On the connection between probability boxes and possibility measures

Authors:
Matthias C. M. Troffaes;Enrique Miranda;Sebastien Destercke
Affiliations:
Durham University, Dept. of Mathematical Sciences, Science Laboratories, South Road, Durham DH1 3LE, United Kingdom;University of Oviedo, Dept. of Statistics and O.R. C-Calvo Sotelo, s/n, Oviedo, Spain;CNRS, HEUDIASYC Joint Research Unit, UMR 7253, Centre de Recherche de Royallieu, UTC, F-60205 Compiegne cedex, France
Venue:
Information Sciences: an International Journal
Year:
2013

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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.