Fuzzy sets and probability: Misunderstandings, bridges and gaps

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Fuzzy sets and probabilityMisunderstandings,
bridges and gaps

• Paper Authors Didier Dubois
• Henri Prade
• Presenter Hao Lac

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Seminar Outline

• Introduction
• probability versus fuzzy set
• Misunderstandings
• Membership Function and Probability Measure
• Fuzzy Relative Cardinality and Conditional
  Probability
• Possibility Theory is not Compositional
• Bridges
• Likelihood Function
• Fuzzy Sets in Statistical Inference
• Gaps
• Possibility as Preference
• Possibility as Similarity
• Possibility-Probability Transformations
• Conclusion

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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

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Misunderstandings Membership Function and
Probability Measure

• Fuzzy set F on a universe U is defined by

is the grade of membership of element u
in F.
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Misunderstandings Membership Function and
Probability Measure

• Probability space (U, 2U, P)
• Probability measure P maps

• assigns a number P(A) to each crisp subset of U.

• Satisfies the Kolmogorov axioms

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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.

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Misunderstandings Fuzzy Relative Cardinality and
Conditional Probability

• The cardinality of a fuzzy set F defined on U is

• An index of inclusion of F in another fuzzy
  

set G is
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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.

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Misunderstandings Fuzzy Relative Cardinality and
Conditional Probability

• 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 n
  B / A ).

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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 0,1U as
  a set of fuzzy events

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Misunderstandings Fuzzy Relative Cardinality and
Conditional Probability

• Then I(F, G) becomes

so, changing I(F, G) into P(Y X)
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Misunderstandings Possibility Theory is not
Compositional

• A possibility measure on a finite set U is a
  mapping from 2U to 0,1 such that

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Misunderstandings Possibility Theory is not
Compositional

• Zadeh equates px(u) µF(u).
• px(u) is short for p(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.

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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

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Bridges Likelihood Function

• The membership function can also be defined as

Why?
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Bridges Likelihood Function

• Consider a population of individuals and a fuzzy
  concept F each individual is then asked whether
  a given element

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.
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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

is to be tested against the opposite
hypothesis
on the basis of
observation O alone, (cont. on next slide)
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Bridges Fuzzy Sets in Statistical Inference

• and that the likelihood functions are P(O
  u),

, then the likelihood ratio
test methodology suggests the comparison between
and

• recall that

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Bridges Fuzzy Sets in Statistical Inference

• Then the Bayesian updating procedure

can be reinterpreted in terms of fuzzy
observations.
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Bridges Fuzzy Sets in Statistical Inference

• Therefore, the a posteriori probability can be
  redefined as

where P(F) is Zadehs probability of a fuzzy
event.
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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.

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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 ? such that A ? B
  means A is at least as possible as B should
  satisfy the basic axiom

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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
  pre-ordering on U that can be viewed as a
  preference relation on the possible values of x.

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Gaps Possibility as Preference

• This means that qualitative possibility
  distributions can be analyzed from the point of
  view of their informational content.

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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 p µF, x u is all the more
  possible as u looks like uF, is close to uF.

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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?

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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(p), where H(p) is the entropy measure
  based on the probability distribution p and NS(p)
  non-specificity measure based on the possibility
  distribution p.

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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.

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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.

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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.

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• Thank You!