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Encyclopedia > Possibility theory

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. D. Dubois and H. Prade further contributed to its development. Earlier in the 50s, economist G.L.S. Shackle proposed the min/max algebra to describe degrees of potential surprise. // Relation between uncertainty, probability and risk In his seminal work Risk, Uncertainty, and Profit, Frank Knight (1921) established the important distinction between risk and uncertainty: … Uncertainty must be taken in a sense radically distinct from the familiar notion of Risk, from which it has never been properly separated. ... Probability theory is the mathematical study of phenomena characterized by randomness or uncertainty. ... Lotfi A. Zadeh (2004) Lotfi Asker Zadeh (in Persian:لطفی علی‌عسکرزاده), (born February 4, 1921) is a mathematician and computer scientist, and a professor of computer science at the University of California at Berkeley. ... Fuzzy sets are an extension of the classical set theory used in Fuzzy logic. ... Fuzzy logic is derived from fuzzy set theory dealing with reasoning that is approximate rather than precisely deduced from classical predicate logic. ... George Lennox Sharman Shackle (14 July 1903 - 3 March 1992) was an English economist. ...

Contents

Formalization of possibility

For simplicity, assume that the universe of discourse Ω is a finite set, and assume that all subsets are measurable. A distribution of possibility is a function operatorname{pos} from Ω to [0, 1] such that: The term universe of discourse generally refers to the entire set of terms used in a specific discourse, i. ... In mathematics, a measure is a function that assigns a number, e. ...

Axiom 1: operatorname{pos}(varnothing) = 0
Axiom 2: operatorname{pos}(Omega) = 1
Axiom 3: operatorname{pos}(U cup V) = max left( operatorname{pos}(U), operatorname{pos}(V) right) for any disjoint subsets U and V.

It follows that, like probability, the possibility measure on finite or countably infinite sets is determined by its behavior on singletons:

operatorname{pos}(U) = max_{omega in U} operatorname{pos}({omega})

provided U is finite or countably infinite.


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 operatorname{pos} 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:

operatorname{pos}(U cup V) = max left( operatorname{pos}(U), operatorname{pos}(V) right) for any subsets U and V.

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:

operatorname{pos}(U cap V) leq min left( operatorname{pos}(U), operatorname{pos}(V) right)

Remark for the mathematicians:


When Ω is not finite Axiom 3 can be replaced by:

For all index sets I, if the subsets U_{i,, i in I} are pairwise disjoint, operatorname{pos}left(cup_{i in I} U_iright) = sup_{i in I}operatorname{pos}(U_i)

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 set U, the necessity measure is defined by Probability theory is the mathematical study of phenomena characterized by randomness or uncertainty. ...

operatorname{nec}(U) = 1 - operatorname{pos}(overline U)

In the above formula, overline U denotes the complement of U, that is the elements of Ω that do not belong to U. It is straightforward to show that:

operatorname{nec}(U) leq operatorname{pos}(U) for any U

and that:

operatorname{nec}(U cap V) = min ( operatorname{nec}(U), operatorname{nec}(V))

Note that contrary to probability theory, possibility is not self-dual. That is, for any event U, we only have the inequality:

operatorname{pos}(U) + operatorname{pos}(overline U) geq 1

However, the following duality rule holds:

For any event U, either operatorname{pos}(U) = 1, or operatorname{nec}(U) = 0

Accordingly, beliefs about an event can be represented by a number and a bit.


Interpretation

There are four cases that can be interpreted as follows:


operatorname{nec}(U) = 1 means that U is certainly true. It implies that operatorname{pos}(U) = 1.


operatorname{pos}(U) = 0 means that U is certainly false. It implies that operatorname{nec}(U) = 0.


operatorname{pos}(U) = 1 means that I would not be surprised at all if U occurs. It leaves operatorname{nec}(U) unconstrained.


operatorname{nec}(U) = 0 means that I would not be surprised at all if U does not occur. It leaves operatorname{pos}(U) unconstrained.


The intersection of the last two cases is operatorname{nec}(U) = 0 and operatorname{pos}(U) = 1 meaning that I believe nothing at all about U. Because it allows for indeterminacy like this, possibility theory relates to the graduation of a three-valued logic, such as intuitionistic logic, rather than the classical two-valued logic. Intuitionistic logic, or constructivist logic, is the logic used in mathematical intuitionism and other forms of mathematical constructivism. ...


Note that unlike possibility, fuzzy logic is compositional with respect to both the union and the intersection operator. 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.

Possibility theory as an imprecise probability theory

  • There is an extensive formal correspondence between probability and possibility theories, where the addition operator corresponds to the maximum operator.
  • A possibility measure can be seen as a consonant plausibility measure in Dempster-Shafer theory of evidence. The operators of possibility theory can be seen as a hyper-cautious version of the operators of the transferable belief model, a modern development of the theory of evidence.
  • Possibility can be seen as an upper probability: any possibility distribution defines a unique set of admissible probability distributions by

This allows one to study possibility theory using the tools of imprecise probabilities. Upper and lower probabilities are representations of imprecise probability. ... The Dempster-Shafer theory is a mathematical theory of evidence [SH76] based on belief functions and plausible reasoning, which is used to combine separate pieces of information (evidence) to calculate the probability of an event. ... In epistemic probability theory, the transferable belief model (TBM) is a model developed to represent quantified beliefs, or equivalently justified beliefs or supports. ...


References

  • 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. (Reprinted in Fuzzy Sets and Systems 100 (Supplement): 9-34, 1999.)

Lotfi A. Zadeh (2004) Lotfi Asker Zadeh (in Persian:لطفی علی‌عسکرزاده), (born February 4, 1921) is a mathematician and computer scientist, and a professor of computer science at the University of California at Berkeley. ...

See also


  Results from FactBites:
 
Possibility theory - Wikipedia, the free encyclopedia (754 words)
Possibility theory is a mathematical theory for dealing with certain types of uncertainty and is an alternative to probability theory.
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.
The operators of possibility theory can be seen as a hyper-cautious version of the operators of the transferable belief model, a modern development of the theory of evidence.
Probability theory - Wikipedia, the free encyclopedia (838 words)
Probability theory is the mathematical study of probability.
Two crucial concepts in the theory of probability are those of a random variable and of the probability distribution of a random variable; see those articles for more information.
Mathematicians usually take probability theory to be the study of probability spaces and random variables — an approach introduced by Kolmogorov in the 1930s.
  More results at FactBites »

 
 

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