Probability - Possibility Transformation Based on Evidence Theory
Koichi YAMADA
Departmentof ManagementandInformationSystemsScience
NagaokaUniversity of Technology
1603-1 Kami-tomioka,Nagaoka,Niigata,940-2188,JAPAN,
Email: [email protected]
Abstract
Transformationbetweenprobability andpossibilityhas
been studied by many researchers.Most of those
studiesexaminedthe principlesthat must be satisfied
for the transformation, and devised an equation
satisfyingthem in a heuristicway. They did not show
that the proposedtransformationwas the only one
satisfyingthe principles.
The paper devises three new transformation
methodsbasedon Evidencetheory. Each startsCorn
some principles in the same way as the previous
research,but finds the only transformationsatisfying
the principles.Oneof them considersthat possibility is
an ordinal scale of uncertainty, and obtains the
possibilisticorderfrom a given probability distribution.
The remainingtwo regardspossibility as a ratio scale,
and transforms a probability distribution into a
possibilityone.
All of the three generate the same ordinal
structureof possibility.However,two of themgenerate
different possibility distributions. Thus, the paper
examineswhich of the two is more valid from the
point of the given principles.
1. Introduction
Transformationbetweenprobability andpossibilityhas
been studied by many researchers,since Zadeh
proposedPossibility theory [ 11.Most of those studies
examinedthe principlesthat must be satisfiedfor the
transformation, and devised an equation satisfying
them in a heuristicway or assuminga certaintype of
equations. However, they did not show that the
proposedtransformationwas the only one satisfying
the principles.
The paper devises and examines three new
transformationmethodsfrom probability to possibility
basedon Evidencetheory. They also startsfrom the
discussion about the principles, but the proposed
transformationsare thosederived uniquely from their
respective principles. One of the methods (Zl)
considers that possibility is an ordinal scale of
uncertainty,and obtainesthe possibilisticorder from a
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given probability distribution.The remainingtwo (72,
23) regardspossibility as a ratio scaleand derivesthe
equationstransforminga probability distributioninto a
possibility one. Especiallythe principles used for 73
lead to an equationtransformablein both directions.
The paperalso showsthat the equationhappensto be
identicalto the onedevisedheuristicallyby Duboisand
Prade[3]. That is, the transformationproposedby them
is shownthe uniqueone satisfyingthe principlesgiven
in this paper.
These three methods generate the identical
ordinal structureof possibilitywhenthe transformation
is in the direction from probability to possibility.
However, the possibility distributionsobtainedby 12
and 73 are different. Thus,the paperexamineswhich
of the two is more valid from the viewpoint of the
given principles, when the transformation is from
probability into possibility. The discussionsuggests
that l3 is more valid than 22. This result might be
welcomed,because73 is a transformationapplicable
in both directions,while T2 is only from probability to
possibility.
2. RelatedWork
2.1 Zadeh’sconsistencyprinciple
In the paper where Zadeh proposed the notion of
possibility [l], he discussedthe relation between
possibility and probability, and proposed the
possibility/probability consistencyprinciple (Zadeh’s
consistencyprinciple) for the transformation from
possibility into probability.
Let U = {or,...,+} be a finite set of discrete
events.X is a variabletaking a value in U. p(%) and
n(ui) are probability and possibility that X = z+,
respectively.Then, Zadeh’s consistencyprinciple can
n(ui)=OjP(Ui)=O
and
be expressed by
means
3t(ui)> n(uj) a~(% ) > p(uj ), where +
implication. He also proposed the degree of
consistency between possibility and probability
distributions, which is applicable in transformation
from possibility into probability. The degree of
consistencyis definedby the following equation:
Page: 70
(1)
Maximizing the degreeof consistency,however,posesus
a very restrictive condition that ;It(ui) c la P(z+) = 0 .
Note that this restriction is equivalentto that between
possibility n(ui ) and necessity v(ui) . THUS, the
restriction might be too strong considering that
n(ui ) 2 p(ui >2 *U i) holds in general[2,3,5]. Dubois and
Pradepointed out that Zaheh’spossibility expressesthe
degree of easeand is different from epistemic possibility
expressingthe uncertaintyof occurrence[2].
where p( u,+i) = 0. DuboisandPradeprovedthat N(Ai)
defined by the above is a necessitymeasurein the
mathematicalsense.Then, they derived the following
equation for transformation from probability into
possibility.
It(ui ) = fl(# i}) = l- N@J- @iI)
(6)
= i” P(Ui)+ IP(“j)9
j==i+lF
2.2 Transformationproposedby DuboisandPrade
Dubois and Prade asserted that the following two
principles must be satisfied for the transformation
betweenprobability andpossibility [33.
where 2 j-n +1.P( Uj) = 0. The equation to
tsmsfom
possibilityinto probability is alsoderivedasfollows.
P(4)>P(Uj)* z(“i)‘It(uj)p
(3)
where e means equivalence. P(A) and II(A)
(AC U) are probability and possibility measuresdefined
respectively. v meansmaximum. Eq.s (2) and (3) are
called principles of probability-possibility consistency
and preference preservation, respectively[5]. Hereafter
in the paper,the consistencyprinciple refersto eq. (2). If
eq. (2) is satisfied, P(A)2 N(A) holds. N(A) is a
necessitymeasuredefinedby N(A)=I- II(U -A).
Oneof the simplesttransformations&om probability
to possibilitysatisfyingeq. (3) might be given by
Jt(Ui) = P(“i)
v PO+)
/ Ui6v
The above equationsatisfies Jt(ui) r p(ui ). However, it
is provedthat eq.(2) cannot be satisfiedin general[2].
Dubois and Prade proposed a transformation
satisfying both the consistency and the preference
preservationprinciples [3]. Assumethat elementsin U
are orderedso that p( ul ) 2 I;(u $2 ... 2 p(u,, ). In the case,
ui might have an extra degree of necessity
P(4)-PC%+11 Over ui+l- Thus, the degreeof necessity
N(Ai) of event Ai ={ul,...,ui)c u might be defined by
SUIIIof p(uj)-p(&+l), uj EAi, where ui+l has the
largestprobability amongelementsin U - 4. That is
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p(4) = 2 :(x(,
i-v
j) -N”j+l
))y
where ;I~(u,+1) = 0.
The transformationis possible in both directions
between probability and possibility, and satisfies the
consistencyand the preferencepreservationprinciples.
However, there is no guaranteethat this is the only
transformationsatisfyingthem.
2.3 Transformationbasedon maximal specificity
Probability is a quantitative ratio scale of uncertainty,
while possibility is a quasi-qualitativeordinal scale [5].
Thus, an idea arisesthat the information included in a
possibility distribution is less than that in probability.
This idea leads to a principle that the possibility
distribution generatedfrom a probability one shouldbe
minimal in the senseof inclusionof fuzzy setsso that the
possibilitydistributionis maximally specific[5].
Assume that a possibility and probability
distributionsare given and that the possibility satisfies
Z(ZQ)2 ... z .7t(h ). In this case, it is shown that the
consistency principle is satisfied iff the following
conditionshold [6].
JG(ui
)a z1P(Uj ), VUi E U.
j=l,n
(8)
Thus, when a probability distribution satisfying
p( ul ) 2 ... z dun) is given, the maximally specific
possibility distribution satisfying eq.s (2) and (3) is
obtainedby the next equation.
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It(Ui
)
=
fi;(u
j),
QUi
E
(9)
U.
j=l,n
2
discreteset.
3.1 Evidence structures: probability and possibility
However,the transformationusing the aboveequationis
not adequateaccordingto a principle given later in the
paper, because ui and u . ( i # j ) satisfying
Callbe
P(%)=P(Uj)
n(Ui)*wT(Ujis
2.4 Transformationbasedon information-preservation
Geer and Klir proposed a “information preserving
transformation”betweenprobability and possibility [4].
They defined two kinds of uncertainty called
nonspeczfkity and discord on the body of evidence (this
is explainedin the next section)in Evidencetheory [7],
and assertedthat the total uncertaintyof the two must be
preservedin the transformationbetweenprobability and
possibility. Then, they found the log-interval scale
transformationsatisfyingthe assertionin both directions.
The transformationfrom probability into possibility is
doneby the next equation.
a
P(“i )
p(uI)
n(“i)=
t
(10)
3
1
where Ul is an elementhaving the largestprobability,
and a is obtainedby solving g( a) = 0.
(fi/~)~
l@)=
i-3n
-
E{@i
log2(i/(i-l))+
log2
I
and A C U satisfying m(A) > 0 is calledfocal element.
m(A) expresses
the degreeof belief committedexactlyto
setA. Let F be the setof all focal elements.Then,the pair
(F, m) is calledbody of evidence.
Belief and Plausibility measuresare defined using
the basicprobability assignmentasshownbelow.
BeZ(A) =
(12)
PI(A) =
(13)
BeZ(A) and PI(A) have the relations; BeZ(A) = l-P&A’)
and PZ(A)=l-Bd(A’),
where AC -U-A.
When all of the focal elements are nested or
consonant, BeZ(A) and PI(A) equal to necessity and
possibility measuresN(A) and n(A), respectively.For
any A,B c u , following equationshold.
LJ(AUB)=l7(A)vII(B).
II(AUAC)=12JA)v17(AC)=1.
17(A) =l-N(A
).
Now, if a setof consonantfocal elementsis givenby
F =j& ,...,F’},
F’ CF~+~, (kl,..., K-l), possibility
It(ui ) can be calculatedusingthe next equation.
I
N”i ) = n(# il) =
m(Fh),if
where pi = p(ui). It is provedthat g( a) = 0 hasa unique
solutionsatisfying 0 < a < 1.
Thereare two questionsin the transformation.One
is the validity of the principle of informationpreservation.
It is incompatiblewith the idea on which the maximal
specificity is basedin the previoussubsection.The other
is the assumptionthatthe transformationfrom probability
pi to possibility ni is given by a function 3ti = f(~) [5].
The transformationis not limited to such functions in
general.
3.
Transformations Based on Evidence Theory
The sectiondiscussesthree transformations(Z’l, 72 and
23) basedon Evidencetheory. U is assumeda finite and
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(14)
(15)
(16)
(17)
Pi
(11)
lpI)a -@i+llPl)“}’
i ~(Pjip1)oO/Xi-9)~
j-1 +lg
1 Ahum (A) = I is calledbasic probability assignment,
N”i ) = n(* i)) *
Pi
i-7 ,n
i=Jn-1
I
log,(l-
A fbnction 112
:z” - [QI] satisfying m(0)= o and
=
IT
h1 0,
m(B)
Bl &‘4 z0
Ui E&
-Fk-l,(k=&...,K),
(l*)
,K
if
Ui EU-Fk,
where &, = 0. From the above,it is derivedthat every
has the samevalue of possibility;
ui h Gk=Fk-Fk-1
n(ui ) = n(Gk ) 7 and that .7t(ui)>Jt(uj) > O if ui 6~k,
uj EGh and 1s k c hs K . Note that the order of
possibilitiesof ui canbe decidedonly by F = &, ....FK} ,
whatever m(e) is.
From eq.(1S),the next equationis derived.
m(fi)=Wd-17(G+d~
(19)
Then, it is known that P(A) = BeZ(A)= PI(A), when
Page: 72
m(A)=0 foranysetAsuchthat IAl >l,where IAl is
the cardinal&yof A. In this case,the following alsoholds.
preservationone.
Now, following lemmaandpropositionareproved,
thoughtheir proofsareomitteddueto lack of space.
Now, assumethat elements in U is ordered so that
p( ul ) z ... 2 P(U,J . uk is definedin the following.
Transformation from E, into E, sat@ the order
preservationprinciple,Ethe next conditionshold.
[Lemma l]
EGk = I;;: - Ffsl, k =l,..., K,
ui EUk *ui
w,=u-
UUh,
(22)
(24)
where K=K,=K,.
h-O,k-1
Pmax(Wk)=
V
P(“i) 3
ui-i
(23)
where u. = 0, and K is the k satisfying pmax(Wk) z 0
and pmax(Wk+l) = 0. Uk is a set of ui having the same
probabilities.If ui EU~, uj ah and 15 k<hsX,then
p(q) >P(u]) >0. It is evident that Uk nub = 0 for
k #th andthat P(Uk)=IUk Ip(ui) if ui EUk.
In the following discussion, E, = (Fp, mp ) and
E, = (F” ,m,)
are bodies of evidence defining
probability and possibility distributions, respectively.
Thus, the transformationbetweenprobability p(h) and
possibility it
can be replacedby the transformation
between E, and E,, where mJ{z+})
= p(ui ),
FP={~i}l~EU~U...UU~}and
Fn=~~,sse,F&}*
[Proposition l]
Transformation from E, into E, satisfy the order
preservationprinciple,iff the next conditionshold.
$
= u
uh, k =l,...,K .
(25)
h=l,k
From the proposition, it is evident that the order
preservation principle determines Fn = FT ,..., Fi} .
Since the order of possibilitiesof ui is given only by
Fld:, m,(e) doesnot have to be specifiedin the caseof
TI. That is, the transformationZl is uniquely determined
by the order preservationprinciple. If there is a needto
assignnumericalvaluesin somereason,you cangive any
satisfying
m(e”)+...+m(&f)=I
and
mn(” 1
?4i3>0*
3.2 Zl : Case where possibility is an ordinal scale
Probability is a ratio scale of uncertainty and their
numericalvalueshave their own meaning.On the other
hand, possibility can be consideredas an ordinal scale.
That is, the essential information of a possibility
distributionis the order of elementsfor possibilities,and
numerical values attached are considered just an
expedient to specify the order. Based on the above
interpretation,the only possibletransformationwe can
consideris the onefrom probability into possibility.
Now, we introducethe following principle called
probabilistic
order preservation WqJlY,
order
preservation principle);
1)
PM= P(Uj) +
n(ui ) =Jt(u j) 9 2, P(%) > P(“j) + Jt(ui) > Jt(uj>, where
--* is not an implicationbut a productionrule. The order
preservationprinciple inherits the order of possibilities
from probabilities.
Sincewe have only thesetwo rules,the following
conditionshold betweenthe given probabilitiesand the
possibilities
rules:
obtained
from
the
l)P(ui)=
P(“j)
e> JlY(Ui)=n(Uj),
2, P(ui) ’ P(“j)
cs n(ui >> n(uj), where - is equivalence. Thus,
hereafter the order preservationprinciple means the
abovetwo formulae.The principle is different both from
Zadeh’s consistency principle and the preference
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3.3 22 : Case where possibility is a ratio scale ( I )
In 72, we regardpossibility as a ratio scale.This means
that valuesof possibility have their own meanings,and
possibility 0.2 is twice as much aspossibility 0.1. In this
case, some new principles have to be introduced in
addition to the order preservationprinciple in order to
specify mA(*). We employ the consistencyprinciple and
the maximal specificity.
At first, assume that the given probability
distribution satisfies p( ul) > ... > P(U ,J . The maximally
specificpossibility distribution satisfyingthe consistency
principle is givenby eq. (9). The distributionalsosatisfies
the order preservationprinciple. If eq. (9) is adoptedas
the equation for transformation, mK(c)
is derived as
follows using eq.s (19) and (25), since uk = {uk},
(kl ,...,n).
m,($> = 4uk)- $&+I) = PW 3
(26)
where JT(u,+~)= 0.
Then, the case is considered where
Ph 1 r- 5:dun) and p(%) = p(q+r) holds at leastfor a
Ui ( IS i S n -
1).
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[Proposition 21
Let ~(4) = p(h+l) for iE{qj,...,rj -1) where qj 9 r.
(1s jsm)
satisfies 1s a <q c... <&I crm sn, and lit
p(q) > p(%+i) for other i. Then,the following equations
give the maximally specific possibility distribution
satisfying the order preservationand the consistency
principles.
44)=(
[ zp(uk) qj si =y(j =L...,m)
ksqj,n
(27)
Since probabilities of ui in Gz are all the same,it is
evidentthat Uk= GF .
Let 24: be an element in Uk= Gr . Then,
m,(g)
is derived inversely when a probability
distributionis given.
where p( %F+l)= o and $ is given by eq.(25).
From the above,possibility distributionis derived
from the probability onein the following way.
k=i,n
proof : Eq. (27) clearly satisfiesthe order preservation
andthe consistencyprinciples.It is alsoclearthat eq.(27)
doesnot satisfyboth of them, if one of the terms p( 4)
in RHS is replaced by a smaller value. Thus, the
propositionholds.(EOP)
kX(Ui) =
n(u, ), if UiE Uk = en - Fcl,
0, otherwise.
n(u,) =
An
=
m,(q)
=
j- T Jc
Now, m,(c)
giving the possibility distribution
in eq.(27) is derivedasfollows.
(28)
where Gi =$ - F’i,
Z$EG: , fl(Gi+i) = 0 and
Jt(UK+1)
= 0.
Since the transformation by eq. (27) is not a
subjective mapping from probability distributions to
possibilityones,the inversetransformationis not possible
in general.
3.4 T3 : Casewhere possibility is a ratio scale( II )
T3 also considersthat possibility is a ratio scale, and
adopts a principle called equidistribution, which was
introduced when Dubois and Prade approximated a
plausibility measure by a probability measure [2].
However,they did not examinethe detailsin the caseof
transformationbetweenprobability andpossibility.
If the idea is applied to the transformationfrom
possibilityto probability,the value of m,(g ) is equally
distributed to all elements in $. This is because
m P(A ) must be 0 for all A satisfjkg v\ > 1. Using the
idea as the principle, we can transform a possibility
distributioninto a probability oneasshownbelow.
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(31)
= IFif I&d)+
j= >;
1F; I&)
K
- p(ui’+‘))(32)
IIuj l~(U,j)j-k+l,K
Assumethat elementsin U areorderedso that p( ui) r ...
up
and that u,={u, ,..., 4}, 1s asisbsn.
Then,
the next equationis derived,becausep( u, ) = ... = p(ub).
X(Ui)=n(uk)=bP(‘i)+
P(‘j)
j-c+Jn
(33)
As the result, it is found that the transformationhappens
to be equal to eq. (6) proposedby [3]. As discussedin
subsection 2.2, the transformation satisfies the
consistencyprinciple and the preferencepreservation.It
also satisfies the order preservation principle, since
Lemma 1 and Proposition 1 hold. In addition, the
transformationis possiblein both directions.
4. An Example and Discussion
Threetypes of transformationsbasedon Evidencetheory
was discussed in the previous section. Tl is a
transformation from a probability distribution into a
possibilistic order based on the order preservation
principle. 22 is one to derive the maximally specific
possibility distribution from a probability one basedon
the consistency principle as well as on the order
preservation principle. 23 is one only based on
equidistribution,but the derived possibility distribution
Page: 74
satisfiesboth the consistencyprinciple and the order
preservationone.
Table 1 shows a simple example of these
transformations. P(4) is the given probability
distributionand Tl : n(ui) , T2 : n(ui ) and T3 : n(ui) are
the transformed possibility ones. oi (i = 1,...,5) are
symbolsto expressthe order of possibilities.The order is
q > ...>os.
As seenin the table, the following relationshold
for any two subsetsA,B C U .
This meansthat all three transformationsgeneratethe
sameordinal structureof possibility from a probability
distribution.
However, there is a problem to decide which is
better 12 or 73, when possibility is regardedas a ratio
scale. Both of them satisfy the consistencyand the
preferencepreservationprinciples as well as the order
preservationprinciple. The differenceis that 2’2usesthe
maximum specificity, while 73 does equidistribution.
Thus,the problemof “which is better” dependson which
principleis more valid.
As Dubois and Pradementioned,it might be true
that information included in a possibility distribution is
lessthan informationin a probability one,if possibility is
an ordinal scale. However, the order of possibility is
determineddependingonly on the set of focal elements,
andnot on the valuesof the basicprobability assignment
of the focal elements. This fact suggests us that
possibility is not regarded as an ordinal scale, when
valuesof possibility or basic probability assignmentare
discussed.From this point, thereis an importantquestion
aboutthe validity for the maximal specificity.
On the other hand, the idea of equidistribution
gives a very clear meaning to transformationbetween
probability andpossibility as an approximationbasedon
Evidencetheory.Thus,it might be concludedthat 23 has
Table 1 An Exampleof Transformation
p(Ui) T1 : nr(Ui) ~ : JC(Ui) ~ : X(UJ
u, 0.25
01
1.oo
1.oo
u, 0.20
0.2
0.75
0.95
u, 0.20
02
0.75
0.95
U” 0.15
02
0.35
0.80
ug 0.15
03
0.35
0.80
u, 0.05
0,
0.05
0.30
247 0.00
Or
0.00
0.00
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more validity than 7’2.This result might be welcomed,
because 23 is a transformation applicable in both
directions, while 72 is one only from probability to
possibility.
5.
Conclusions
The paper discussed some prominent work for
transformationbetweenprobability and possibility, and
showed that none of them found the unique
transformationsatisfyingthe principles discussedin the
work. Then, it examined three new ideas based on
Evidencetheory: one regardspossibility as an ordinal
scaleof uncertainty,andthe othersasa ratio scale.
When regarded as an ordinal scale, the paper
showed that the order preservationprinciple gives the
unique ordinal structureof possibility. When considered
asa ratio scale,it examinedtwo methodsandshowedthat
one of them basedon equidistributionleadsto the unique
transformationsatisfyingthe consistency,the preference
preservationand the order preservationprinciples. The
transformationis possiblein both directionsandhappens
to be the sameasthat proposedin [3].
6.
References
[l] L. A. Zadeh: Fuzzy sets as a basis for a theory of
possibility,Fuzzy Sets and Systems1,3-28 (1978)
[2] D. Dubois, H. Erade:On severalrepresentationsof
uncertainbody of evidence,in Fuzzy Information and
Decision Processes (eds.M. M. Gupta,E. Sanchez),
North-HollandRub.,pp.167-181(1982)
[3] D. Dubois, H. Prade: Unfair coins and necessity
measures:Towards a possibilistic interpretationof
histograms,Fuzzy Sets and Systems 10, pp.15-20
(1983)
[4] J. F. Geer, G. J. Klir: A mathmaticalanalysis of
information-preserving transformations between
probabilistic and possibilistic formulations of
uncertainty, Int. J General Systems 20, pp. 143-176
(1992)
[5] D. Dubois, H. Prade, S. Sandri : On
possibility/probability transformations, in Fuzzy
Logic : State of the Art (eds. R. Lowen, M. Lowen),
Kluwer AcademicPub.,pp. 103-l 12(1993)
[6] M. Delgado,S. Moral: On the conceptof possibilityprobability consistency,Fuzzy Sets and Systems21, pp.
311-318(1987)
[7] G. Shafer: A mathematical theoy of evidence,
PrincetonUniversityPress(1976)
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