Transcript Document

Fuzzy sets and probability:
Misunderstandings, bridges and
gaps
Paper Authors: Didier Dubois
Henri Prade
Presenter: Hao Lac
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Seminar Outline
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Introduction
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Misunderstandings
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Likelihood Function
Fuzzy Sets in Statistical Inference
Gaps
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Membership Function and Probability Measure
Fuzzy Relative Cardinality and Conditional Probability
Possibility Theory is not Compositional
Bridges
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probability versus fuzzy set
Possibility as Preference
Possibility as Similarity
Possibility-Probability Transformations
Conclusion
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Introduction
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Address “probability versus fuzzy set” challenge.
Main points:
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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
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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.
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Misunderstandings: Membership
Function and Probability Measure
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Probability space (U, 2U, P)
U
Probability measure P maps 2  [0,1]
• assigns a number P(A) to each crisp subset of U.
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Satisfies the Kolmogorov axioms:
P(U )  1; P()  0
if A  B   P(A  B)  P(A)  P(B)
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Misunderstandings: Membership
Function and Probability Measure
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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
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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|
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Misunderstandings: Fuzzy Relative
Cardinality and Conditional Probability
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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
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 )
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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 |).
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Misunderstandings: Fuzzy Relative
Cardinality and Conditional Probability
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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)
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Misunderstandings: Fuzzy Relative
Cardinality and Conditional Probability
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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 )
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Misunderstandings: Possibility
Theory is not Compositional
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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 uA ({u})
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Misunderstandings: Possibility
Theory is not Compositional
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Zadeh equates πx(u)= µF(u).
πx(u) is short for π(x = u | F)
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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)
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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
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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))
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Bridges: Likelihood Function
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The membership function can also be
defined as:
μF (u )  P(' F ' | u ), u  U ,
where ' F' is a non - fuzzy event.
Why?
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Bridges: Likelihood Function
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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.
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Bridges: Fuzzy Sets in Statistical
Inference
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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)
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Bridges: Fuzzy Sets in Statistical
Inference
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and that the likelihood functions are
P(O | u), u U, then the likelihood ratio
test methodology suggests the comparison
between max uA P(O | u) and max uA P(O | u )
recall that
μF (u)  P(' F '| u), u U .
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Bridges: Fuzzy Sets in Statistical
Inference
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Then the Bayesian updating procedure:
P(O | u )  p (u )
p (u | O) 
P(O)
can be reinterpreted in terms of fuzzy
observations.
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Bridges: Fuzzy Sets in Statistical
Inference
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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.
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Gaps: Possibility as Preference
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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
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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.
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Gaps: Possibility as Preference
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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.
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Gaps: Possibility as Preference
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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
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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:
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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.
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Possibility-Probability
Transformations
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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
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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:
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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.
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Possibility-Probability
Transformations
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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
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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
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Investigations of the relationships between fuzzy
set, possibility and probability may be fruitful:
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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!
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