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Fuzzy sets and probability: Misunderstandings, bridges and gaps Paper Authors: Didier Dubois Henri Prade Presenter: Hao Lac 1 Seminar Outline • Introduction • • Misunderstandings • • • • • • • Likelihood Function Fuzzy Sets in Statistical Inference Gaps • • Membership Function and Probability Measure Fuzzy Relative Cardinality and Conditional Probability Possibility Theory is not Compositional Bridges • • probability versus fuzzy set Possibility as Preference Possibility as Similarity Possibility-Probability Transformations Conclusion 2 Introduction • • Address “probability versus fuzzy set” challenge. Main points: • • • • Consistent body of mathematical tools Several bridges to reconcile opposite points of view (possibility theory) Fuzzy sets in probability are not random objects Alternative point view of fuzzy sets and possibility theory 3 Misunderstandings: Membership Function and Probability Measure • Fuzzy set F on a universe U is defined by: m F : U [0,1] mF (u) is the grade of membership of element u in F. 4 Misunderstandings: Membership Function and Probability Measure • • Probability space (U, 2U, P) U Probability measure P maps 2 [0,1] • assigns a number P(A) to each crisp subset of U. • Satisfies the Kolmogorov axioms: P(U ) 1; P() 0 if A B P(A B) P(A) P(B) 5 Misunderstandings: Membership Function and Probability Measure • • For P(A), the set A is well defined while the value of the underlying variable x, to which P is attached is, unknown (and moves). For µF(u), the element u is fixed and known and the set is ill-defined. 6 Misunderstandings: Fuzzy Relative Cardinality and Conditional Probability • • The cardinality of a fuzzy set F defined on U is: | F | u μF (u ) An index of inclusion of F in another fuzzy set G is: | F G | I( F , G ) |F| 7 Misunderstandings: Fuzzy Relative Cardinality and Conditional Probability • Bart Kosko claims that there is an analogy that exists between I(F, G) and Bayes’ conditional probability P(B | A), where B and G play the same role. 8 Misunderstandings: Fuzzy Relative Cardinality and Conditional Probability I (G, F ) I (U , G) I ( F , G) I (G, F ) I (U , G) I (G , F ) I (U , G ) P( A | B) P( B) P( B | A) P( A | B) P( B) P( A | B ) P( B ) • That I(F, G) implies P(B | A) is debatable because this would mean the former is a special kind of conditional probability; that is, P is uniformly distributed on U (i.e. P(B | A) = | A ∩ B | / | A |). 9 Misunderstandings: Fuzzy Relative Cardinality and Conditional Probability • To generalize both I(F, G) and P(B | A) we need probability measure P on U and consider [0,1]U as a set of fuzzy events: P( F ) p(u) μF (u) 10 Misunderstandings: Fuzzy Relative Cardinality and Conditional Probability • Then I(F, G) becomes: P( F G ) P (G | F ) P( F ) so, changing I(F, G) into P(Y | X): P( F | G) P(G | U ) P(G | F ) P( F | G) P(G | U ) P( F | G ) P(G | U ) 11 Misunderstandings: Possibility Theory is not Compositional • A possibility measure on a finite set U is a mapping from 2U to [0,1] such that: () 0 (U ) 1 if A B, then ( A) ( B) (monotonicity) ( A) sup uA ({u}) 12 Misunderstandings: Possibility Theory is not Compositional • • Zadeh equates πx(u)= µF(u). πx(u) is short for π(x = u | F) • • It estimates the possibility that the variable x is equal to u, knowing the incomplete state of knowledge “x is F” µF(u) is short for µ(F | x = u) • • It estimates the degree of compatibility of the precise information x = u with the statement to evaluate “x is F” Similar to likelihood functions. 13 Misunderstandings: Possibility Theory is not Compositional • • Controversy between possibility measures is related to the one about fuzzy sets. Union and intersection is not compositional due to monotonicity: ( A B) min( ( A), ( B)) ( A B) max( ( A), ( B)) 14 Bridges: Likelihood Function • The membership function can also be defined as: μF (u ) P(' F ' | u ), u U , wh ere ' F' is a non - fuzzy event. Why? 15 Bridges: Likelihood Function • Consider a population of individuals and a fuzzy concept F; each individual is then asked whether a given element u U can be called an F or not. The likelihood function P(‘F’ | u) is then obtained and represents the proportion of individuals that answered yes to the question. Thus, ‘F’ must be a non-fuzzy event. 16 Bridges: Fuzzy Sets in Statistical Inference • • Likelihood functions are treated as possibility distributions in classical statistics for so-called likelihood ratio tests. Consider some hypothesis of the form u A is to be tested against the opposite hypothesis u A on the basis of observation O alone, (cont. on next slide) 17 Bridges: Fuzzy Sets in Statistical Inference • • and that the likelihood functions are P(O | u), u U, then the likelihood ratio test methodology suggests the comparison between max uA P(O | u) and max uA P(O | u ) recall that μF (u) P(' F '| u), u U . 18 Bridges: Fuzzy Sets in Statistical Inference • Then the Bayesian updating procedure: P(O | u ) p(u ) p(u | O) P(O) can be reinterpreted in terms of fuzzy observations. 19 Bridges: Fuzzy Sets in Statistical Inference • Therefore, the a posteriori probability can be redefined as: μF (u ) p(u ) p(u | F ) P( F ) where P(F) is Zadeh’s probability of a fuzzy event. 20 Gaps: Possibility as Preference • • • It is not always meaningful to relate uncertainty to frequency. Some events can be rare, unrepeatable, or statistical data may be unavailable. However, this does not prevent us from thinking that some events are more possible, probable or certain than others. 21 Gaps: Possibility as Preference • • Comparative possibility is a recent theory that allows comparing events by defining a complete pre-ordering on 2U. The complete pre-ordering ≥Π such that A ≥Π B means A is at least as possible as B should satisfy the basic axiom: C, A B A C B C. 22 Gaps: Possibility as Preference • • Dubois proved that the only numerical counterparts of comparative possibility are possibility measures. The significance of this is that a comparative relation on 2U describing the location of an unknown variable x induces a complete preordering on U that can be viewed as a preference relation on the possible values of x. 23 Gaps: Possibility as Preference • This means that qualitative possibility distributions can be analyzed from the point of view of their informational content. 24 Gaps: Possibility as Similarity • • • The degree of membership µF(u) reflects the similarity between u and an ideal prototype uF of F (for which µF(uF) = 1). Relation to distance and not probability. Example: • If a variable x is attached a possibility distribution π = µF, x = u is all the more possible as u looks like uF, is close to uF. 25 Possibility-Probability Transformations • • Possibility-Probability Transformations are meaningful in the scope of uncertainty combination with heterogeneous sources (some supplying statistical data, other linguistic data, for instance). Issue with transformation: does some consistency exists between possibilistic and probabilistic representations of uncertainty? 26 Possibility-Probability Transformations • • Assumptions made: the translation between languages are neither weaker or stronger than the other (Klir and Parviz). Leads to transformation that respect the principle of uncertainty and information invariance, on the basis that: • H(p) = NS(π), where H(p) is the entropy measure based on the probability distribution p and NS(π) nonspecificity measure based on the possibility distribution π. 27 Possibility-Probability Transformations • • • Another view is that possibility and probability theories have distinct roles in describing uncertainty but do not have the same descriptive power. Probability theory can describe total randomness while possibility theory cannot. Possibility theory can express ignorance while probability theory cannot. 28 Possibility-Probability Transformations • • However, mathematically possibility representation is weaker than probability representation due to the fact that the former represents a set of probability measures (i.e. a weaker knowledge than the one of a single probability measure). Implications: • Going from possibility to probability leads to an increase in the informational content of the considered representation. • Going from probability to possibility leads to a loss in information. 29 Conclusion • Investigations of the relationships between fuzzy set, possibility and probability may be fruitful: • • • • Correcting some misunderstandings which are quite prevalent in the literature. Fuzzy set-theoretic operations can be justified from probabilistic viewpoint such as a likelihood function. Possibilistic nature of likelihood seems to be in accordance with the way statisticians have used them. Encouraging conjoint use of fuzzy sets and probability in applications. 30 Thank You! 31