Theory of possibility

The possibility theory (Engl. Possibility , "possibility"), often with users possibility theory called, is a mathematical theory of incomplete information derived uncertainty modeled. In a way, it complements the more popular probability theory , which deals with uncertainty caused by chance . Of: it differs from probability theory by using not only a lot of function (the probability), but a pair of mutually dual lot of features Possibility and Necessity (Engl. Necessity , "necessity"). The possibility of an event is always at least as great as its probability. Colloquially: What is probable is even more possible, even the improbable is possible.

Historical

The theory of possibility (as a mathematical theory) goes back to Lotfi Zadeh (1978). Didier Dubois and Henri Prade in 1988 contributed significantly to its popularization. Before Zadeh, for example, the economist GLS Shackle in 1961, and the philosophers D. Lewis in 1973 and LJ Cohen in 1977 dealt with the formalization of the concept of possibility.

Possibility

Designate the so-called universe, i.e. H. the basic set containing all conceivable events for the problem at hand. For the sake of simplicity, let it be a finite set. Then the following axioms apply to the set function Possibility, abbreviated to : ${\ displaystyle {\ Omega}}$ ${\ displaystyle {\ Omega}}$ ${\ displaystyle pos}$

{\ displaystyle {\ begin {aligned} & pos ({\ Omega}) = 1 \\ & pos ({\ emptyset)} = 0 \\ & pos (A {\ cup} B) = {\ max} [pos (A) , pos (B)], \ quad A {\ subseteq} {\ Omega}, B {\ subseteq} {\ Omega}, \ quad A {\ cap} B = {\ emptyset} \ end {aligned}}}

Favorable for calculations it follows from the axioms that for arbitrary not necessarily disjoint holds. It also follows that one can calculate the possibility of an event from the possibilities of the elementary events according to ${\ displaystyle pos (A {\ cup} B) = \ max [pos (A), pos (B)]}$ ${\ displaystyle A, B {\ subseteq} {\ Omega}}$ ${\ displaystyle A}$ ${\ displaystyle {\ omega} {\ in} {\ Omega}}$

{\ displaystyle pos (A) = {\ underset {{\ omega} {\ in} A} {\ max}} \ pos [\ {{\ omega} \}]}

The possibility of elementary events, considered as a function of , is also called the possibility distribution function . It is often referred to as, i.e. H. ${\ displaystyle {\ omega}}$ ${\ displaystyle {\ pi} ({\ omega})}$

{\ displaystyle {\ pi} ({\ omega}) = pos [\ {{\ omega} \}]}

For example, the membership function of a fuzzy set can serve as a possibility distribution function. It describes the degree of possibility that belongs to the fuzzy set . This is L. Zadeh's approach to the theory of possibility, see. ${\ displaystyle m_ {A}}$ ${\ displaystyle A}$ ${\ displaystyle {\ omega} {\ in} {\ Omega}}$ ${\ displaystyle A}$

Necessity

The necessity is the set function dual to the possibility and is denoted by. It arises through ${\ displaystyle nec}$

{\ displaystyle nec (A) = 1-pos ({\ overline {A}})}

where the complementary set is to. It is always , so it holds ${\ displaystyle {\ overline {A}}}$ ${\ displaystyle A}$ ${\ displaystyle nec (A) {\ leqslant} pos (A)}$

{\ displaystyle pos (A) + pos ({\ overline {A}}) {\ geqslant} 1, \ quad nec (A) + nec ({\ overline {A}}) {\ leqslant} 1}

,

in contrast to the (self-dual) probability for which applies. ${\ displaystyle P}$ ${\ displaystyle P (A) + P ({\ overline {A}}) = 1}$

Possibility theory as a probability theory with incomplete information

If you know too little to state about a probability distribution, e.g. B. if one only knows the probabilities of some events , then one can only give lower and upper bounds for the true probability for any events due to the calculation rules for probabilities. This problem was addressed by AP Dempster in 1967, see and created the terms lower probability and upper probability . Possibility is an upper probability, Necessity is a lower probability. In addition, the Possibility is a special plausibility and the Necessity a special Beliefunction in the sense of the evidence theory of Dempster and G. Shafer, see. In general, in the sense of Dempster plausibility, upper probabilities and belief functions are lower probabilities. ${\ displaystyle {\ Omega}}$ ${\ displaystyle {\ Omega}}$ ${\ displaystyle {\ Omega}}$ ${\ displaystyle pos}$ ${\ displaystyle nec}$

Individual evidence

↑ ^a ^b Zadeh, L .: Fuzzy Sets as the Basis for a Theory of Possibility , Fuzzy Sets and Systems 1 (1978) 3-28. doi : 10.1016 / S0165-0114 (99) 80004-9
↑ Dubois, D. and Prade, H .: Possibility Theory: An Approach to Computerized Processing of Uncertainty , New York: Plenum Press 1988
^ Shackle, GLS: Decision, Order and Time in Human Affairs , 2nd edition, Cambridge University Press
^ Lewis, DL: Counterfactuals , Oxford: Basil Blackwell 1973
^ Cohen, LJ: The Probable and the Provable , Oxford: Clarendon Press 1977
^ Dempster, AP: Upper and lower probabilities induced by a multivalued mapping , Annals of Mathematical Statistics 38 (1967) 325-339
^ Shafer, G .: A Mathematical Theory of Evidence , Princeton, University Press 1976

[Zadeh-1] Zadeh, L .: Fuzzy Sets as the Basis for a Theory of Possibility , Fuzzy Sets and Systems 1 (1978) 3-28. doi : 10.1016 / S0165-0114 (99) 80004-9

[2] Dubois, D. and Prade, H .: Possibility Theory: An Approach to Computerized Processing of Uncertainty , New York: Plenum Press 1988

[3] Shackle, GLS: Decision, Order and Time in Human Affairs , 2nd edition, Cambridge University Press

[4] Lewis, DL: Counterfactuals , Oxford: Basil Blackwell 1973

[5] Cohen, LJ: The Probable and the Provable , Oxford: Clarendon Press 1977

[6] Dempster, AP: Upper and lower probabilities induced by a multivalued mapping , Annals of Mathematical Statistics 38 (1967) 325-339

[7] Shafer, G .: A Mathematical Theory of Evidence , Princeton, University Press 1976