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  Possibility Theory

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free men
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التوقيع : رئيس ومنسق القسم الفكري

عدد الرسائل : 1500

الموقع : center d enfer
تاريخ التسجيل : 26/10/2009
وســــــــــام النشــــــــــــــاط : 6

 Possibility Theory Empty
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مُساهمة Possibility Theory

Possibility theory (Dubois & Prade 1988) is based on fuzzy set theory (Zadeh 1978). According to the latter theory, an element need not belong to a given set either completely or not at all, but may be a member of the set to a certain degree. For instance, Sophia may belong to the set of politically active people to a degree of .88. This is represented by a membership function [ltr]μ[size=13]A:W→[0,1]μA:W→[0,1][/ltr], where [ltr]μA(w)μA(w)[/ltr] is the degree of membership to which person [ltr]w∈Ww∈W[/ltr] belongs to the set of politically active people [ltr]AA[/ltr].[/size]
Furthermore, the degree [ltr]μ[size=13]WAμW∖A[/ltr](Sophia) to which Sophia belongs to the set [ltr]W∖AW∖A[/ltr] of people who are not politically active equals [ltr]1−μA1−μA[/ltr](Sophia). Moreover, if [ltr]μM:W→[0,1]μM:W→[0,1][/ltr] is the membership function for the set of philosophically minded people, then the degree of membership to which Sophia belongs to the set [ltr]A∪MA∪M[/ltr] of politically active or philosophically minded people is given by[/size]
[ltr]μ[size=13]AM(Sophia)=max{μA(Sophia),μM(Sophia)}.μA∪M(Sophia)=max{μA(Sophia),μM(Sophia)}.[/ltr][/size]
Similarly, the degree of membership to which Sophia belongs to the set [ltr]A∩MA∩M[/ltr] of politically active and philosophically minded people is given by
[ltr]μ[size=13]AM(Sophia)=min{μA(Sophia),μM(Sophia)}.μA∩M(Sophia)=min{μA(Sophia),μM(Sophia)}.[/ltr][/size]
[ltr]m[size=13]AM(Sophia)mA∩M(Sophia)[/ltr] is interpreted as the degree to which the vague statement “Sophia is a politically active and philosophically minded person” is true (for vagueness see Égré & Barberousse 2014, Raffman 2014, Williamson 1994 as well as the entry on vagueness; Field (forthcoming) discusses uncertainty due to vagueness in a probabilistic setting). Degrees of truth belong to the philosophy of language. They do not (yet) have anything to do with degrees of belief, which belong to epistemology. In particular, note that degrees of truth are usually considered to be truth functional (the truth value of a compound statement such as [ltr]A∧BA∧B[/ltr] is a function of the truth values of its constituent statements [ltr]A,BA,B[/ltr]; that is, the truth values of [ltr]AA[/ltr] and [ltr]BB[/ltr] determine the truth value of [ltr]A∧B)A∧B)[/ltr]. Degrees of belief, on the other hand, are hardly ever considered to be truth functional. For instance, probabilities are not truth functional, because the probability of [ltr]A∩BA∩B[/ltr]is not determined by the probability of [ltr]AA[/ltr] and the probability of [ltr]BB[/ltr]. That is, there is no function [ltr]ff[/ltr]such that for all probability spaces [ltr]⟨W,A,Pr⟩⟨W,A,Pr⟩[/ltr] and all propositions [ltr]A,BA,B[/ltr] in [ltr]AA[/ltr]: [ltr]Pr(A∩B)=f(Pr(A),Pr(B))Pr(A∩B)=f(Pr(A),Pr(B))[/ltr].[/size]
Suppose someone says that Sophia is tall. How tall is a tall person? Is a person with a height of [ltr]5[size=13]′9′′5′9″[/ltr] tall? Or does a person have to be at least [ltr]510′′5′10″[/ltr] in order to be tall? Although you know that Sophia is tall, your knowledge is incomplete due to the vagueness of the term ‘tall’. Here possibility theory enters by equipping you with a (normalized) possibility distribution, a function[ltr]π:W→[0,1]π:W→[0,1][/ltr] with [ltr]π(w)=1π(w)=1[/ltr] for at least one [ltr]w∈Ww∈W[/ltr]. The motivation for the latter requirement is that at least (in fact, exactly) one possibility is the actual possibility, and hence at least one possibility must be maximally possible. Such a possibility distribution [ltr]π:W→[0,1]π:W→[0,1][/ltr]on the set of possibilities [ltr]WW[/ltr] is extended to a possibility measure [ltr]Π:A→RΠ:A→ℜ[/ltr] on the field [ltr]AA[/ltr]over [ltr]WW[/ltr] by defining for each [ltr]AA[/ltr] in [ltr]AA[/ltr],[/size]
[ltr]Π(∅)=0,Π(A)=sup{π(w):w∈A}.Π(∅)=0,Π(A)=sup{π(w):w∈A}.[/ltr]
This entails that possibility measures [ltr]Π:A→RΠ:A→ℜ[/ltr] are maximitive (and hence sub-additive), i.e., for all [ltr]A,B∈AA,B∈A[/ltr]:
[ltr]Π(A∪B)=max{Π(A),Π(B)}.(8)(8)Π(A∪B)=max{Π(A),Π(B)}.[/ltr]
The idea is that, roughly, a proposition is at least as possible as each of the possibilities it comprises, and no more possible than the “most possible” possibility. Sometimes, though, there is no most possible possibility (i.e., the supremum is no maximum). For instance, this is the case when the degrees of possibility are [ltr][size=13]1/2,3/4,7/8,…,2n1/2n,…1⁄2,3⁄4,7⁄8,…,2n−1⁄2n,…[/ltr] In this case the degree of possibility for the proposition is the smallest number which is at least as great as all the degrees of possibilities of its elements. In our example this is 1. (As will be seen below, this is the main formal difference between possibility measures and non-conditional ranking functions.)[/size]
We can define possibility measures without recourse to an underlying possibility distribution as functions [ltr]Π:A→RΠ:A→ℜ[/ltr] such that for all [ltr]A,B∈AA,B∈A[/ltr]:
[ltr]Π(∅)Π(W)Π(A∪B)=0,=1, and =max{Π(A),Π(B)}.Π(∅)=0,Π(W)=1, and Π(A∪B)=max{Π(A),Π(B)}.[/ltr]
It is important to note, though, that the last clause is not well-defined for disjunctions or unions of infinitely many propositions (in this case one would have to use the supremum operation sup instead of the maximum operation max). The dual notion of a necessity measure [ltr]N:A→RN:A→ℜ[/ltr] is defined for all [ltr]AA[/ltr] in [ltr]AA[/ltr] by
[ltr]N(A)=1−Π(W∖A).N(A)=1−Π(W∖A).[/ltr]
This implies that
[ltr]N(A∩B)=min{N(A),N(B)}.N(A∩B)=min{N(A),N(B)}.[/ltr]
The latter equation can be used to start with necessity measures as primitive. Define them as functions [ltr]N:A→RN:A→ℜ[/ltr] such that for all [ltr]A,B∈AA,B∈A[/ltr]:
[ltr]N(∅)N(W)N(A∩B)=0,=1,=min{N(A),N(B)}.N(∅)=0,N(W)=1,N(A∩B)=min{N(A),N(B)}.[/ltr]
Then possibility measures [ltr]Π:A→RΠ:A→ℜ[/ltr] are obtained by the equation:

[ltr]Π(A)=1−N(W∖A).[/ltr]
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Possibility Theory :: تعاليق

free men
رد: Possibility Theory
مُساهمة الثلاثاء مارس 15, 2016 11:11 am من طرف free men
Although the agent’s doxastic state in possibility theory is completely specified by either [ltr]ΠΠ[/ltr] or [ltr]NN[/ltr], the agent’s epistemic attitude towards a particular proposition [ltr]AA[/ltr] is only jointly specified by [ltr]Π(A)Π(A)[/ltr] and [ltr]N(A)N(A)[/ltr]. The reason is that, in contrast to probability theory, [ltr]Π(W∖A)Π(W∖A)[/ltr] is not determined by [ltr]Π(A)Π(A)[/ltr]. Thus, degrees of possibility (as well as degrees of necessity) are not truth functional either. The same is true for DS belief and plausibility functions.
In our example, let [ltr]W[size=13]HWH[/ltr] be the set of values of the random variable [ltr]H=H=[/ltr] Sophia’s height in inches between [ltr]0′′0″[/ltr] and [ltr]199′′199″[/ltr], [ltr]WH={0,…,199}.WH={0,…,199}.[/ltr] [ltr]πH:WH→[0,1]πH:WH→[0,1][/ltr] is your possibility distribution. It is supposed to represent your doxastic state concerning Sophia’s height, which contains the knowledge that she is tall. For instance, your [ltr]πHπH[/ltr] might be such that [ltr]πH(n)=1πH(n)=1[/ltr] for any natural number [ltr]n∈[60,72]⊆Wn∈[60,72]⊆W[/ltr]. In this case your degree of possibility for the proposition that Sophia is at least [ltr]510′′5′10″[/ltr] is [ltr]ΠH(H≥70)=sup{πH(n):n≥70}=1ΠH(H≥70)=sup{πH(n):n≥70}=1[/ltr].[/size]
The connection to fuzzy set theory is that your possibility distribution [ltr]π[size=13]H:WH→[0,1]πH:WH→[0,1][/ltr], which is based on the knowledge that Sophia is tall, can be interpreted as the membership function [ltr]μT:WH→[0,1]μT:WH→[0,1][/ltr] of the set of tall people. So the epistemological thesis of possibility theory is that your degree of possibility for the proposition that Sophia is [ltr]510′′5′10″[/ltr] given the vague and hence incomplete knowledge that Sophia is tall should equal the degree of membership to which a[ltr]510′′5′10″[/ltr] tall person belongs to the set of tall people. In more suggestive notation,[/size]
[ltr]π[size=13]H(H=n∣T)=μT(n).πH(H=n∣T)=μT(n).[/ltr][/size]
Let us summarize the accounts we have dealt with so far. Subjective probability theory requires degrees of belief to be additive. An ideal doxastic agent’s subjective probability Pr: [ltr]A→RA→ℜ[/ltr] is such that for any [ltr]A,BA,B[/ltr] in [ltr]AA[/ltr] with [ltr]A∩B=∅A∩B=∅[/ltr]:
[ltr]Pr(A)+Pr(B)=Pr(A∪B)Pr(A)+Pr(B)=Pr(A∪B)[/ltr]
The theory of DS belief functions requires degrees of belief to be super-additive. An ideal doxastic agent’s DS belief function Bel: [ltr]A→RA→ℜ[/ltr] is such that for any [ltr]A,BA,B[/ltr] in [ltr]AA[/ltr] with [ltr]A∩B=∅A∩B=∅[/ltr]:
[ltr]Bel(A)+Bel(B)≤Bel(A∪B)Bel(A)+Bel(B)≤Bel(A∪B)[/ltr]
Possibility theory requires degrees of belief to be maxitive and hence sub-additive. An ideal doxastic agent’s possibility measure [ltr]Π:A→RΠ:A→ℜ[/ltr] is such that for any [ltr]A,BA,B[/ltr] in [ltr]AA[/ltr]:
[ltr]Π(A)+Π(B)≥max{Π(A),Π(B)}=Π(A∪B)Π(A)+Π(B)≥max{Π(A),Π(B)}=Π(A∪B)[/ltr]
All of these functions are special cases of real-valued plausibility measures Pl: [ltr]A→RA→ℜ[/ltr], which are such that for all [ltr]A,BA,B[/ltr] in [ltr]AA[/ltr]:
[ltr]Pl(A)≤Pl(B) if A⊆B.Pl(A)≤Pl(B) if A⊆B.[/ltr]
We have seen that each of these accounts provides an adequate model for some doxastic situation (plausibility measures do so trivially). We have further noticed that subjective probabilities do not immediately give rise to a notion of qualitative belief that is consistent and deductively closed (unless qualitative belief is identified with a subjective probability of 1). Therefore the same is true for the more general DS belief functions and plausibility measures. Besides the accounts of Leitgeb (2013; 2014) and Lin & Kelly (2012) discussed above it should be noted that Roorda (1995, Other Internet Resources) provides a definition of belief in terms of sets of probabilities. (As will be mentioned in the next section, there is a notion of belief in possibility theory that is consistent and deductively closed in a finite sense.)

Moreover, we have seen arguments for the thesis that degrees of belief should obey the probability axioms. Smets (2002) tries to justify the corresponding thesis for DS belief functions. To the best of my knowledge nobody has yet published an argument for the thesis that degrees of belief should be plausibility or possibility measures, respectively (in the sense that all and only plausibility respectively possibility measures are rational degree of belief functions). However, there exists such an argument for ranking functions, which are formally similar to possibility measures. Ranking functions also give rise to a notion of belief that is consistent and deductively closed (indeed, this very feature is the starting point for the argument that doxastic states should obey the ranking calculus). They are the topic of the next section.
 

Possibility Theory

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