Supremum Preserving Upper Probabilities

We study the relation between possibility measures and the theory of imprecise probabilities, and argue that possibility measures have an important part in this theory. It is shown that a possibility measure is a coherent upper probability if and only if it is normal. A detailed comparison is given between the possibilistic and natural extension of an upper probability, both in the general case and for upper probabilities defined on a class of nested sets. We prove in particular that a possibility measure is the restriction to events of the natural extension of a special kind of upper probability, defined on a class of nested sets. We show that possibilistic extension can be interpreted in terms of natural extension. We also prove that when either the upper or the lower cumulative distribution function of a random quantity is specified, possibility measures very naturally emerge as the corresponding natural extensions. Next, we go from upper probabilities to upper previsions. We show that if a coherent upper prevision defined on the convex cone of all non-negative gambles is supremum preserving, then it must take the form of a Shilkret integral associated with a possibility measure. But at the same time, we show that such a supremum preserving upper prevision is never coherent unless it is the vacuous upper prevision with respect to a non-empty subset of the universe of discourse.