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Possibility theory is a mathema)cal theory for dealing with certain types of uncertainty and is an alterna)ve to probability theory ... when first introduced possibility was an extension of the theory of fuzzy sets and fuzzy logic, others added to this a proposed min/max algebra to describe degrees of poten)al surprise.
Formalization of possibility
For simplicity, assume that the universe of discourse is a finite set (Ω), and assume that all subsets are measurable. A distribution of possibility is a func)on such that: It follows that, like probability, the possibility measure on finite set is determined by its behavior in singletons (unit X): provided U is finite or infinite.
Axiom 1 can be interpreted as the assump)on that Ω is an exhaus)ve descrip)on 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 assump)on that the evidence from which possibilities are constructed, is free of any contradic)on. Technically, it implies that there is at least one element in Ω with possibility 1.
Axiom 3 corresponds to the addivity axiom in probabili)es. However there is an important practical difference. Possibility theory is computa)onally more convenient because Axioms 1–3 imply
Because one can know the possibility of the union from the possibility of each component, it can be said that possibility is composi)onal with respect to the union operator. Note however that it is not composi)onal with respect to the intersec)on operator. Generally:
When Ω is not finite, Axiom 3 can be replaced for all index sets, if the subset variables, are pairwise disjoint.
Necessity
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. For any event X, the necessity measure is defined by the complement of the elements of Ω that do not belong to axiom 3.
Note that contrary to probability theory, possibility is not self-dual. For any event, we only have the inequality: However, the following duality rule holds:
For any event, possibility = / < 1 but > 0, or necessity = 0
Accordingly, beliefs about an event can be represented by a number and a bit. ... Interpretations ...
variable Z is necessary, true. It implies possibility is 1.
variable Z is necessary, and false. It implies possibility is 0.
variable Z is necessary, impossible. It implies possibility is 0.
variable Z is necessary, possible. It implies possibility is 0 - 1.
variable Z is unnecessary, true. It implies possibility is <1 (1 – drag of unnecessary V Z)
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