norms

Baets, Bernard De; Tsiporkova, Elena; Mesiar, Radko

doi:10.1016/s0165-0114(97)00245-5

Cited by 62 publications

(23 citation statements)

References 5 publications

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“…Then N (B | A) = 1 − Π(B c | A). See De Baets et al [13] for a complete mathematical study of possibilistic conditioning, leading to the unicity of the product-based notion, in the infinite setting. The possibilistic counterpart to Bayes theorem looks formally the same as in probability theory:…”

Section: Conditioning In Possibility Theorymentioning

confidence: 99%

Possibility theory and statistical reasoning

Dubois¹

2006

Computational Statistics & Data Analysis

372

243

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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: Conditioning In Possibility Theorymentioning

confidence: 99%

Possibility theory and statistical reasoning

Dubois¹

2006

Computational Statistics & Data Analysis

372

243

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“…De Baets et al [33] provide a mathematical justification of this notion in an infinite setting, as opposed to the min-based conditioning of qualitative possibility theory. Indeed, the maxitivity axiom extended to the infinite setting is not preserved by the min-based conditioning.…”

Section: Conditioningmentioning

confidence: 99%

Possibility Theory and Its Applications: Where Do We Stand?

Dubois

Prade

2015

Springer Handbook of Computational Intelligence

131

101

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This paper provides an overview of possibility theory, emphasizing its historical roots and its recent developments. Possibility theory lies at the crossroads between fuzzy sets, probability and non-monotonic reasoning. Possibility theory can be cast either in an ordinal or in a numerical setting. Qualitative possibility theory is closely related to belief revision theory, and common-sense reasoning with exception-tainted knowledge in Artificial Intelligence. Possibilistic logic provides a rich representation setting, which enables the handling of lower bounds of possibility theory measures, while remaining close to classical logic. Qualitative possibility theory has been axiomatically justified in a decision-theoretic framework in the style of Savage, thus providing a foundation for qualitative decision theory. Quantitative possibility theory is the simplest framework for statistical reasoning with imprecise probabilities. As such it has close connections with random set theory and confidence intervals, and can provide a tool for uncertainty propagation with limited statistical or subjective information.

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“…This is quite unnatural, and all the more so if A = 0 (see [10]. Note that when moving to an infinite setting, the use of TM for constructing conditional possibility measures becomes debatable as well because the conditioning may not preserve O"-maxitivity (De Baets et al [2]). …”

Section: Pseudo-additive Measuresmentioning

confidence: 99%

“…Assuming that Tl = min we have solved (2) completely in [8] and exhibited the analytical forms of (IL, Vl, V2) ..…”

Section: Hybrid Possibilistic/probabilistic Event Treesmentioning

confidence: 99%

Pseudo-additive Measures and the Independence of Events

Dubois

Pap

Prade

2002

Technologies for Constructing Intelligent Systems 1

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Abstract. This paper is devoted to the investigation of independence (separability) of events with respect to a pseudo-additive measure that combine probabilistic and possibilistic behaviours via a threshold. It is based on a restricted family of t-conorm/ t-norm pairs which are very special ordinal sums. Any practically useful theory of pseudo-additive measures must use such special pairs of operations in order to conjointly extend the additivity property, and the notion of probabilistic independence.

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Conditioning in possibility theory with strict order norms

Cited by 62 publications

References 5 publications

Possibility theory and statistical reasoning

Possibility theory and statistical reasoning

Possibility Theory and Its Applications: Where Do We Stand?

Pseudo-additive Measures and the Independence of Events

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