4
Possibility Theory for Treating Uncertainty
Possibility theory deals with uncertainty characterization in the case of incomplete information. It differs from probability theory (which uses a single probability measure) in that it uses a pair of dual set functions called possibility and necessity measures. The background and scope of possibility theory is described as follows by Dubois and Prade (2007):
The name Theory of Possibility was coined by (Zadeh, 1978), inspired by (Gaines and Kohout, 1975). In Zadeh's view, possibility distributions were meant to provide a graded semantics to natural language statements. However, possibility and necessity measures can also be the basis of a full-fledged representation of partial belief that parallels probability (Dubois and Prade, 1988). Then, it can be seen either as a coarse, non-numerical version of probability theory, or as a framework for reasoning with extreme probabilities, or yet as a simple approach to reasoning with imprecise probabilities (Dubois, Nguyen, and Prade, 2000).
In the following we first review some basics of possibility theory, largely taken from or based on Aven (2011c), and then present some approaches to constructing possibility distributions.
4.1 Basics of Possibility Theory
A central component in possibility theory is the possibility function 
. For each in a set , expresses the degree of possibility of
