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Probability boxes on totally preordered spaces for multivariate modelling.

Troffaes, Matthias C. M. and Destercke, Sébastien (2011) 'Probability boxes on totally preordered spaces for multivariate modelling.', International journal of approximate reasoning., 52 (6). pp. 767-791.


A pair of lower and upper cumulative distribution functions, also called probability box or p-box, is among the most popular models used in imprecise probability theory. They arise naturally in expert elicitation, for instance in cases where bounds are specified on the quantiles of a random variable, or when quantiles are specified only at a finite number of points. Many practical and formal results concerning p-boxes already exist in the literature. In this paper, we provide new efficient tools to construct multivariate p-boxes and develop algorithms to draw inferences from them. For this purpose, we formalise and extend the theory of p-boxes using Walley's behavioural theory of imprecise probabilities, and heavily rely on its notion of natural extension and existing results about independence modeling. In particular, we allow p-boxes to be defined on arbitrary totally preordered spaces, hence thereby also admitting multivariate p-boxes via probability bounds over any collection of nested sets. We focus on the cases of independence (using the factorization property), and of unknown dependence (using the Fréchet bounds), and we show that our approach extends the probabilistic arithmetic of Williamson and Downs. Two design problems - a damped oscillator, and a river dike - demonstrate the practical feasibility of our results.

Item Type:Article
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Publisher statement:NOTICE: this is the author’s version of a work that was accepted for publication in International Journal of Approximate Reasoning. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in International Journal of Approximate Reasoning, 52, 6, September 2011, 10.1016/j.ijar.2011.02.001.
Date accepted:No date available
Date deposited:01 December 2014
Date of first online publication:September 2011
Date first made open access:No date available

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