Supremum preserving upper probabilities
The main purpose of the paper is to study the complex relationship between possibility measures and the theory of imprecise probabilities, and to point out the importance of possibility measures (i.e. supremum preserving set functions) within the behavioural theory of imprecise probabilities. Section 2 gives an introduction into the announced topic. Section 3 studies the coherence of possibility measures interpreted as upper probabilities, and their natural extension to larger domains of events and gambles. The role of Choquet integral in this extension is also discussed. Section 4 investigates the possibilistic extension problem in its full generality, and compares the possibilistic and natural extensions. In particular, there is proved that if possibilistic extension is possible, then there exists a special relationship between natural and possibilistic extension methods. In Sections 5 and 6 these two extension methods are studied for upper probabilities defined on classes and events where possibilistic extension is especially interesting, namely the collections of nested set. The main conclusion is that for upper probabilities defined on these chains of sets, natural and possibilistic extensions coincide under certain additional continuity conditions. Section 7 shows that upper probabilities on classes of nested sets appear naturally in a number of situations, this fact emphasizing the role of the supremum preserving upper probabilities within the theory of imprecise probabilities. The final Section 8 argues that if a coherent upper prevision defined on the convex cone of all nonnegative gambles is supremum preserving, then it must take the form of a Shilkret integral associated with a possibility measure. However, such a supremum preserving upper prevision is proved to be never coherent to a nonempty subset of the universe of discourse, being thus less promising in this respect.
scientific article; zbMATH DE number 1394680
scientific article