18

Inference with Probabilistic and Fuzzy Information

Giulianella Coletti and Barbara Vantaggi

18.1 Introduction

We adopt the interpretation of fuzzy sets in terms of coherent conditional probabil-ities, introduced in [2–4] and presented in this issue [7] by R. Scozzafava. Aim ofthis chapter is to discuss (from a syntactical point of view) which concepts of fuzzysets theory [9] are naturally obtained simply by using coherence. In particular, wefocus on operations among fuzzy subsets (and relevant t-norms and t-conorms), andon Bayesian inference procedures, when statistical and fuzzy information must betaken into account. It is obvious that in the case of inference with hybrid informationthe proposed interpretation of the membership provides a general and well foundedframework (that of coherent conditional probability ) for merging and managing allthe available information. For instance, in this frame the simplest inferential prob-lem (to find the most probable element of a data base, starting from a probabilitydistribution on the single elements and a fuzzy information expressed by a member-ship function defined on the elements of the data base) is referable to a Bayesianupdating of an initial probability. The only remarkable question is that the Bayesformula is applied in an unusual semantic way: the distribution, which plays the roleof “prior” probability, is here usually obtained by statistical data, whereas the mem-bership function, which plays the role of “likelihood”, is a subjective evaluation.

We refer (see Section 18.2) to the results about coherent conditional probabilityrecalled in [7] to find the class of t-norms and t-conorms such that the membership ofthe union and intersection of two fuzzy sets obtained by them is a coherent extensionof the two coherent conditional probabilities modeling the initial fuzzy sets. In Sec-tion 18.3 hybrid information is handled by maintaining consistence with the modeland this gives rise to a general inferential model able to deal with different kinds ofapplications.

18.2 Operations among Fuzzy Sets

Let X be a (not necessarily numerical) variable, with range CX , and , for any x ∈CX , let Ax = {X = x}. For any property ϕ related to the variable X we considerthe conditional event Eϕ |Ax = {You claim (that X has the property) ϕ under thehypothesis that X assumes the value x }, recalled in [7]. The membership μϕ of a

R. Seising et al. (Eds.): On Fuzziness: Volume 1, STUDFUZZ 298, pp. 115–119.DOI: 10.1007/978-3-642-35641-4_18 © Springer-Verlag Berlin Heidelberg 2013

116 18 Inference with Probabilistic and Fuzzy Information

fuzzy set E∗ϕ = (Eϕ ,μϕ ) can be reinterpreted by means of the conditional probabilityP(Eϕ |·) in fact there is complete freedom in assessing it (see [4]).

We recall that the events Ei (i= 1, ...,n) are logically independent if all the disjunc-tions E∗1 ∧ ...∧E∗n (where E∗i represents the event Ei or its contrary Ec

i ) are possible,that is the atoms generated by events Ei are 2n. The following result concerns theglobal coherence of a set of probability assessments and it is useful for a family offuzzy subsets such that the events Eϕi are logically independent in order to show thatthe probability rules do not impose constrains to the membership functions (see [1]).

Theorem 1. Let C = {E j|Hji}i=1,...,n j ; j=1,...,n be a set of conditional events suchthat H j = {Hj1 , ...,Hjn j

} is a partition of Ω , for every j, and the events of E =

{E j}i=1,...,n are logically independent. For every j, let Pj : H j �→ [0,1] be a proba-bility distribution and p(E j|·) : H j �→ [0,1] a coherent conditional probability.

If the probability distributions Pj’s are “globally” coherent on H ∗=⋃

j H j , thenthe assessment {Pj, p(E j|·)} j=1,...,n is “globally” coherent in E ×H ∗.

This result is important since generally events Eϕ representing fuzzy sets are logi-cally independent, even those seemingly linked: as an example we consider Eϕ andEψ , with ψ = ¬ϕ which are logical independent, since we can claim both “X hasthe property ϕ” and “X has the property ¬ϕ ′′.

Now we are going to introduce the operations between fuzzy sets by referringto [3]: the definitions of the binary operations of union and intersection and that ofcomplementation can be obtained directly by using the rules of coherent conditionalprobability. For this aim let us denote by ϕ ∨ψ , ϕ ∧ψ , respectively, the properties“ϕ or ψ ” , “ϕ and ψ ”.

As proved in [3], for any given x in the range of X , the assessmentP(Eϕ ∧Eψ |Ax) = v is coherent if and only if it takes values in the interval

max{P(Eϕ |Ax)+P(Eψ |Ax)− 1,0} ≤ v≤min{P(Eϕ |Ax),P(Eψ |Ax)}. (18.1)

Now we need to define Eϕ∨ψ = Eϕ ∨Eψ , Eϕ∧ψ = Eϕ ∧Eψ .Let us consider two fuzzy subsets E∗ϕ , E∗ψ , corresponding to the same variable x,

with the events Eϕ , Eψ logically independent. As proved in [3], for any given X inthe range of X , the assessment P(Eϕ ∧Eψ |Ax) = v is coherent if and only if takesvalues in the interval

max{P(Eϕ |Ax)+P(Eψ|Ax)− 1,0} ≤ v≤min{P(Eϕ |Ax),P(Eψ |Ax)} (18.2)

and moreover any choice of the values for P(Eϕ ∧Eψ |Ax) in the corresponding in-tervals is a coherent conditional probability assessment. From the probability rules,given P(Eϕ ∧Eψ |Ax), we get automatically also the value of P(Eϕ ∨Eψ |Ax). Then,it is possible to put

E∗ϕ ∪E∗ψ = {Eϕ∨ψ , μϕ∨ψ} , E∗ϕ ∩E∗ψ = {Eϕ∧ψ , μϕ∧ψ} , (18.3)

with μϕ∨ψ (x) = P(Eϕ ∨Eψ |Ax) , μϕ∧ψ(x) = P(Eϕ ∧Eψ |Ax).

18.3 Inference with Fuzzy and Probabilistic Information 117

Three possible coherent choices for the value of the conditional probability v giverise to different well-known (see, e.g., [5]) t-norms and t-conorms: in [3] (seealso [2]) the choice of the so-called TM and SM as t-norm and t-conorm, of theLukasiewicz t-norm and t-conorm, and, finally, of the so-called probabilistic sumSP and product TP is discussed also with semantic implications. As it is well knownthese three coherent choices correspond to the particular values λ = 0, λ = 1, λ = ∞,respectively, of the fundamental (archimedean) Frank (see [6]) t-norms Tλ and t-conorms Sλ , with λ ∈ [0,∞], which are in fact all and only the possible coherentchoices for v and u, [3]. We notice that the condition of logical independence ofevents Eϕ ,Eψ is crucial for proving the above assertions. So if we have a family oflogically independent events Eϕi and consider the algebra B spanned by them, wecan use any Frank’s t-norm and its dual t-conorm to compute any union and inter-section between to relevant fuzzy sets (Eϕi ,μi); coherence rules the extension of theconditional probability P(·|Ax) to the other events of the algebra (for instance to theevents Ec

ϕ ), which do not support a fuzzy set. Starting from these considerations wecan define the complement of a fuzzy set

(E∗ϕ)′ = {E¬ϕ , μ¬ϕ}. (18.4)

We recall that: E¬ϕ = (Eϕ)c . Then, while Eϕ ∨ (Eϕ)

c = Ω , one hasEϕ ∨E¬ϕ ⊂ Ω . So the membership μϕ∨¬ϕ (x) = P(Eϕ ∨E¬ϕ |Ax) can be differentfrom 1 for some x ∈ CX . In other words E∗ϕ∨¬ϕ is a fuzzy set.

The case of two fuzzy subsets E∗ϕ , E∗ψ , corresponding to the random quantities X1

and X2, respectively, has been studied in [3] by assuming the following conditionalindependence condition: for every (x,x′) belonging to the range of the random vector(X1,X2)

P(Eϕ |Ax∧Ax′) = P(Eϕ |Ax) , P(Eψ |Ax∧Ax′) = P(Eψ |Ax′) . (18.5)

In [1] the same problem has been studied without independence conditions. In bothcases it is possible to conclude that the following choice for the membership of con-junction and disjunction is coherent:

μϕ∨ψ(x,x′) = P(Eϕ ∨Eψ |Ax∧Ax′), μϕ∧ψ(x,x

′) = P(Eϕ ∧Eψ |Ax∧Ax′) . (18.6)

with the only constraints

max{μϕ(x)+ μψ(x′)− 1 , 0} ≤ μϕ∧ψ (x,x

′)≤min{μϕ(x)+ μψ(x′)} . (18.7)

andμϕ∨ψ (x,x

′) = μϕ(x)+ μψ(x′)− μϕ∧ψ(x,x

′) . (18.8)

18.3 Inference with Fuzzy and Probabilistic Information

Our first aim is the following: if we have a probability distribution on the elements ofCX and a fuzzy information, expressed by a membership function μϕ(·) = P(Eϕ |·),we would choose the most probable element x ∈ CX under the hypothesis Eϕ .

118 References

By Theorem 1 the global assessment {P,μϕ} is coherent and so we can compute, forevery x ∈ CX the value P(Ax|Eϕ). We can compute the extension by Bayes formula:

P(Ax|Eϕ) = αP(Ax)μϕ(x) (18.9)

where α = (∑x μϕ (x)P(Ax))−1.

So, to reach our goal it is sufficient to find the events Ax∗ with maximum posterior,i.e.

P(Ax∗ |Eϕ) = α maxx{P(Ax)μϕ (x)} (18.10)

In [1] more general situations have been studied in order to give some algorithmsfor finding the most probable element of CX also when the statistical information isrelated to a family different from {Ax}. Moreover, in the same paper this inferentialmodel is applied for the virtual representation of a female avatar based also on thesimilarities studied in this context in [8].

A particular kind of inference is that at the basis of the ”perception based proba-bilistic reasoning" introduced by Zadeh in [10]). We sketch a solution alternative tothat given by Zadeh, based on our interpretation of fuzzy set, starting from a simpleexample.

A box contains n balls of various sizes s1, ...,sm, with m≤ n, with unknown per-centages. Consider an experiment consisting in drawing a ball from the box, and letEϕ be the event (referred to the drawn ball) “You claim that the size is large”. Con-sider also the event Eψ=“You claim that the size of most ball is large”The problemis: what is the probability of Eϕ |Eψ?

The fuzzy subset Eϕ is related to the variable (the size) S and if the compositionof the urn were known, then the probability of Eϕ would be computed by disintegra-tion formula. Otherwise we need to refer to the possible compositions Hk, and theprobability αk = P(Eϕ |Hk) is obtained again by disintegration formula with respectto the possible values of S. Let P be the variable taking the possible values αk.

Note that Eψ can also be expressed by the sentence (referred to the ball to bedrawn) “You claim that the probability of being claimed large is high”, and the mem-bership of the fuzzy subset of claiming high EH is P(EH |Aαr) where Ar = {P =αr}.

Concerning the conditional event Eϕ |Eψ , by assuming conditional independenceof Eϕ and Eψ given the possible compositions Hk, we have:

P(Eϕ |Eψ) = ∑k

P(Eϕ |Hk)P(Hk|Eψ) , (18.11)

where P(Hk|Eψ) is obtained by Bayes’ formula.

References

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2. Coletti, G., Scozzafava, R.: Probabilistic Logic in a Coherent Setting. In: Trends in Logic,vol. 15. Kluwer, Dordrecht (2002)

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