fuzzy systems - possibility theoryfuzzy.cs.ovgu.de/.../lehre.fs0910/fs0910lecture07.pdf ·...

Fuzzy SystemsPossibility Theory

Rudolf Kruse Christian Moeweskruse,[email protected]

Otto-von-Guericke University of MagdeburgFaculty of Computer Science

Department of Knowledge Processing and Language Engineering

R. Kruse, C. Moewes Fuzzy Systems – Possibility Theory 2009/12/16 1 / 61

http://www.ovgu.de/

http://www.fin.ovgu.de/

http://iws.cs.ovgu.de/

Outline

1. Introduction

Problems with Probability TheoryUpper and Lower Probabilities

2. Dempster-Shafer Theory of Evidence

3. Possibility Theory


Problems with Probability TheoryRepresentation of Ignorance

• standard method to handle uncertainty is probability theory

• however, it has some problems regarding ignorance

• we are given one die with faces 1, . . . , 6

• what is the certainty of showing up face i?

1. conduct statistical survey (roll the die 10,000 times) andestimate relative frequency: P(i) = 1

62. use subjective probabilities (i.e., often normal case):

we don’t know anything (especially and explicitly we don’t haveany reason to assign unequal probabilities),so most plausible distribution is uniform

⇒ problem: uniform distribution because of ignorance or extensivestatistical tests


Problems with Probability TheoryRepresentation of Ignorance

• experts analyze aircraft shapes• 3 aircraft types A, B, C

• “It is type A or B with 90% certainty.”

• “About C , I don’t have any clue and I don’t want to commitmyself.”

• “No preferences for A or B.”

⇒ problem: propositions hard to handle with Bayesian theory


Upper and Lower Probabilities

• a way out to solve this problem are upper and lower probabilities• “It is type A or B with 90% certainty.”• “About C , I don’t have any clue and I don’t want to commit

myself.”• “No preferences for A or B.”

A ∅ A B C A, B A, C A, B, CP∗(A) 0 0 0 0 .9 0 1P∗(A) 0 1 1 .1 1 1 1

• consider P∗(A) and P∗(A) as lower and upper probability bounds

• total ignorance: P∗(A) = 0 and P∗(A) = 1 for A 6= ∅, A 6= Ω


Properties of Upper and Lower Probabilities I

1. P∗ : P(Ω) → [0, 1]

2. 0 ≤ P∗ ≤ P∗ ≤ 1, P∗(∅) = P∗(∅) = 0, P∗(Ω) = P∗(Ω) = 1

3. A ⊆ B ⇒ P∗(A) ≤ P∗(B) and P∗(A) ≤ P∗(B)

4. A ∩ B = ∅ 6⇒ P∗(A) + P∗(B) = P∗(A ∪ B)

5. P∗(A ∪ B) ≥ P∗(A) + P∗(B) − P∗(A ∩ B)

6. P∗(A ∪ B) ≤ P∗(A) + P∗(B) − P∗(A ∩ B)

7. P∗(A) = 1 − P∗(Ω\A)


Properties of Upper and Lower Probabilities II

• one can prove the following generalized equation

P∗(n

⋃

i=1

Ai) ≥∑

∅6=I:I⊆1,...,n

(−1)|I|+1 · P∗(⋂

i∈I

Ai)

• these set functions play important role in theoretical physics (i.e.,capacities [Choquet, 1954])

• thoughts have been generalized and developed into theory ofbelief functions [Shafer, 1976]


Outline

1. Introduction


Belief MeasurePlausibility MeasureBasic Belief AssignmentFocal ElementTotal IgnoranceDempster’s Rule of CombinationExampleConversion Formulas



Dempster-Shafer Theory of Evidence

• like Bayesian approaches, the Dempster-Shafer theory ofevidence [Shafer, 1976] wants to model and quantify uncertaintyby degrees of belief

• Bayesian approaches assign degrees of belief to single hypothesis

• theory of evidence assigns degrees of belief to sets of hypotheses

• let X be finite “frame of discernment”

⇒ each proposition “x is x0” can be represented by A ⊂ X

• sets including one element correspond to elementary propositions

• proposition is true if it contains the true elementary proposition

• belief in “x is x0” related to A ⊂ X is measured by Bel(A) ∈ [0, 1]


Belief Measure

Definition

Given a finite universe X , a belief measure is a function

Bel : P(X ) → [0, 1]

s.t. Bel(∅) = 0, Bel(X ) = 1 and

Bel(A1 ∪ A2 ∪ . . . ∪ An) ≥n

∑

i=1

Bel(Aj) −∑

i ,j:1≤i<j≤n

Bel(Ai ∩ Aj)

+ . . . + (−1)n+1 Bel(A1 ∩ A2 ∩ . . . ∩ An)

for all possible families of subsets of X .

• e.g., Bel(A1 ∪ A2) ≥ Bel(A1) + Bel(A2) − Bel(A1 ∩ A2)

• difficult to test ⇒ basic belief assignment is introduced


Belief Measure

• for each A ∈ P(X ), Bel(A) is interpreted as degree of belief

(based on evidence) that given x ∈ X belongs to A

• if sets A1, . . . , An are pairwise disjoint, then

Bel(A1, . . . , An) ≥n

∑

i=1

Bel(Ai)

⇒ weaker version of additivity property of probability measures

⇒ probability measures are special case of belief measures whereequality in definition of Bel is always satisfied

• fundamental property: let n = 2, A1 = A and A2 = AC , then

Bel(A) + Bel(AC ) ≤ 1


Plausibility Measure

• associated with each Bel is plausibility measure Pl defined by

Pl(A) = 1 − Bel(AC ), ∀A ∈ P(X )

Definition

Given a finite universe X , a plausibility measure is a function

Pl : P → [0, 1]

s.t. Pl(∅) = 0, Pl(X ) = 1 and

Pl(A1 ∩ A2 ∩ . . . ∩ An) ≤∑

j

Pl(Aj) −∑

j<k

Pl(Aj ∪ Ak)

+ . . . + (−1)n+1 Pl(A1 ∪ A2 ∪ . . . ∪ An)

for all possible families of subsets of X .


Basic Belief Assignment

• fundamental property: let n = 2, A1 = A and A2 = AC , then

Pl(A) + Pl(AC ) ≥ 1

• Bel and Pl can be characterized by basic belief assignment

m : P(X ) → [0, 1]

s.t. m(∅) = 0 and∑

A∈P(X)

m(A) = 1

• for each A ∈ P(X ), value m(A) expresses proportion to which allevidence supports “particular element of X belongs to A”

• m(A) does not imply any claim regarding subsets of A• definition of m resembles probability distribution function, but

• probability distribution functions are defined on X• basic belief assignments are defined on P(X)



• observe the following facts

1. m(X) need not to be 12. if A ⊆ B, then m(A) need not to be less or equal m(B)3. there does not need to be any relation between m(A) and m(AC )

• given m, Bel and Pl are uniquely determined for all A ∈ P(X ) by

Bel(A) =∑

B|B⊆A

m(B) and Pl(A) =∑

B|A∩B 6=∅

m(B)

• note that inverse is also possible

• e.g., m(A) =∑

B|B⊆A(−1)|A\B| Bel(B)

• each of three functions is sufficient to determine other two



• Bel(A) =∑

B|B⊆A m(B) means the following:

• m(A) characterizes degree of evidence that x belongs only to A• Bel(A) represents evidence that x belongs to A and subsets of A

• Pl(A) =∑

B|A∩B 6=∅ m(B) represents both kinds of evidence:• total evidence that x belongs to A or to any subset of A

• evidence associated with sets that overlap with A

⇒ for all A ∈ P(X ),Pl(A) ≥ Bel(A)


Focal Element

• every A ∈ P(X ) for which

m(A) > 0

is called focal element of m

• they are subsets of X on which available evidence focuses

• X is finite ⇒ m can be fully described by list F of its focalelements

• pair 〈F , m〉 is called body of evidence


Total Ignorance

• total ignorance is expressed by

m(X ) = 1 and m(A) = 0 ∀A 6= X

• i.e., we know that element is in universal set but we have noevidence about is location in any A ⊂ X

• in terms of belief it is exactly the same:

Bel(X ) = 1 and Bel(A) = 0 ∀A 6= X

• in terms of plausibility it is different:

Pl(∅) = 0 and Pl(A) = 1 ∀A 6= ∅


Dempster’s Rule of Combination

• now, consider two independent sources of evidence (e.g., fromtwo experts) expressed by m1 and m2 on some P(X )

• they must be appropriately combined to obtain joint basicassignment m1,2

• generally, evidence can be combined in many ways

• standard way is expressed by Dempster’s rule of combination

m1,2(A) =

∑

B∩C=A m1(B) · m2(C)

1 − K

for all A 6= ∅ and m1,2(∅) = 0 where

K =∑

B∩C=∅

m1(B) · m2(C)


Dempster’s Rule of Combination

• m1(B) (focusing on B ∈ P(X )) and m2(C) (focusing onC ∈ P(X )) are combined by m1(B) · m2(C) (focusing on B ∩ C)

⇒ exactly the same for joint probability distribution which iscalculated from two independent marginal distributions

• since some intersections of focal elements may result in A, wemust add corresponding product to obtain m1,2

• also, some intersections may be empty

• since, m1,2(∅) = 0, K is thus not included in definition of m1,2

• to ensure∑

A∈P(X) m(A) = 1, each product is divided by 1 − K


Example

• assume that an old code fragment was discovered by Rudolf Kruse

• it strongly resembles code fragments of Christian Moewes

• several questions can come up:

1. Is the discovered code fragment a real fragment of Christian?2. Is the discovered code fragment a product of one of Christian’s

students?3. Is the discovered code fragment a fake?


Example

• let C , S, F denote subsets of universe X of all code fragments

• C denotes set of all code fragments by Christian

• S represents set of all code fragments by Christian’s students

• F represents set of all fakes of Christian’s code fragments

• two experts (named Georg and Matthias) performed carefulexaminations of the code fragment

• afterwards, they provided Rudolf Kruse with basic assignments m1

and m2 as follows


Two Independent Sources

experts Georg Matthias

focalelements m1 Bel1 m2 Bel2

C .05 .05 .15 .15S 0 0 0 0F .05 .05 .05 .05

C ∪ S .15 .2 .05 .2C ∪ F .1 .2 .2 .4S ∪ F .05 .1 .05 .1

C ∪ S ∪ F .6 1 .5 1

• degrees of evidence obtained by examination and supported bydifferent claims that code fragment belongs to one of the sets

• given m1 and m2, we can easily compute Bel1 and Bel2, resp.


Calculation of Normalization Factor

K = m1(C) · m2(S) + m1(C) · m2(F ) + m1(C) · m2(S ∪ F )

+ m1(S) · m2(C) + m1(S) · m2(F ) + m1(S) · m2(C ∪ F )

+ m1(F ) · m2(C) + m1(F ) · m2(S) + m1(F ) · m2(C ∪ S)

+ m1(C ∪ S) · m2(F ) + m1(C ∪ F ) · m2(S) + m1(S ∪ F ) · m2(C)

= .03

• so, the normalization factor is then 1 − K = .97


Calculation of Joint Basic Assignment

m1,2(C) = [m1(C) · m2(C) + m1(C) · m2(C ∪ S)

+ m1(C) · m2(C ∪ F ) + m1(C) · m2(C ∪ S ∪ F )

+ m1(C ∪ S) · m2(C) + m1(C ∪ S) · m2(C ∪ F )

+ m1(C ∪ F ) · m2(C) + m1(C ∪ F ) · m2(C ∪ S)

+ m1(C ∪ S ∪ F ) · m2(C)]/.97

= .21

m1,2(S) = [m1(S) · m2(S) + m1(S) · m2(C ∪ S) + m1(S) · m2(S ∪ F )

+ m1(S) · m2(C ∪ S ∪ F ) + m1(C ∪ S) · m2(S)

+ m1(C ∪ S) · m2(S ∪ F ) + m1(S ∪ F ) · m2(S)

+ m1(S ∪ F ) · m2(C ∪ S) + m1(C ∪ S ∪ F ) · m2(S)]/.97

= .01


Calculation of Joint Basic Assignment

m1,2(C ∪ F ) = [m1(C ∪ F ) · m2(C ∪ F )

+ m1(C ∪ F ) · m2(C ∪ S ∪ F )

+ m1(C ∪ S ∪ F ) · m2(C ∪ F )]/.97

= .2

m1,2(C ∪ S ∪ F ) = [m1(C ∪ S ∪ F ) · m2(C ∪ S ∪ F )]/.97

= .31


Joint Basic Assignment

combinedGeorg Matthias evidence

focalelements m1 Bel1 m2 Bel2 m1,2 Bel1,2

C .05 .05 .15 .15 .21 .21S 0 0 0 0 .01 .01F .05 .05 .05 .05 .09 .09

C ∪ S .15 .2 .05 .2 .12 .34C ∪ F .1 .2 .2 .4 .2 .5S ∪ F .05 .1 .05 .1 .06 .16

C ∪ S ∪ F .6 1 .5 1 .31 1


Projection

• consider now basic assignment m : P(X × Y ) → [0, 1]

• each focal element of m is binary relation R on X × Y

• let RX denote projection of R on X , then

RX = x ∈ X | (x , y) ∈ R for some y ∈ Y

• projection mX of m on X is

mX (A) =∑

R|A=RX

m(R) for all A ∈ P(X )

• adding values of m(R) for all focal elements R whose RX is A

• similarly, computation can be done for RY and mY , resp.


Marginal Bodies of Evidence

• mX and mY are also called marginal basic assignments

• 〈FX , mX 〉 and 〈FY , mY 〉 are called marginal bodies of evidence

• 〈FX , mX 〉 and 〈FY , mY 〉 are called noninteractive if and only iffor all A ∈ FX and B ∈ FY

m(A × B) = mX (A) · mY (B)

andm(R) = 0 for all R 6= A × B

• i.e., joint basic assignment m is uniquely determined by marginalbasic assignments


Example

• consider body of evidence given in table• focal elements are subsets of X × Y = 1, 2, 3 × a, b, c

focal X × Y

element 1a 1b 1c 2a 2b 2c 3a 3b 3c m(Ri)R1 = 0 0 0 0 1 1 0 1 1 .0625R2 = 0 0 0 1 0 0 1 0 0 .0625R3 = 0 0 0 1 1 1 1 1 1 .125R4 = 0 1 1 0 0 0 0 1 1 .0375R5 = 0 1 1 0 1 1 0 0 0 .075R6 = 0 1 1 0 1 1 0 1 1 .075R7 = 1 0 0 0 0 0 1 0 0 .375R8 = 1 0 0 1 0 0 0 0 0 .075R9 = 1 0 0 1 0 0 1 0 0 .075

R10 = 1 1 1 0 0 0 1 1 1 .075R11 = 1 1 1 1 1 1 0 0 0 .15R12 = 1 1 1 1 1 1 1 1 1 .15


Example

• margin bodies of evidence are obtained as follows

mX (2, 3) = m(R1) + m(R2) + m(R3) = .25

mX (1, 2) = m(R5) + m(R8) + m(R11) = .3

mY (a) = m(R2) + m(R7) + m(R8) + m(R9) = .25

mY (a, b, c) = m(R4) + m(R10) + m(R11) + m(R12) = .5

X

1 2 3 mX (A)

A = 0 1 1 .251 0 1 .151 1 0 .31 1 1 .3

Y

a b c mY (B)

B = 0 1 1 .251 0 0 .251 1 1 .5

• here, marginal bodies of evidence are noninteractive


Conversion Formulas

given m(A) = Bel(A) = Pl(A) =

m m(A)∑

B⊆A

m(B)∑

B∩A6=∅m(B)

Bel∑

B⊆A

(−1)|A\B| Bel(B) Bel(A) 1 − Bel(AC )

Pl∑

B⊆A

(−1)|A\B|(1 − Pl(BC )) 1 − Pl(AC ) Pl(A)


Outline

1. Introduction



Necessity and PossibilityPossibility DistributionBasic DistributionMathematical PropertiesFuzzy Sets and Possibility TheoryPossibility versus ProbabilityComparison of both TheoriesInterpretations


Possibility Theory

• possibility theory is special kind of evidence theory

• bodies of evidence contain nested focal elements

• belief and plausibility are called necessity and possibility, resp.

Theorem

Let a given finite body of evidence 〈F , m〉 be nested. Then, the

associated belief and plausibility measures have the following

properties for all A, B ∈ P(X )

Bel(A ∩ B) = minBel(A), Bel(B)

Pl(A ∪ B) = maxPl(A), Pl(B)


Necessity and Possibility

• let necessity and plausibility be denoted by Nec and Pos, resp.

• last theorem provides us with basic equation of possibility theory

Nec(A ∩ B) = minNec(A), Nec(B)

Pos(A ∪ B) = maxPos(A), Pos(B)

• can be also defined for arbitrary universal set (e.g., infinite ones)

Nec

⋂

k∈K

Ak

= infk∈K

Nec(Ak)

Pos

⋃

k∈K

Ak

= supk∈K

Pos(Ak)

where K is arbitrary index set



• since necessity and possibility are special belief and plausibilitymeasures, resp., they satisfy fundamental properties

Nec(A) + Nec(AC ) ≤ 1

Pos(A) + Pos(AC ) ≥ 1

1 − Pos(AC ) = Nec(A)

• from the definitions of Nec and Pos, it follows that

min

Nec(A), Nec(AC )

= 0

max

Pos(A), Pos(AC )

= 1



• necessity and possibility measures constrain each other

Theorem

For every A ∈ P(X ), any necessity measure Nec on P(X ) and the

associated possibility measure Pos satisfy the following implications:

Nec(A) > 0 ⇒ Pos(A) = 1,

Pos(A) < 1 ⇒ Nec(A) = 0.


Possibility Distribution Function

• given possibility measure Pos on P(X )• let function r : X → [0, 1] s.t. ∀x ∈ X : r(x) = Pos(x) be called

possibility distribution function associated with Pos

Theorem

Every possibility measure Pos on a finite power set P(X ) is uniquely

determined by a possibility distribution function

r : X → [0, 1]

via the formula

Pos(A) = maxx∈A

r(x)

for each A ∈ P(X ).

• when X is infinite, then Pos(A) = supx∈A r(x)


Possibility Distribution

• let possibility distribution function r be defined onX = x1, . . . , xn

⇒ possibility distribution associated with r is n-tuple

r = (r1, . . . , rn)

where ri = r(xi ) for all xi ∈ X

• it is useful to order r s.t. ri ≥ rj when i < j


Example

• now, let Pos be defined on P(X ) by m

• w.l.o.g., focal elements are some or all of subsets in completesequence of nested subsets

A1 ⊂ A2 ⊂ . . . ⊂ An = X

where Ai = x1, . . . , xi i ∈ INn

• i.e., m(A) = 0 for each A 6= Ai (i ∈ IN) and∑n

i=1 m(Ai) = 1


Example

• complete sequence of nested focal elements of Pos on P(X )where X = x1, . . . , xn

x1 x2 x3 x4 . . . xn

m(A1)

m(A2)

m(A3)m(A4) . . . m(An)

r(x1)

r(x2)r(x3) r(x4)

. . . r(xn)


Basic Distribution

• every Pos on finite set can be represented by n-tuple

m = (m1, . . . , mn)

for some finite n ∈ IN where mi = m(Ai) for all i ∈ INn

• clearly,∑n

i=1 mi = 1 and mi ∈ [0, 1] for all i ∈ INn

• m is called basic distribution

• each m represents exactly one possibility distribution r

∀xi ∈ X : ri = r(xi) = Pos(xi ) = Pl(xi )

∀i ∈ INn : ri = Pl(xi) =n

∑

k=i

m(Ak) =n

∑

k=i

mk

• solving all n equations for mi (i ∈ INn) leads to

mi = ri − ri+1

• one-two-one correspondence between r and m


Example

• one possibility measure defined on X with A1 ⊂ A2 ⊂ . . . ⊂ A7

x1 x2 x3 x4 x5 x6 x7

m(A2) = .3

m(A3) = .4

m(A6) = .1

m(A7) = .2

r(x1) = 1

r(x2) = 1

r(x3) = .7

r(x4) = .3

r(x5) = .3

r(x6) = .3

r(x7) = .2

•

∑ni=1 m(Ai ) = 1 is satisfied, m(A1) = m(A4) = m(A5) = 0


Perfect Evidence and Total Ignorance

• smallest possibility distribution r has form

r = (1, 0, 0, . . . , 0)

• r represents perfect evidence with no uncertainty involved

• largest possibility distribution r can be written as

r = (0, 0, . . . , 0, 1)

• r denotes total ignorance, i.e., no relevant evidence is available

• the larger r , the less specific the evidence is (the more ignorant)


Joint Possibility Distribution

• consider joint possibility distributions r defined on X × Y

• projections rX , rY are called marginal possibility distribution

∀x ∈ X : rX (x) = maxy∈Y

r(x , y)

∀y ∈ Y : rY (y) = maxx∈X

r(x , y)

• formula for rX (x) follows from

PosX (x) = Pos((x , y) | x ∈ Y )

PosX (x) = rX (x)

Pos((x , y) | x ∈ Y ) = maxy∈Y

r(x , y)

• same argumentation can be done for rY


Possibilistic Noninteraction

Definition

Nested bodies of evidence X and Y represented by rX and rY , resp.,are called noninteractive if and only if

r(x , y) = minrX (x), rY (y)

for all x ∈ X and all y ∈ Y

• recall product rule m(A × B) = mX (A) · mY (B)

• this definition is not based upon product rule of evidence theory

⇒ does not conform to general noninteractive bodies of evidence

• here, product rule would not preserve nested structure


Possibilistic Noninteraction

• let rX and rY be marginal poss. dist. functions on X and Y , resp.

• let r be joint poss. dist. function on X × Y defined by rX and rY

• assume PosX , PosY and Pos correspond to rX , rY and r , then

Pos(A × B) = minPosX (A), PosY (B)

for all A ∈ P(X ) and all B ∈ P(Y ) where

PosX (A) = maxx∈A

rX (x),

PosY (B) = maxy∈B

rY (y),

Pos(A × B) = maxx∈A,y∈B

r(x , y)


Possibilistic Independence

• two marginal possibilistic bodies of evidence are independent ifand only if conditional possibilities equal marginal possibilities,i.e., for all x ∈ X and all y ∈ Y

rX |Y (x |y) = rX (x),

rY |X (y |x) = rY (y)

• rX |Y (x |y) and rY |X (y |x) are conditional possibilities on X × Y

• possibilistic independence is stronger than poss. noninteraction

• possibilistic independence ⇒ possibilistic noninteraction

• possibilistic noninteraction 6⇒ possibilistic independence


Fuzzy Sets and Possibility Theory

• possibility theory can be also formulated using fuzzy sets insteadof nested bodies of evidence [Dubois and Prade, 1993]

• fuzzy sets (similar to bodies of evidence) are based on families ofnested sets, i.e., α-cuts

• Pos measures are connected with fuzzy sets via possibilitydistribution function

• e.g., let X denote variable taking values in set X

• X = x describes “X is x” where x ∈ X

• fuzzy set F on X expresses constraint on values assigned to X

• given value x ∈ X , F (x) is degree of compatibility of x

described by F


Fuzzy Sets and Possibility Theory

• given “X is F”, F (x) is degree of possibility that X = x

• i.e., given F on X and “X is F”,

∀x ∈ X : rF (x) = F (x)

• given rF , associated possibility measure is

∀A ∈ P(X ) : PosF (A) = supx∈A

rF (x)

• for normal fuzzy sets, NecF can be calculated by

∀A ∈ P(X ) : NecF (A) = 1 − PosF (AC )


Example

• let variable T be temperature in C (only integers)

• actual value is given in terms of “T is around 21C”

0

0.25

0.50

0.75

1.00

17 18 19 20 21 22 23 24 25

F = rF

b b

b

b

b

b

b

b b b

19 20 21 22 23

rF : 1/3 2/3 1 2/3 1/3

• incomplete information induces possibility distribution function rF

• rF is numerically identical with membership function F

• nested α-cuts play same role as focal elements

• Question: Which values do Nec and m take here?


Summary

• if rF is derived from normal fuzzy set F , then both formulationsof possibility theory are equivalent

• full equivalence breaks when F is not normal

⇒ basic probability assignment function m is not directly applicable

• all other properties remain equivalent in both formulations

• possibility theory is measure-theoretic counterpart of fuzzy settheory based upon standard fuzzy operations

• it provides tools for processing incomplete information expressedby fuzzy propositions

• it plays major role in fuzzy logic and approximate reasoning


Possibility versus Probability

• probability theory and possibility theory are distinct

• to show that, let us examine both theories by evidence theory

• probability theory and possibility theory are special branches of it

⇒ introduce probabilistic counterparts of possibilistic characteristics

• probability measure Pro must satisfy

Pro(A ∪ B) = Pro(A) + Pro(B)

for all A, B ∈ P(X ) s.t. A ∩ B = ∅

• Pro is special type of belief measure


Probability Distribution Function

Theorem

A belief measure Bel on a finite power set P(X ) is a probability

measure if and only if the associated basic probability assignment

function m is given by m(x) = Bel(x) and m(A) = 0 for all

subsets of X that are no singletons.

⇒ probability measures on finite sets are thus fully represented by

p : X → [0, 1]

s.t. p(x) = m(x)

• p is called probability distribution function

• let p = (p(x) | x ∈ X ) be probability distribution on X


Probability Measure

• recall definition of belief and plausibility

Bel(A) =∑

B|B⊆A

m(B) and Pl(A) =∑

B|A∩B 6=∅

m(B)

• if basic probability assignment focuses only on singletons, then

∀A ∈ P(X ) : Bel(A) = Pl(A) =∑

x∈A

m(x)

• Bel and Pos are equal ⇒ it is convenient to use one symbol Pro

∀A ∈ P(X ) : Pro(A) =∑

x∈A

p(x)

• Pro is called probability measure


Comparison of both Theories

Property Probability Theory Possibility Theory

measures probability Pro possibility Pos, necessity Nec

body of evidence singletons family of nested subsets

representation p : X → [0, 1] via r : X → [0, 1] viaPro(A) =

∑

x∈Ap(x) Pos(A) = maxx∈A r(x)

normalization∑

x∈Xp(x) = 1 maxx∈X r(x) = 1

total ignorance ∀x ∈ X : p(x) = 1|X |

∀x ∈ X : r(x) = 1

noninteraction p(x , y) = pX (x) · pY (y) r(x , y) = minrX (x), rY (y)

independence pX |Y (x |y) = pX (x) rX |Y (x |y) = rX (x)pY |X (y |x) = pY (y) rY |X (y |x) = rY (y)

independence ⇔ noninteraction independence ⇒ noninteraction


Comparison of both Theories

Probability Theory Possibility Theory

Pro(A ∪ B) = Pro(A) + Pro(B) − Pro(A ∩ B) Pos(A ∪ B) = maxPos(A), Pos(B)Nec(A ∩ B) = minNec(A), Nec(B)

not applicable Nec(A) = 1 − Pos(AC )Pos(A) < 1 ⇒ Nec(A) = 0Nec(A) > 0 ⇒ Pos(A) = 1

Pro(A) + Pro(AC ) = 1 Pos(A) + Pos(AC ) ≥ 1Nec(A) + Nec(AC ) ≤ 1maxPos(A), Pos(AC ) = 1minNec(A), Nec(AC ) = 0


FUZZY MEASURES: monotonic and continuousor semi continuous

BELIEF

MEASURES:

superadditiveand continuous

from above

PLAUSIBILITY

MEASURES:

subadditiveand continuous

from below

PROBABILITY

MEASURES:

additive

NEC.

MEASURES

POS.

MEASURES


Discussion

• possibility, necessity and probability measures do not overlap

• except for one special measure characterized by only one focalelement (i.e., singleton)

⇒ distribution functions of possibility and probability become equal

• one element of universal set is assigned 1, all others assigned 0

⇒ this measure represents perfect evidence


Discussion

• differences in mathematical properties make each theory suitablefor modeling certain types of uncertainty

• probability theory is ideal for formalizing uncertainty in situations• where class frequencies are known or• evidence is based on outcomes of many independent random

experiments

• possibility theory is ideal for formalizing incomplete informationexpressed by fuzzy propositions

• necessity and possibility can be seen as lower and upperprobabilities [Dempster, 1967]


Interpretations of Possibility Theory

based on similarity

• possibility r(x) reflects degree of similarity between x andprototype xi having r(xi) = 1

• i.e., r(x) is expressed by distance between x and xi

• the closer x to xi , the more possible its interpretation

• closeness comes from measurement or subjective judgement

based on preference

• due to ordering ≤Pos defined on P(X )

• ∀A, B ∈ P(X ), A ≤Pos B means B is at least as possible as A

• A ≤Pos B ⇔ AC ≤Nec BC


Literature for the Lecture

Choquet, G. (1954).Theory of capacities.Annales de l’Institut Fourier, 5:131–295.

Dempster, A. P. (1967).Upper and lower probabilities induced by a multivalued mapping.The Annals of Mathematical Statistics, 38(2):325–339.

Dubois, D. and Prade, H. (1993).Fuzzy sets and probability: Misunderstanding, bridges and gaps.In Proceedings of the Second IEEE International Conference on Fuzzy Systems, pages1059–1068, San Francisco, CA, USA.

Shafer, G. (1976).A Mathematical Theory of Evidence.Princeton University Press.


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