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Concept
Possibility theory
Summary
Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. It uses measures of possibility and necessity between 0 and 1, ranging from impossible to possible and unnecessary to necessary, respectively. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. Didier Dubois and Henri Prade further contributed to its development. Earlier, in the 1950s, economist G. L. S. Shackle proposed the min/max algebra to describe degrees of potential surprise.
For simplicity, assume that the universe of discourse Ω is a finite set. A possibility measure is a function from to [0, 1] such that:
Axiom 1:
Axiom 2:
Axiom 3: for any disjoint subsets and .
It follows that, like probability on finite probability spaces, the possibility measure is determined by its behavior on singletons:
Axiom 1 can be interpreted as the assumption that Ω is an exhaustive description of future states of the world, because it means that no belief weight is given to elements outside Ω.
Axiom 2 could be interpreted as the assumption that the evidence from which was constructed is free of any contradiction. Technically, it implies that there is at least one element in Ω with possibility 1.
Axiom 3 corresponds to the additivity axiom in probabilities. However there is an important practical difference. Possibility theory is computationally more convenient because Axioms 1–3 imply that:
for any subsets and .
Because one can know the possibility of the union from the possibility of each component, it can be said that possibility is compositional with respect to the union operator. Note however that it is not compositional with respect to the intersection operator. Generally:
When Ω is not finite, Axiom 3 can be replaced by:
For all index sets , if the subsets are pairwise disjoint,
Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur, possibility theory uses two concepts, the possibility and the necessity of the event.
Official source
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