Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory. Professor Lotfi Zadeh first introduced possibility theory in 1978 as an extension of his theory of fuzzy sets and fuzzy logic. Whereas fuzzy logic describes imprecision, possibility theory describes uncertainty. The idea was not entirely new: early in the 50s economist G.L.S. Shackle proposed the min/max algebra to describe the degree of potential surprise.
Contents
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* 1 Mathematical elements
o 1.1 Independent fuzzy events
* 2 Conditional possibility and necessity
o 2.1 Properties of conditionals
o 2.2 Set-theoretic operations with conditionals
* 3 Relationship with other imprecise probability theories and fuzzy logic
* 4 See also
* 5 References
* 6 External links
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Mathematical elements
Whereas probability theory uses a single number, the probability, to describe how likely an event is to occur possibility theory uses two numbers, the possibility of the event and the necessity of the event. Like probabilities, possibilities and necessities are set measures. Formally, a distribution of possibility is a function π from Ω to [0, 1] normalized such that at least one element has possibility 1. We start with a finite universe of discourse Ω. This is the set of all possible true states of affairs. By analogy with probability,
\Pi(\empty)=0, \Pi(\Omega)=1
For any collection of subsets
\omega_1 \in \mathcal{P}(\Omega),\ldots,\omega_n \in \mathcal{P},
\Pi\left(\cup_{i=1}^{n}\omega_i\right)=\sup_{i=1,\ldots,n}\left(\omega_i\right)
(\mathcal{P}(\Omega) denotes the powerset of Ω, i.e, the set of all subsets). For any set S, the necessity measure is defined by:
\Nu(S) = 1 - \Pi(\overline S) = \inf_{\omega \in S} (1 - \pi(\omega))\,
Note that contrary to probability theory it is generally not true that \Pi(S) + \Pi(\overline S) = 1. Possibility is not self dual. However, the following duality rule holds: for any event S, either the possibility is 1, or the necessity is 0.
Possibility measures obey the following composition rule, which holds for any two subsets S and T even if they are not disjoint (compare with the additivity axiom in probabilities): \Pi(S \cup T) = \max (\Pi(S), \Pi(T))\,
The possibility is a non-additive set measure (mathematics) when it is applied to the membership function of a set.
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Independent fuzzy events
Two fuzzy events S and T are independent iff:
\Pi(S \cap T) = \min(\Pi(S),\Pi(T))\,
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Conditional possibility and necessity
Conditional possibility and necessity play similar role as conditional probabilities in probability theory do. For any two, possibly fuzzy, subsets A and B of the same universal set Ω, with the membership functions \mu_A, \mu_B:\Omega\rightarrow[0,1] the conditional possibility is defined as:
\operatorname{pos}(A|B) = \sup_{\omega \in \Omega} \min (\mu_A(\omega), \mu_B(\omega)) = \sup_{\omega \in \Omega} \mu_{A \cap B}\,
The conditional possibility tells how possible is A given the set B. When B is normal, i.e. its membership function reaches 1, then \operatorname{pos}(\cdot|B) becomes a possibility measure. The conditional necessity:
\operatorname{nec}(A|B) = \inf_{\omega \in \Omega} \max (\mu_A(\omega), 1 - \mu_B(\omega)) = \inf_{\omega \in \Omega} \mu_{A \cup \overline{B}}\,
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Properties of conditionals
\operatorname{nec}(A|B) \equiv 1 - \operatorname{pos}(\overline{A}|B)\,
The next two results are the possibilistic equivalent of Bayes' theorem:
\operatorname{pos}(A|B) \equiv \operatorname{pos}(B|A)\,
\operatorname{nec}(A|B) \equiv \operatorname{nec}(\overline{B}|\overline{A})\,
If B is normal, then
\operatorname{nec}(A|B) \le \operatorname{pos}(A|B)\,
The pair of conditionals
\{\operatorname{pos}(A|B), \operatorname{nec}(A|B)\}\,
characterize B as a subset of A, i.e. B \subseteq A. For A and B crisp sets, if B is a subset of A then pos(A | B) = nec(A | B) = 1. If B intersects A then pos(A|B) = 1 and nec(A|B) = 0. If B is outside A then pos(A|B) = nec(A|B) = 0.
Let {ω} be a singleton subset of Ω. Then
\operatorname{pos}(A|\{\omega\})\equiv \operatorname{nec}(A|\{\omega\})\equiv \mu_A(\omega)\,
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Set-theoretic operations with conditionals
\operatorname{pos}(A\cup B|S) \equiv \max (\operatorname{pos}(A|S), \operatorname{pos}(B|S))\,
\operatorname{pos}(A\cap B|S) \le \min (\operatorname{pos}(A|S), \operatorname{pos}(B|S))\,
\operatorname{pos}(T|A\cup B) \equiv \max (\operatorname{pos}(T|A), \operatorname{pos}(T|B))\,
\operatorname{pos}(T|A\cap B) \le \min (\operatorname{pos}(T|A), \operatorname{pos}(T|B))\,
\operatorname{nec}(A\cup B|S) \ge \max (\operatorname{nec}(A|S), \operatorname{nec}(B|S))\,
\operatorname{nec}(A\cap B|S) \equiv \min (\operatorname{nec}(A|S), \operatorname{nec}(B|S))\,
\operatorname{nec}(T|A\cup B) \equiv \min (\operatorname{nec}(T|A), \operatorname{nec}(T|B))\,
\operatorname{nec}(T|A\cap B) \ge \max (\operatorname{nec}(T|A), \operatorname{nec}(T|B))\,
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Relationship with other imprecise probability theories and fuzzy logic
* There are many formal correspondences between probability and possibility theories. For example, just like probability measures, possibility measures (on subsets) can be represented by distributions (on singletons).
* Necessity can be seen as an upper probability: any necessity distribution defines a unique set of admissible probability distributions by
\left\{\, p: \forall S, p(S)\leq \operatorname{nec}(S)\,\right\}.
* Possibility measure can be seen as a consonant plausibility measure in Dempster-Shafer theory of evidence (when the focal sets are nested like Russian dolls). Possibility theory as an hyper-cautious transferable belief model.
The relationship with fuzzy theory can be explained with the following classical example.
* Fuzzy logic: When a bottle is half full, it can be said that the level of truth of the proposition "The bottle is full" is 0.5. The word "full" is seen as a fuzzy predicate describing the amount of liquid in the bottle.
* Possibility theory: There is one bottle, either completely full or totally empty. The proposition "the possibility level that the bottle is full is 0.5" describes a degree of belief. One way to interpret 0.5 in that proposition is to define its meaning as: I am ready to bet that it's empty as long as the odds are even (1:1) or better, and I would not bet at any rate that it's full.
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See also
* logical possibility
* probability theory
* Fuzzy measure theory
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References
* Dubois, Didier, and Prade, Henri, Possibility Theory, Plenum Press, New York, 1988.
* Joslyn, Cliff, "Possibilistic Measurement and Set Statistics", in Proceedings of the 1992 NAFIPS Conference 2:458-467, NASA, 1992.
* Joslyn, Cliff, "Possibilistic Semantics and Measurement Methods in Complex Systems", in Proceedings of the 2nd International Symposium on Uncertainty Modeling and Analysis, Bilal Ayyub (editor), IEEE Computer Society 1993.
* Dubois, Didier and Prade, Henri, Possibility theory, probability theory and multiple-valued logics: A clarification, Annals of Mathematics and Artificial Intelligence 32:35-66, 2001.
* Zadeh, Lotfi, "Fuzzy Sets as the Basis for a Theory of Possibility", Fuzzy Sets and Systems 1:3-28, 1978.
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External links
* http://lrb.cs.uni-dortmund.de/fd7/dubois.pdf
* http://www.dmitry-kazakov.de/ada/fuzzy.htm
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