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 dened 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, dened 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 specied, possibility measures very naturally emerge as the corresponding natural extensions. Next, we go from upper probabilities to upper previsions...

| The relationship is studied between possibility and necessity measures dened on arbitrary spaces, the theory of imprecise probabilities, and elementary random set theory. It is shown how special random sets can be used to generate normal possibility and necessity measures, as well as their natural extensions. This leads to interesting alternative formulas for the calculation of these natural extensions. Keywords|Upper probability, upper prevision, coherence, natural extension, possibility measure, random sets. I. Introduction P OSSIBILITY measures were introduced by Zadeh [1] in 1978. In his view, these supremum preserving set functions are a mathematical representation of the information conveyed by typical armative statements in natural language. For recent discussions of this interpretation within the behavioural framework of the theory of imprecise probabilities, we refer to [2], [3], [4]. Supremum preserving set functions can also be found in the literature under a number o...

The paper deals with a possibilistic imprecise second-order probability model. It is argued that such models appear naturally in a number of situations. They lead to the introduction of a new type of previsions, called possibilistic previsions, which formally generalise coherent upper and lower previsions. The converse problem is also looked at: given a possibilistic prevision, under what conditions can it be generated by a second-order possibility distribution? This leads to the definition of normality, representability and natural extension of possibilistic previsions. Finally, some attention is paid to the special class of full possibilistic previsions, which can be formally related to Zadeh's fuzzy probabilities. The results have immediate applicability in decision making and statistical reasoning. 1 Introduction Consider an unstable radioactive nucleus. Its probability of decay in a given time interval t is given by 1 \Gamma e \Gammat , where the parameter is the decay rate o...

Probability assessments of events are often linguistic in nature. We model them by means of possibilistic probabilities (a version of Zadeh's fuzzy probabilities with a behavioural interpretation) with a suitable shape for practical implementation (on a computer). Employing the tools of interval analysis and the theory of imprecise probabilities we argue that the verication of coherence for these possibilistic probabilities, the corrections of non-coherent to coherent possibilistic probabilities and their extension to other events and gambles can be performed by nite and exact algorithms. The model can furthermore be transformed into an imprecise rst-order model, useful for decision making and statistical inference. 1 Introduction Consider a football match in which the three possible outcomes are win (w), draw (d) and loss (l) for the home team. Suppose we have the following probability judgements for a specic match: win is likely to occur, draw and loss both have a chan...

The paper discusses integration in possibility theory, both in an ordinal and in a numerical (behavioral) context. It is shown that in an ordinal context, the fuzzy integral has an important part in at least three areas: the extension of possibility measures to larger domains, the construction of product measures from marginals and the denition of conditional possibilities. In a numerical (behavioral) context, integration can be used to extend upper probabilities to upper previsions. It is argued that the role of the fuzzy integral in this context is limited, as it can only be used to dene a coherent upper prevision if the associated upper probability is 0-1-valued, in which case it moreover coincides with the Choquet integral. These results are valid for arbitrary coherent upper probabilities, and therefore also relevant for possibility theory. It follows from the discussion that in a numerical context, the Choquet integral is better suited than the fuzzy integral for producing coherent upper previsions starting from possibility measures. At the same time, alternative expressions for the Choquet integral associated with a possibility measure are derived.

Abstract: Decision making under uncertainty is a key issue in information fusion and logic based reasoning approaches. The aim of this paper is to show noteworthy theoretical and applicational issues in the area of decision making under uncertainty that have been already done and raise new open research related to these topics pointing out promising and challenging research gaps that should be addressed in the coming future in order to improve the resolution of decision making problems under uncertainty.