On the connection between probability boxes and possibility measures

Troffaes, Matthias C.M.; Miranda, Enrique; Destercke, Sebastien

Abstract

We explore the relationship between possibility measures (supremum preserving normed measures) and p-boxes (pairs of cumulative distribution functions) on totally preordered spaces, extending earlier work in this direction by De Cooman and Aeyels, among others. We start by demonstrating that only those p-boxes who have 0-1-valued lower or upper cumulative distribution function can be possibility measures, and we derive expressions for their natural extension in this case. Next, we establish necessary and sufficient conditions for a p-box to be a possibility measure. Finally, we show that almost every possibility measure can be modelled by a p-box, simply by ordering elements by increasing possibility. Whence, any techniques for p-boxes can be readily applied to possibility measures. We demonstrate this by deriving joint possibility measures from marginals, under varying assumptions of independence, using a technique known for p-boxes. Doing so, we arrive at a new rule of combination for possibility measures, for the independent case.

Citation

Troffaes, M. C., Miranda, E., & Destercke, S. (2013). On the connection between probability boxes and possibility measures. Information Sciences, 224, 88-108. https://doi.org/10.1016/j.ins.2012.09.033

• Journal Article Type: Article
• Publication Date: Mar 1, 2013
• Deposit Date: Nov 17, 2011
• Publicly Available Date: Oct 17, 2014
• Journal: Information Sciences
• Print ISSN: 0020-0255
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 224
• Pages: 88-108
• DOI: https://doi.org/10.1016/j.ins.2012.09.033
• Keywords: Probability boxes, Possibility measures, Maxitive measures, Coherent lower and upper probabilities, Natural extension.
• Public URL: https://durham-repository.worktribe.com/output/1502498
• Publisher URL: http://dx.doi.org/10.1016/j.ins.2012.09.033
• Related Public URLs: http://arxiv.org/abs/1103.5594