Probabilistic arithmetic. I. Numerical methods for calculating convolutions and dependency bounds

Williamson, Robert C.; Downs, T.

doi:10.1016/0888-613x(90)90022-t

Cited by 322 publications

(212 citation statements)

References 38 publications

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“…This is in agreement with similar treatments done with precise probabilities when dependencies between variables are not known [25]. This indicates that belief functions alone are perhaps not always sufficient to treat some problems.…”

Section: Discussionsupporting

confidence: 79%

“…A possible alternative is to search for subsets of conjunctively merged belief structures satisfying a weaker contour function principle, thus working with sets of belief functions rather than with a single one. This goes in the sense of propositions made by other authors in order to deal with situations where dependencies or exact features of belief structures are not precisely known [1,25,5]. Such an alternative is explored in the next section.…”

Section: Proposition 3 (S-coherence) Letmentioning

confidence: 99%

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Can the Minimum Rule of Possibility Theory Be Extended to Belief Functions?

Destercke

Dubois

2009

Lecture Notes in Computer Science

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Abstract. When merging belief functions, Dempster rule of combination is justified only when sources can be considered as independent. When dependencies are ill-known, it is usual to ask the merging operation to satisfy the property of idempotence, as this property ensures a cautious behaviour in the face of dependent sources. There are different strategies to find such rules for belief functions. One strategy is to rely on idempotent rules used in either more general or more specific frameworks and to respectively study their particularisation or extension to belief functions. In this paper, we try to extend the minimum rule of possibility theory to belief functions. We show that such an extension is not always possible, unless we accept the idea that the result of the fusion process can be a family of belief functions.

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Section: Discussionsupporting

confidence: 79%

Section: Proposition 3 (S-coherence) Letmentioning

confidence: 99%

Can the Minimum Rule of Possibility Theory Be Extended to Belief Functions?

Destercke

Dubois

2009

Lecture Notes in Computer Science

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“…Frank et al (1987) proved the best-possible nature of these bounds. Williamson and Downs (1990) translated these results into bounds for the value-at-risk of the sum of two risks using the duality principle. The n-dimensional formulation of this result is stated formally in the next theorem.…”

Section: Bounds When the Marginal Distributions Are Knownmentioning

confidence: 99%

Bounds on the value-at-risk for the sum of possibly dependent risks

Mesfioui

Quessy

2005

Insurance: Mathematics and Economics

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In this paper, explicit lower and upper bounds on the value-at-risk (VaR) for the sum of possibly dependent risks are derived when only partial information is available about the dependence structure and the individual behaviors. When the marginal distributions are known, a reformulation of a result of Embrechts et al. [Finan. Stoch. 7 (2003) 145-167] makes it possible, under some regularity conditions, to compute explicit bounds for the VaR under various dependence scenarios. In the case where only the means and the variances of the risks are available, explicit bounds are obtained from an optimization over all possible values of the correlation matrix associated with the vector of risks. Analytical and numerical investigations are presented in order to investigate the quality of these bounds.

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“…Arithmetic rules on P-boxes were defined in e.g. [42], and implementations are available, for instance the DSI Toolbox [2] based on Matlab and INTLAB [34], Statool [4] implementing the arithmetic of [3] and RiskCalc [16]. They were not designed to be used for static analysis of programs (there is no consideration on semantics of programs nor join operators defined, typically) as we do in this paper but are rather designed for making numerical simulations or optimizations [19] for instance for risk assessment [18].…”

Section: Related Workmentioning

confidence: 99%

Static Analysis of Programs with Imprecise Probabilistic Inputs

Adjé

Bouissou

Goubault-Larrecq

et al. 2014

Verified Software: Theories, Tools, Experiments

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Abstract. Having a precise yet sound abstraction of the inputs of numerical programs is important to analyze their behavior. For many programs, these inputs are probabilistic, but the actual distribution used is only partially known. We present a static analysis framework for reasoning about programs with inputs given as imprecise probabilities: we define a collecting semantics based on the notion of previsions and an abstract semantics based on an extension of Dempster-Shafer structures. We prove the correctness of our approach and show on some realistic examples the kind of invariants we are able to infer.

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Probabilistic arithmetic. I. Numerical methods for calculating convolutions and dependency bounds

Cited by 322 publications

References 38 publications

Can the Minimum Rule of Possibility Theory Be Extended to Belief Functions?

Can the Minimum Rule of Possibility Theory Be Extended to Belief Functions?

Bounds on the value-at-risk for the sum of possibly dependent risks

Static Analysis of Programs with Imprecise Probabilistic Inputs

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