◎ Possibility Theory is a strict correlate to Probability Theory.
◎ Possibility measures are also a class of non-additive fuzzy measures.
◎ The Fuzzy world has a lot of correlates to the Classical world.
◎ Fuzzy sets and possibility theory are nothing more than probability theory is rebutted directly by Klir [1989]. This view might be related to the fact that Shafer's Belief and
Plausibility measures are derived from a probability measure on the power set of the power set of the universe, and that possibility measures are a class of plausibility measures.
◎ Two reasons to introduce the concept of a fuzzy measure
。Providing a broad framework within which it is convenience to introduce and examine possibility theory
。Explicating difference between fuzzy set theory and probability theory
◎ Comparing Fuzzy set and Fuzzy measure
Fuzzy set Fuzzy measure
Assigning value to … Each element of the universal set
Degree Membership grade
Each crisp set of the universal set
The degree of evidence
or belief
◎ Categories of Fuzzy Measures dual plausibility measure Evidence theory
belief measure
probability measure probability theory dual possibility measure possibility theory
necessity measure

◎ e.g. : Diagnose an ill patient ( incomplete data)
0.45
0.3
0.75 pnecumonia bronchitis emphysema cold
measurements

 m m
1
2 

 m n
 Fuzzy measure:
0 . 4 5 0 . 3 0
, , , p n e u m o n i a b r o n c h i t i s m p h y s e m a c o l d
◎ e.g. : Criminal trial
Not with the degree to which the defendant is guilty, but with the degree to which the evidence proves his membership in either the crisp set of guilty people or the crisp set of innocent people
◎ e.g. : Measurements (length, weight, electric current, light intensity)
。A measurement range [a, b] ,
。Partition it into n disjoint intervals
 a a
1

,
 a a
1 2

, ,
 a n

1
, b

。Given a measurement with
i, no observational error  fit exactly into one of the intervals
ii, measurement error  coincides with or is in close proximity to one of the boundaries a a
1
, , n

1

◎ Definition 7.1 X


: infinite universal set , C: a family of subsets of X

,
Fuzzy measure g: C
 , satisfies the following requirement :
(g1) g ( )

0, ( )


1 (boundary requirement)
(g2) If A

B , 
( )

( ) (increasing monotonicity)
(g3) A
1

A
2


If i

1
A i

C ,
in C

 i lim (
 g A i
) g
 i

1
A i


(continuity from below)

(g4) A
1

A
2

If
 i

1
in C
A i

C , 
 i lim (
 g A i
) g
 i

1
A i


(continuity from above)
※ (g3),(g4)  g : continuous function  i lim (
 i
) = g (lim ) i

A i

※ C: Borel field of X

, i,e, 
,

U

C
If A C , then

A

C
If ,

C , then A

,
 
C
◎ Fuzzy measure g satisfies the following inequalities :
。
 
B
  min

( ), ( )

。
 
B
  max

( ), ( )


◎ Evidence theory
。An uncertain reasoning method with powerful computational tools for data analysis, aims at developing effective reasoning method involving uncertainty, to derive what is behind data even data is incomplete, inconsistent, or with other problems.
。Namely Dempster-Shafer (D-S) theory, aims to provide a theory of partial belief, which extend traditional probability theory.
◎ Based on two dual nonadditive measures : belief and plausibility measures
。Belief measure : Bel:

( )

  
(1) Bel ( ) 0,

( )


1
(2)
(
1

A
2
 
A n
)
  i
Bel A i
)
 
( i

A j
) ( 1) n

1
(
1

A
2
 
A n
)
(3) ( )

: the degree of belief that an element of X

belongs to A
。Plausibility measure : Pl:

( )

  
(1) Pl ( ) 0,

( )


1
(2) (
1

A
2
 
A n
)
  i
Pl A i
)
 
( i

A j
) ( 1) n

1
(
1

A
2
 
A n
)
。Remark:
(1)  
 
( ),
 
Pl A
(2) ( )




。Duality property :
(1) A


A
(2) , ,
(3) Bel

Pl
◎ Basic Probability Assignment (BPA)
。Belief and plausibility measures can be characterized by a function

: ( )

   m
 
0,



( )

( )

1
。Observations
(1) Not required (


)

1
(2) Not required ( )

( ) when B

A
(3) Not relationship between ( )

and ( )
。Given m
 






 
|

A

|
 
( )

( )
。Given Bel
  
  
  
◎ Focal Element: A If ( )

0 or 
0
◎ Body of evidence: ( , )
。 F : a set of focal elements
。 m : the associated basic assignments

◎ Joint Basic Assignment m
1,2
。Dempster Shafer’s rule of combination m A
1,2
( )


 


0,

A m B m C
1
( )
2
( )
1
 
 k
, A
  where K
  m B m C
1
( )
2
( )
。 m B m C
1
( ),
2
( ) : degree of evidence measured from two independent sources focus on B,C, respectively.
。 m B m C
1
( )
2
( ) : focuses on B

C
◎ Basic assignment defined on Binary Relation
。


 m P X Y

)
  
。 R R x
, y
:

The projections of R on X

, Y

, resp.
(1) R x

{ x


X

|
 

, . .( , )

R }
(2) R y

{ y

Y

|
 


, . .( , )

R }
。 m m x , y
:

The projection of m on X


, Y ,resp.
(1) x
 i
 i m R i
( ) ,

A

(

) , where ' are defined on (

X Y



)
(2) y
 i
 i m R i
( ) ,

B

( )

(3)Call , y
: marginal basic assignments
(4) ( F m x
, x
), ( F m y
, y
) : marginal bodies of evidence
◎ ( F m x
, x
), ( F m y
, y
) are noninteractive iff  
F x
and  
F y
。 (

)

( ) y
( ) and ' , ( ')

0
◎ Commonality function
。  
※ ( )

( ), ( )

( )

◎ Possibility theory
。 New form of information theory , related to but independent of both fuzzy sets and probability theory
。A possibility distribution is a normal fuzzy set (at least one membership grade equals 1).For example, all fuzzy numbers are possibility distributions.
。The rules of possibility theory are similar to probability theory.
◎ A branch of evidence theory deals with only nested focal elements.(consonant)
◎ possibility theory evidence theory
necessity measures belief measures
possibility measures plausibility measures
◎ Possibility and Necessity Measures
。nested A
1

A
2
 
A n
。Consonant : If the focal elements of ( , ) are nested,
the associated Bel and Pl are called consonant.
◎ Theorem7.1 : ( , ) : consonant (nested)
i, (

B )
 min[ Bel A Bel B ,


A B

(

)
ii, (

B )
 max[ ( ), ( )]

Compare these equations with the general properties of fuzzy measures,
(

B )
 m i n [ ( ) , ( ) ]
(

B )
 m a x [ ( ) , ( ) ]
Possibility theory is based on extreme values of fuzzy measures.
◎ Definition 7.2: Nec ( A k
)
 inf Nec A k
)
◎ Definition 7.3: Pos ( A k
)
 sup Pos A k
)
◎ Relations between Nec, Pos
。 ( )


( ) 1
。 ( )


( ) 1

。  
。 min[ ( ),

( )]

0
。 max[ ( ),


◎ Theorem 7.2 :

 
( ),

i ( )
 
( )

1
ii ( ) 1 ( )

0
◎ Possibility distribution function 
。 
:

X


[0,1] , s.t. 
( )

({ })
。Every possibility measure is uniquely represented by the associated 

◎ Theorem 7.3 :

Pos , it can uniquely be determined by a possibility distribution function 

 m a x

 x sup ( )
( )

X


X

: f i n i t e
: i n f i n i t e

◎ Possibility Distribution
。 
:

X


[0,1] : Possibility distribution function where

X


{ ,
1 2
, , x n
} finite r r
1 2 r n
: possibility distribution with length n
where r i

( ) i
◎ Ordered possibility distribution --- r
1
 
2
 r n
。 n
R : the set of all ordered possibility distribution of length n
。 R
 n
R
。
1 r

(
1 r r
1
,
1
2
, ,
1 r n
)

2 r

(
2 r
1
,
2 r
2
, ,
2 r n
)
 n
R n
R
 1  2 r r iff  1 r i
 2 r i
,
 i
。  : partial and forms a lattice on n
R in which GLB, LUB are defined by
。Join  : i r
 j r

(max( i
1
, j r r
1
), max( i
2
, j r r
2
), , max( i
, j r r n n
))
。Meet  : i r
 j r

(min( i
1
, j r r
1
), min( i
2
, j r r
2
), , min( i
, j r r n n
))
。 n R ,

: Lattice of possibility distributions of length n
◎ notations
。

X


{ ,
1 2
, , x n
}
。 A s i
 x x
1 2
, x i s focal elements of m 
A
1

A
2
 

A X n 


( ) A A i
( A not a focal element) n  i

1 m A i
)

1 ,


( )
(

◎ Basic distribution
。 m

( m m
1
,
2
,..., m n
) where m i

( i
), i n 

1 m i

1
。 n
M : the set of all basic distributions with n components
。 M
  n
M
。Each m

M represents exactly one possibility distribution r

R , r i

( i
)

P o s i x
 p l i x
( )
 
|
  
(
 n  m A k
(

) n  m k
i.e., m i r i r i

1 r n

1

0 i

N n
(7.33)
by convention
( 7 . 3 3 ) m
 t r one-to-one from R

M
◎ Consider
。 The smallest possibility distribution : r

(1, 0,..., 0) , m i r i r i

1
It’s basic distribution : m

( )
 <= perfect evidence
(i.e., no uncertainly involved)
。 The largest possibility distribution : r

( 1 , 1 , . . . . , 1 )
ˆ  ˆ 
(0, 0,....,1)
。 The larger the possibility distribution , the less specific the evidence
◎ Joint possibility distribution r defined on X Y
。Marginal possibility distribution
(1) r x
X
 r x y
(2) r y
Y
( )
 max[ ( , )]
。Nested bodies of evidence on , : noninteractive
Iff ( , )
 min[ r
X
( ), ( )]
Y
, 
,

。Definition of noninteractive
(1) ( , )
 m
X
( ) m y
Y
( )
(2) ( , )
 min[ r
X
( ), ( )]
Y
(3) (2) preserves the nested structure of force element
。Defines the largest joint distribution
(1)
X
( )
 r x y
 r x y
 r x ---- (A)
(2)
Y
( )
 r x y

( , )

Y
( ) ---- (B)
( ), ( )

( , )
 min[ r
X
( ), ( )]
Y

( , )
 m i n [
X
( ) ,
Y
( ) ]
◎ Pos , Pos Pos
X Y
。 The possibility measures corresponding to the possibility distribution function r r r
X Y
。 ( )
 min[ Pos ( ),
X Y
( )]
◎ Cartesian Coproduct A B A B
◎ Conditional measures of possibility Pos B A
。 (

)

Pos ( | ) *
Y
( )
[ P o s ( B (
Y

)

Y
( ) B
。 Pos A B

(

) (

)

Y
( ) B
Pos ( ) A B

X
 
◎ Conditional probability distribution function r x y
。 ( , )
 r ( | ) * ( )
Y
。 r x y
 min[ r x y r y
Y
 min[ r y x r
X x

◎ Possibility independence
。 r ( | )
 r
X
( )
。 r ( | )

Y
( )
。If r r
X Y
: independent
  r x y )
 r x ( y |
Y r y

X r x
Y r y ( ) ]
i.e., r r
X Y
: noninteractive
。If r r
X Y
: noninteractive
 r x y
 min[ r
X x r y
Y
。 r x y
 min[ r x y r y
Y
( )]
 min[ r
X x r y
Y
 min[ r x y r y
Y
( )]
。 r x y
 r
X
( ) r
X
( )

Y
( )
[ r y
Y
( ) , 1 ] r
X
( )

Y
( )
。 r y x

( ) ( )
 r
X
( )
[ r
X
( ) , 1 ]
Y
( )
 r
X
( )
◎ Degree of confirmation C
。 ( )

( )
   
C : ( )
 
。 ( )
 0 if ( )

0
( ) if ( )

0
 m a x [ 0 , ( ) ]
。 ( )

( )

1 if ( )

0
1 if ( )

0
 m i n [ 1 , 1 ( ) ]

( )
 m a x [ 0 , (

) ] m i n [ 1 , 1

( ) ] 1
If ( )

0

( ) 0 ( ) 1 1 ( )
If ( )

0

( )

( ) 1 1 ( )

◎ Fuzzy set theory has at least two advantages
。Imprecise knowledge to be modeled
(1) An approximation of a numerical value (``about 3.80 meters'')
(2) A vague information expressed
。The only way to deal with knowledge numerically provided by measuring devices, and knowledge symbolically expressed by a human observer for instance.
◎ A reasoning based on imprecise knowledge often generates uncertainties on the final decision that must be taken. E.g. :
。Being over 18, makes one eligible to vote.
。We know that Peter is about 18, is he allowed to vote?
。It is quite possible, but not certain.
◎ Possibility theory, in connection with the fuzzy set theory
。Allowing a reasoning to be carried out on imprecise or vague knowledge
。Making it possible to deal with uncertainties on this knowledge
◎ From a mathematical perspective, fuzzy sets and probability
。Exist as parts of a greater Generalized Information Theory which includes many formalisms for representing uncertainty :
(1) random sets
(2) Demster-Shafer evidence theory,
(3) probability intervals
(4) possibility theory
(5) general fuzzy measures
(6) interval analysis, etc.

◎ Semantically, the distinction between fuzzy logic and probability theory has to do with the difference between the notions of probability and a degree of membership.
。Probability statements are about the likelihoods of outcomes: an event either occurs or does not, and you can bet on it.
。Fuzziness, one cannot say unequivocally whether an event occurred or not, and instead you are trying to model the EXTENT to which an event occurred.
◎ Due to nested structures of
  cuts of fuzzy sets.
◎ Possibility theory can also be formulated in terms of fuzzy sets
◎ Fuzzy sets Possibility theory
X universal set

A
   i
A n

A

A
( )
  
nested focal elements
fuzzy
  cuts

basic probability assignment
membership function

possibility distribution function
i.e., ( )

( )

◎ Comparison between Possibility and Probability
。P ossibility measures reflect a vague but coherent knowledge.
。 Probability measures summarize a body of precise and varied knowledge
◎
Comparison of the main formulae of possibility theory with those of probability theory.
Possibility Probability
Distribution , measure
Distribution p, probability P si if independent
Ignorance: Ignorance:

◎ Probability Measure Pro
。 
,

( ) =>Pro( A

B )

Pro( )

B

◎ Theorem 7.4
: Bel

Pro iff the associated m satisfies
( ) 0, A X , A : not singleton
A

B )
◎ Probability Distribution function
。 :

[0,1] , s.t. ( )

({ })
。 Bel becomes a probability , measure iff its ({ })

({ }) and
A X , A not a singleton , ( )

0 , m focuses only on singletons

( )

(

)
 x

A
 
 x A

i.e., Bel and Pl merge under the additivity axiom if probability measure
。 Total ignorance

1
| X |
◎ Joint probability distribution – a probability distribution p defined on X Y
。 Marginal probability distributions
P x
X
( )
  p x y P y
Y
( )
 
P P
X
,
Y
: nonunteractive iff ( , )

X
( ) P y
Y
( )
。 Conditional probability distributions
P ( | )

P ( | )

( , )
Y
( , )
X
X
,
Y
: independent iff P ( | )

X
( ) , P ( | )

Y
( )
◎ Possibility-Probability Transformation
。 Possibility-Probability Consistency
( , )
 
P x ( ) r x ( )

The strongest consistency condition
P r o ( ) 0 (

) 1
i.e., any event with nonzero probability must be fully possible
。 X

{ ,
1 2
,..., x n
}
Let X s where i
,
are ordered , s.t. , r i
 r i

1
P i

( ), i r i

( ) i r

( , ,..., )
1 2 r n
, P i

P i

1
, P

( ,
1 2
,..., P n
)
If max ( )
 
1 , n  
1 : normalization i

1
P i
(1) ratio-scale transformation :
(2) interval-scale transformation :
(3) r i
 j n 

1 min( , i j
) , P i
 n  j

1 r i

P i
 r i

P
1
  r i
 r j

1 j
◎ r i
'
 n 
P j
=> yields the smallest probability distribution that satisfies the weakest probability-possibility consistency condition
 
P
 r ' , r '
 p ' , p '

(not information-preserving)