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Belief functions generated by signed measures
FUZZY
sets and systems ELSEVIER
Fuzzy Sets and Systems 92 (1997) 157-166
Belief functions generated by signed measures Ivan Kramosil* Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod voddrenskou yogi 2, 18207 Praha 8, Czech Republic
Received April 1997
Abstract It is a well-known fact that the usual and already classical combinatorial definition of belief function over (the powerset of) a finite set can be generalized in such a way that belief function is defined by the quantile function of a set-valued (generalized) random variable defined over an abstract probability space. In this contribution we shall investigate a further stage of generalization resulting when the probability space in question is replaced by a measurable space equipped by a signed measure; signed measure is a a-additive set function which can take values also outside the unit interval, including the negative and infinite ones. An assertion analogous to the Jordan decomposition theorem for signed measures is stated and proved, according to which each signed belief function restricted to its finite values can be defined by a linear combination of two classical probabilistic belief functions, supposing that the basic set is finite. © 1997 Elsevier Science B.V. Keywords: Dempster-Shafer theory; Belief function; Signed measure; Signed belief function; Hahn decomposition theorem
I. Introduction Dempster-Shafer ( D - S ) approach to uncertainty quantification and processing or, as it is often called, D - S theory, has been developed for more than twenty years now (the basic Dempster paper [1] dates back to 1967). It represents, in our days, an interesting and promising mathematical model which can be seen, from the purely formal point o f view, as a non-traditional application o f probability theory, but which offers interpretations going out o f the frameworks o f the usual interpretations o f the probability theory. The belief function has been introduced and investigated in the D - S theory as a numerical characteristic o f uncertainty ascribing to each set of possible answers to a question (or set o f possible states o f * Fax: +42-2 85 85 789; e-mail: [email protected]. 0165-0114/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PH S01 65-01 14(97)00167-X
an investigated system, under another interpretation) the value which can be taken as the probability with which the obtained random empirical data (observations) are such that the true answer (or the actual state o f the system) can be proved to belong to the set in question. In other words, this value is defined by the probability that the set o f all answers or states compatible with the randomly obtained data forms a subset o f the set the (degree of) belief of which is to be defined. As the aim o f this paper is to present some generalizations of the classical belief functions and to investigate some purely mathematical issues connected with these generalizations, we omit here discussions concerning the philosophical and methodological problems as well as the motivation and we begin with the most simple and the most common definition of belief functions. We shall present this definition at an
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informal level, as it will be covered, as a special case, by a more general definition below. Let S be a finite nonempty set of possible answers or states and let us adopt the assumption of the closed world according to which the true answer to the given question or the true state of the investigated system is in S. Let m :~(S)---~ (0, 1) be the so called basicprobability assignment, i.e., a probability distribution on the power-set ~ ( S ) of all subsets of S. Hence, 0 ~
bel(m'A)=(1-m(O))-I Z
m(B). ¢48 c A
(1)
Taking the value m(A) as the probability that just A C S is the subset of all answers (states) compatible with the random data at hand, the interpretation of the value bel(m,A) agrees with that one outlined above. The relation (I) can be easily rewritten in a way which is equivalent to the just presented one but which admits a generalization to the case when S is infinite. Let (~,~d, Pl be a probability space, i.e. ~d is a nonempty ~-field of subsets of a set ~ and P is a probability measure defined on ~d. Let U be a measurable mapping taking I~,~d, Pl into (~(S), ~ ( ~ ( S ) ) ) , i.e., U : f2 --+ ~ ( S ) is such that for all ~' C ~ ( S ) , {co E f2 : U(~o) E ..~} E ~d holds. As S is finite, the condition {o9 E f2 : U(co) =A} E ~ ' for all A C S obviously suffices. Moreover, let U be such that the equality m(A) =P({co C f2: U(co) = A } ) holds for each A C S. Then (1') obviously converts into
bel(m,A) P({co: ~o E (2, 0 # U(og) CA})
P({~: ~En, 0-¢ u(o~)}) = B({co E 12: U(co) C A}l{co E 12: U(co) ¢ 0}) (2) using the elementary definition of conditional probability P(A/B) = P(A fqB)/P(B), if P(B) > O. In this approach we do not avoid the case when m(0) is positive, i.e., when the random event U(co) = 0 can occur with a positive probability. This case means that no state or answer from S is compatible with the observed data and, under the assumption of the closed
world, the only interpretation is that the data are logically inconsistent and that the original results of observations and measurements have been deformed in a confusing way in an information channel: D-S theory solves this problem by a simple re-normalization of the values of the belief function to the set of cases when the data are consistent. This solution can be discussed and criticized from different points of view and the problem of inconsistent data processing in D--S theory deserves a more detailed investigation, but it is not our goal in this paper. Abandoning the assumption of the closed world, the event U(og)= 0 can be interpreted in such a way that the true answer to the question or the actual state of the system is beyond the set S. Very often the simplifying assumption that m(0) = P({o9 E f2: U(co) = 0}) = 0 holds is accepted; in this case obviously
bel(m,A) =P({co E f2: U(og) cA})--- Z
m(B). (3)
BCA
In what follows, we shall also often use this simplification. There are at least three following ways how to generalize the simple and already classical model just described. (i) We can suppose that the set S is infinite, so that the set ~ ( S ) is uncountable. In this case, however, it is not too reasonable to take the whole power-set ~ ( ~ ( S ) ) as the G-field ~ , as in this case the class of possible probability measures on d is rather restricted (cf. [3], e.g.). So, a better solution is to choose a afield SP C ~ ( ~ ( S ) ) as a new free parameter in our constructions, and to define bel(m,A) by (2) only for those A C S for which ~(A) E 5~ holds (and supposing that {0} E 5 a and P({coEY2: U ( c o ) = 0 } ) < l hold). This approach is developed, in more detail, in [4]. (ii) Another generalization arises when we admit that the set-theoretic inclusion U(og) C A, the probability of validity which defines the value bel(m,A), needs not be always effectively decidable and that we can be sure that U(co) c A holds only if it holds for all the class of subsets of S equivalent with (or indistinguishable from) the set A. This model leads to some reasonable, "pessimistic", or "by the worst-case analysis motivated" approximations of belief functions. Some more detailed results in this direction can be found in [5].
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159
(iii) In this paper we shall investigate the case when the probability measure P defined on the measurable space (f2, ~¢) is replaced by a more general measure, namely, by a measure keeping the properties of a-additivity, but taking also values outside the unit interval of real numbers. We shall define appropriate generalizations of the notions of belief functions and their elementary properties as defined in the classical D - S theory over finite spaces S (cf., e.g. [6]).
( - 1 ) . cx~= - cxD, ( - 1 ) . ( - c x ~ ) = oo. We shall also suppose that a/cx~ = a / ( - c ~ ) = 0 for all a E ( - o c , cx~). No conventions concerning the expressions like 0/0, 0. (-4-oc), ( ± o e ) / ( ± o o ) , etc. are adopted at this instant; in the case of necessity we shall do so later. We denote by R + = (0, e~) and by R - = (-cx~,O) the corresponding subsets of (-cx~,o~). All these conventions can be easily seen to be consistent with the elementary properties of the corresponding operations on ( - o o , ~ ) .
2. From probability measure to signed measures and to the corresponding belief functions. Hahn decomposition theorem
Definition 2. Let ( f 2 , J ) be a measurable space. A mapping # : ~ ¢ ~ (0, e~) is called a (a-additive) measure, if #(0) = 0 (0 is the empty subset off2 which evidently is in z¢), and if #([_J~= 1Ai) = ~ o i= ~ 11"t(Ai ) for each sequence Al, A2 .... of mutually disjoint sets from ~¢.
Let us start with the well-known definition of probability measure. Definition 1. Let f2 be a nonempty set, let d be a nonempty a-field of subsets of f2, so that, for each A, A 1, A2 .... E d , f 2 - A E s,¢ and [.Ji~ l Ai E ~ holds as well. A mapping # : ,~¢ ~ (0, 1) is called probability measure on the measurable space (f2, d ) , if it is nonnegative, normalized and a-additive, i.e., if (i) # ( f 2 ) = 1 (obviously, f 2 E z ¢ holds for each nonempty a field ~¢), (ii) for each sequence A1, A2,... of mutually disjoint oo o(3 sets from ~¢, #(Ui=l A~) = ~-~i=1 #(Ai). During the following generalizations we shall work with the extended real line ( - o o , oo). Elementary arithmetical operations and relations are extended from R = ( - e o , oo) to R* according to the usual conventions. Namely, Y'~7= 1Xi = 0(3, if xi = oo for at least one i and xi = - o o for no index i, dually, n ~ i = x xi = - e ~ , if xi = - ~ for at least one i and OQ xi = ~ for no index i, ~ i = 1xi = c~, if xi = oo for at least one i, xi = - c x z for no i, and if there exists i0 such that xi>~O holds for all i>~io. Dually, 0<3 E i = 1Xi = - - 0 0 , if xi = - - o o f o r at l e a s t o n e i, X i = (DO for no i, and if there exists i0 such that xi ~<0 holds for all i >~i0. If there are i, j such that xi = ~ and xj = - e ~ , ~7(=°~l)x i is not defined. The inequality < is extended from ( - o o , oo) to { - o o , oe) in such a way that - c x D < x < o o holds for each x CR. Moreover, a • ~ = o o for all 0 < a ~ < o ~ , a • ~ = - o o for all - o o ~ < a < 0 , a • ( - e ~ ) = - o o for all 0 < a ~ < o o , and a - ( - c x ~ ) = o o for all - o o ~ < a < 0 . Namely,
Definition 3. Let ( f 2 , d ) be a measurable space. A mapping # : d - - + ( - o c , oo) is called a (aadditive) signed measure, if # ( 0 ) - - 0 , if for each sequence A1, A2,... of mutually disjoint sets from d the series ~ i ~ t l d ( A i ) is defined and the equaloo o~ ity #(Ui=lAi)= ~-~i=l~(Ai) holds, and if there are no sets A1,A2 in d such that / ~ ( A 1 ) = ~ and # ( A 2 ) = - c ~ (i.e., # assumes at most one of the infinite values to avoid expressions like oo - e~z). The two definitions above copy the definitions of the corresponding notions from [2] up to the difference that in [2] both the measures are supposed to be defined on a-rings of subsets of Q, not on a-fields. A system d of subsets off2 is a a-ring, if for each A1, A2,... E ~¢ also the sets A1 - A 2 and Ui= l Ai are in ze'. As each afield is, due to the de Morgan laws, obviously closed with respect to finite and countable intersections, and for each A1, A2 E d , Al - A 2 -----AtA ( f 2 - A 2 ) E d as well, each a-field is also a a-ring, so that the definitions in [2] are more general than our ones. On the other side, it is generally accepted that the complement of a random event, consisting in the fact that this random event has not occurred, is also a random event, so that the closedness of the system of random events with respect to the operation of complement seems to be quite natural. As it is not our goal, in this paper, to generalize the structure of the system of random events, or to propose an alternative structure, but rather to extend the system of possible numerical
160
I. Kramosil/Fuzzy Sets and Systems 92 (1997) 15~166
quantifications ascribed to the random events, we shall suppose, throughout all this paper, that random events form a a-field and we shall adapt all our definitions, here and below, to this case. It is almost obvious that each probability is also a measure, the only we have to prove is that #(0) = 0 for each probability measure/~. But, (A1,A2.... ) = (12, 0, ~, ~3.... ) is a sequence of disjoint sets from ~¢, so that OO 1 = # ( ( 2 ) = ~--~i=1 #(Ai)>f#( A1 ) + #(A2) = # ( ( 2 ) + #((~)~>#(O) = 1, hence, #(~) = 0 . An important, and very useful for our further purpose, property of signed measures is that they can be defined by differences of two (non-negative) measures. This property is formally stated and proved by the so called Hahn decomposition theorem which can be found, e.g., in [2]. Theorem 1 (Hahn decomposition theorem). Let (g2,
~¢) be a measurable space, let # be a signed measure defined on (f2, ~¢). Then there exist disjoint subsets (2+, f2- of f2 such that f2+ W (2- = O, (2+E ~¢, hence, O - E s~, and both the mappings #1, #z :du¢~ ( - ~ , oo) definedby #I(A) = #(A M f2 +) and # 2 ( A ) = - # ( A M f 2 - ) for each AC~¢, are measures in the sense of Definition 2, so that #i(A) E (0, cxD)for both i = 1,2. So, for each signed measure # defined on (f2, ~¢) and for each A E su¢ the value # ( A ) = #1(A) - # 2 ( A ) is the difference of the values ascribed to A by two (non-negative) measures defined on (f2, ~¢). The pair (f2+, f2-) is called the Hahn decomposition of f2 and it need not be defined uniquely, the pair (#1,#2) is called Jordan decomposition of the signed measure # and it is defined uniquely. Proof. Cf. [2] for more general case of signed measures defined on a-rings of subsets of 12. [] Let us close this chapter by the definition of the basic notion of this paper, namely, of the three variants of belief functions corresponding to the three types of measures defined on the basic measurable space. Definition 4. Let (f2, ~¢) be a measurable space, let S be a nonemp~ set, let 5a C ~ ( ~ ( S ) ) be a nonempty a-field of systems of subsets of S, let U : f2 --~ ~ ( S ) be a measurable mapping from the measurable space
(O, d ) into the measurable space (~(S), Sf), hence, let the inclusion {{co E f2: U(co) E N}: N E 5 °} C ~¢ hold. Let {co E 12: U(co) = ~} = ~. (i) Let # be a probability measure defined on (f2, ~¢). Then the (partial) mapping bel(U,#): ~(S)--* (0, 1), ascribing to each A C S such that ~(A) E 5e the value
bel(U, #) (A) = #({co E (2: U(co) E ~(A)}) = #({co E f2: U(co) c A } )
(4)
is called the (classical or probabilistic) belief function defined by U and # in ~ ( S ) (if ~ ( A ) ~ 5 a, bel(U, #)(A) is not defined). (ii) Let # be a measure defined on (f2, d ) . Then the (partial) mapping bel(U, #) : ~ ( S ) --+ (0, ~ ) defined by (4) is called the generalized belief function defined by U and # in ~(S). (iii) Let # be a soned measure defined on ((2, d ) . Then the (partial) mapping bel(U, #) : ~ ( S ) --~ (-cxD, ~ ) defined by (4) is called the signed belief function defined by U and # in ~(S). As can be easily seen, if the set S is finite and S P = ~ ( ~ ( S ) ) , (4) reduces to (3), hence, Definition 4 corresponds to the simplified version of the usual definition of belief function when m ( ~ ) = 0 is supposed to hold, in fact, our assumption that {co ~ O: U(og) = 0} = 0 is still stronger. The reason for our restriction to the case when the random empirical data are supposed to be logically consistent is that in the more general case corresponding to (2) we have to define: first of all, conditional measures and conditional signed measures, and to investigate the conditions under which they are defined. Such an investigation can be interesting and useful, but it would bring us beyond the scope to which this paper should be oriented. Moreover, such an approach would charge our further reasoning by many technical difficulties and this is why we have decided to follow, at least now, the most simple way suggested by Definition 4. In the next chapter we shall try to arrive at some decompositions of signed belief functions into generalized or even into classical probabilistic belief functions inspired by, and similar to, the Jordan decompositions of signed measures. We shall also prove that generalizations of basic probability assignments, generated
L KramosillFuzzy Sets and Systems 92 (1997) 157-166
on ~ ( S ) with a finite S by signed belief functions, are also signed measures on the measurable space (~(S), ~ ( ~ ( S ) ) ) .
161
the same relation - ~ o < m * ( ~ ) = ~-'m(A)<~oo
(6)
AE2#
3. Decompositionof signed belief functions Theorem 2. Let (0, ag) be a measurable space, let # be a signed measure on (fLag), let S be a finite nonempty set, let U be a measurable mapping which takes the measurable space (f2, ag) into the measurable space ( ~ ( S ) , ~ ( ~ ( S ) ) ) . Set, for each A c S , m(A) = #({o9 E f2: U(co) =A}) (i.e., #({~o E f2: U(o9) E {A} E ~a(~(S))})) and set, for each ~ c ~ a ( S ) , m * ( ~ ) = }--~aE~m(A) with the convention that m*(O)=0 for the empty subset of ~a(~(S)). Then m* : ~(~(S))--+ (-oo, oo) is a signed measure on (~(S), ~a(~(S))}. Set ~ ( S ) = {A C S: m(A) > 0}, ~a2(S) = {A C S: m(A) < 0}. Let ~+(S) C ~(S), ~ - ( S ) C ~ ( S ) be such that ~ ( S ) C .~+(S), °o~2(S)C~.~-(S), ~ + ( S ) O ~ - ( S ) = ~ , ~+(S) U ~ - ( S ) = ~ ( S ) . Then (~+(S), ~ - ( S ) ) is a Hahn decomposition of ~ ( S ) with respect to the signed measure m*. Remark. Obviously, this Hahn decomposition is not uniquely defined, as m(O)= 0, so that we can take either ~ ~ ~+(S) or ~ E ~ - ( S ) . Proof. The equality m*(O)=0 follows immediately from the definition of m*. Due to the finiteness of S and, consequently, of ~a(S), a-additivity reduces to finite additivity and this property obviously holds for m*. Or, if ~1, ~2 are disjoint subsets of ~a(S), then m*(°~lU~2) -----
Z
re(A)= Zm(A)+
AE g~'lU.~82
AE.~I
Zm(A) AE°~2
holds for each ~3 C ~(S). Hence, m* is a signed measure on the measurable space (~(S), ~(~(S))}. Let ( ~ + ( S ) , ~ - ( S ) } be a decomposition of ~ ( S ) such that, for all A E ~+(S) (A E ~a-(S), resp.), the relation m(A) >10 (m(A) <~O, resp.) holds. Then, obviously, the inclusions ~ (S) = {A C S: re(A) > 0} C ~+(S) and ~2(S) = {A C S: m(A) <0} C ~ - ( S ) are valid. Let B C ~a(S). Then the inequalities m*(~' M/~+(S)) =
Z
m(A)>~O
(7)
m(A)<,O
(8)
AE~A,~+(S)
and m*(~ n ~a-(S)) =
Z AE.~rq~-(S)
obviously hold, as m(A)>~O for each A E ~ ( S ) and m(A)~O for each A E ~ - ( S ) . Hence, (~+(S), ~ - ( S ) ) is a Hahn decomposition of ~ ( S ) with respect to the signed measure m* and the (nonnegative and a-additive) measures m]'(-)= m*(. N ~a+(S)) and rn~(.) = - m * ( - N ~ - ( S ) ) on ~a(~a(S)) represent the (obviously uniquely defined) Jordan decomposition of m*. [] Obviously, each Hahn decomposition (~+(S), ~a-(S)) o f ~ ( S ) i s such that ~l(S) C ~+(S) C {A C S: m(A)>~0} and ~a2(S) c ~ - ( S ) c {A C S: m(A)~<0} hold. Or, if there exists A C S such that m(A)<0 and A E ~ + ( S ) hold simultaneously, then for ~ = {A} C ~ ( S ) we obtain that m * ( M N ~ + ( S ) ) = m*({A})=m(A) 0 is treated analogously.
(5)
Theorem 3. Let the notations and conditions of
Let the signed measure # be such that - o o < #(E) ~
Theorem 2 hold. Let ago C ag be the minimal a-field of subsets of f2 containing all the sets { co E f2: U ( ag) = A } , ag C S. Let x be an object different from all subsets of S. Then there exist probability measures P1,P2, defined on the measurable space (f2, do), two random variables U1, U2 taking
= m*(~l ) + m*(~2).
I. Kramosil/Fuzzy Sets and Systems 92 (1997) 157-166
162
(O,S~¢o) into (.~(S) U {x},,~(~@(S) U {x})), and two finite nonnegative real numbers ~, fl such that, for all A c S with -oo
(9)
Remark. The values c~ and fl are independent of A. Hence, the value ascribed to A C S by the signed belief function bel(U, 12) can be obtained as a linear combination of the values ascribed to the same set A by two (classical probabilistic) belief functions bel(Ut,P~) and bel(U2, P2). The relation (9) can be taken as something like a Jordan decomposition of signed belief functions. If bel(U, 12)(A) is infinite, (9) obviously cannot hold for finite ~, fl, as bel(Ui, Pi)(A), i = 1,2, are probability values, hence, values embedded within the unit interval of reals. Proof of Theorem 3. Let (N+(S), ~ - ( S ) ) be a Hahn decomposition of ~ ( S ) with respect to m*. Define the mappings Ui:g2~(S)t_J{x}, i= 1,2, in this way: UI(CO)=AcS iff U(co)=A and AE~+(S), Ul(co)=x otherwise, U2(co)=A iff U(co)=A and A E ~ - ( S ) , U2(co) = x otherwise. Obviously, both U1,U2 are measurable mappings, i.e., (generalized, non-numerical) random variables, as {co E (2: Ui(CO)=A} E ~¢0 holds for both i = 1,2 and for aliA C S orA = x . ForA C S it is clear, forA = x we obtain that {co E f2: Ul(co) = x } = UA e ~-(s){co E £2: U(co)=A} and {co E £2:U2(co)=x}=UA~+(s){co EO:U(CO)=A}, 'and both these sets are in ~¢0 due to the fact that ~ + ( S ) and ~ - ( S ) are finite systems of sets. Moreover, if Ul(co)=A for some A c S , then m(A)=12({coEO: U(co)=A})/>0, and if U2(CO)=ACS, then rn(A)<~O, due to the definitions of U1 and (-/2. As {co E £2: U(co)= 13} = ~, also {co E f2: Ui(co) = ~} = 13for b o t h / = 1,2. Let us suppose that there exist A, B C S such that 0 < m ( A ) < e c and 0 > m ( B ) > - e ~ hold, in other words, suppose that ~l(S)~Fin+(m,S)¢13 and ~2(S) ~ Fin- (m, S) ¢ ~, where Fin+(m, S) = {A C S: re(A) < oo} and fin-(m, S) = {A C S: re(A) > - ec}; let us recall that ~ ( S ) = { A C S : m ( A ) > O } and ~2(S) = {A C S: re(A)<0}. Set
o¢=
~ A~+(S)AFin+(m,S)
m(A ),
(10)
re(A). AE~-(S)MFin-(m,S)
By assumptions ( ~ ( S ) ) finite, ~1(S) and ~2(S) nonempty) we obtain that 0 < ~ < oc and 0 < fl < oc hold. Define P/: s¢0 ~ (0, 1), i = 1,2, in this way:
P ({co c 8: u(co) =A})
= (~m(A)/a,
i
ifA E ~ + ( S ) , m(A)
0,
P2({co c o: g(co) =A})
S -m(A)/fl,
L0,
(11)
i f A E ~ - ( S ) , m(A)> - oc, for other A C S.
Both the mappings PI,P2 can be uniquely extended to a-additive probability measures on the a-field d 0 , as nonnegativity is clear and ~-~Ac s/~({co E £2: U(co) = A } ) = 2AEc~+(S)NFin+(m,S ) m(A)= 1 (similarly for P2). Hence, for both i = 1,2, (£2,~¢0,P~.) is a probability space and Ui: (2 ~ ~(S) U {x} is a random variable (measurable mapping with respect to the a-field ~ ( N ( S ) U {x})). Consequently, we can define the classical probabilistic belief functions bel(U1,Pl) and bel(U2,P:) on ~(St_J{x}), setting bel(Ui,Pi) (A) =/],.({co E £2: Ui(co) CA}) for each A C S U {x}. In particular, for A C S, an easy calculation yields that
bel(U1,Pl ) (A) = P1 ({co E (2:U1 (co) CA}) =
e ({coE BCA
=
((coE
U (co) =B})
B C A,BE~+(S), re(B) < oo
=
Z
m(B)/~
B C A,BE~+(S),m(B) <
E Q:
N~+ (S)NFin+ (m, S)}),
(12)
so that #({co E f2: U(CO) E ~(A) M~ + ( S ) [-1Fin+(m, S)})
=ocbel(U~,P1) (A).
(13)
I. Kramosil/ Fuzzy Sets and Systems 92 (1997) 157-166
Analogously,
bel( U2, P2) (A) =
Z
( m(B )//3 )
= - (1//3) #({co • K2: U(co) • ~(A) f-q~ - ( S ) (14)
and #({co • f2: U(co) • ~(A) Yl~ - ( S ) MFin-(m, S)})
= - fibel(U2,Pz)(A).
(15)
Let A C S be such that
bel(U, #)(A) = #({co • f2: U(co) • A}) = Z
m(B)
Let us consider, now, the case when ~1 (S) or ~2(S) is empty. If ~1 ( S ) = ~ 2 ( S ) = 0, then m ( A ) : 0 for all A CS, so that bel(U,#)(A)=O for all A CS, so that
bel(U, #) (A) = O. bel(U~, Pt ) (A) - O. bel(U2, P2) (A) for no matter which probability distributions P1,P2
B C A,BE~-(S),m(B)< oc)
A Fin-(m, S)}),
163
(16)
BCA
on d 0 . Let ~ l ( S ) # 0 , ~2(S)~---0, so that m (A)>~0 holds for each A C S. Then (~(S),0) is also a Hahn decomposition of ~@(S) with respect to m* (here 0 is the empty subset of ~(S), i.e. the empty system of subsets of S). As the Jordan decomposition of m* is independent of the Hahn decomposition of ~(S), we can replace ( ~ + ( S ) , ~ - ( S ) ) by (~(S), (~) and apply the same way of reasoning as above. Let there exist at least one A C S such that 0
bel(UI,P1)(A) = (1/c~) #({co E ~2: U(co)
If there exists B CA such that m ( B ) = o c , then it cannot exist B ' C S such that m ( B ' ) = - o o , as in this case m * ( { B } ) = o c and m * ( { B ' } ) = - o o , what contradicts the proved fact that m* is a signed measure on ~ ( ~ ( S ) ) . Hence, m ( B ) = e c for some B CA implies that re(A)= o~ as well, in other words, bel(U,#)(A) - oc, then m(B) > - ec for all B C A, hence, :~(A)cFin-(m,S). So, if bel(U,#)(A) is finite, then ~(A) CFin+(m, S)nFin-(m, S) and
bel(U, #) (A) =#({co • ~: u(co) cA}) =#({co • f2: U(~o) • ~(A)})
E~(A)NFin+(m, S)}).
(18)
In the same way as in (11) we prove that if bel(U, #) (A) < oc, then ~(A) C Fin+(m, S), so that
bel(U,p)(A) = #({co E I2: U(co) CA}) =#({co E f2: U(co) E ~(A) N Fin+(m, S)})
=~bel(U1,P1)(A) - O. bel(U2,P2)(A),
(19)
so that (9) holds again with f l = 0. The case when ~1(S) = 0 and ~2(S) MFin-(m, S) # 0 is processed analogously. Finally, if re(A)= ~ for all A c S (or if re(A) = - ~ for all A C S), then there is no A C S with finite and non-zero value bel( U, #) (A) so that (9) holds either trivially or with ~ = fl = 0. The theorem is proved. H
=#({co • f2: U(co) • ~(A)
A ~+(S) fq Fin+(m, S)})
4. Conditioned belief functions
+#({co • o: u(co) • ~(A)
fq ~@-(S) fq Fin- (m, S)}) =abel(UbP1)(A) - fl bel(Uz,Pz)(A) by (13) and (15), so that (9) holds.
(17)
First of all, let us extend the definition of conditioned belief functions also to the generalized versions of belief functions defined above. The assertion following this definition proves that if the basic space S is finite, the well-known combinatorial expressions for the conditioned belief functions are valid in the
1. Kramosil/Fuzzy Sets and Systems 92 (1997) 157-166
164
generalized case as well. As elsewhere in this paper, we shall limit ourselves to the purely formalized and mathematical aspects of conditioning operations for belief functions, leaving aside the motivation and interpretation (the reader is kindly invited to consult [6] or other sources for this sake).
As S and, consequently, ~ ( S ) are finite sets, the tr-additivity of # immediately yields that, for A C T,
bel(A[T)
c
=
u( o) = B}).
(22)
B C S: O~BATCANT
Definition 5. Let (~, .~¢) be a measurable space, let # be a probability measure (a generalized measure, a signed measure, resp.) defined on (f2,~¢), let S be a nonempty finite set, let U : (~, ~¢)---~(~(S), ~ ( ~ ( S ) ) ) be a measurable mapping which takes (2 into ~(S), let T be a nonempty subset of S. The conditioned classical probabilistic (generalized, signed, resp.) belief function bel(U,#)(.IT ) is the mapping of ~ ( S ) into (0, 1) defined, for each A C S, by the relation: bel(U, #) (AIT) = 0, ifA g~ T, otherwise
Each B c S such that 0 ¢ B N T C A N T can be uniquely decomposed into disjoint subsets 0 ~ B1 = B n T and B2 = B n (S - T), conversely, for each 0 CBl C A N T and B2 C S - T, the set B = B1 U B2 belongs to the class of sets over which the summation in (22) is defined. As the mapping B ~ (B1,B2) is one-to-one, (22) immediately yields that
bel(U,#)(A[T)
and the equality between the first and the third item in (21 ) is proved. In fact, it is just this equality which serves, as a rule, as the definition of conditioned belief function in the works dealing with belief functions over finite spaces S. By the definition of the (unconditioned) signed belief function we obtain that
(20)
Obviously, bel(U, #) (.) = bel(U, #) (. IS) for all the three kinds of belief functions.
Theorem 4. Let the notations and conditions of Definition 5 hold. Then for the conditioned classical probabilistic (generalized, signed, resp.) belief function bel(U, #) (.IT) the relation bel(U,/2) (A[ T)
m(B, UB2),
(23)
bel(A U (S - T)) - bel(S - T) =
Z
m(B)-
BcAU(S-- T)
=
~ m(BUX)
~
O~Bi CANT B2CS-- T
Z
m(B)
BCS-- T
Z
re(B)
BCAU(S--T),B~S-T
=bel(U,#)(A U(S - T)) - bel(U,#)(S - T)
= Z
~
bel(AIT)=
(21)
=
Z
m(B)
BE { CcS: CAANT} ~O, CC(AAT)U(S-- T)
05~BCANT X C S-- T
Z
holds for each A C T, here m(C) = #({co c ~2: U(co) = C } ) , C c S , as above. Proof. Because of the evident fact that probability measures are a special case of (generalized) measures and the later ones are a special case of signed measures, it is sufficient to prove the assertion for the case when # is a signed measure. For the sake of simplicity we omit the parameters U and p throughout this proof.
m(B1 U B2)
B=BI UB2, O~Bi CANT, B2 C S-- T
=
~
Z
m(B1 UB2).
(24)
O~BI CANT B2cS-- T
The assertion is proved.
[]
Corollary 1. Let the notations and conditions of Theorem 2 hold, let ~o and x be as in Theorem 3, let
bel(U,p)(AIT ) be defined as in Definition 5. Then
I. Kramosil/Fuzzy Sets and Systems 92 (1997) 157-166 there exist probability measures P1, P2, random variables Ul, U2, and finite nonnegative real numbers c~,t~ with the properties claimed in Theorem 3 and such that, for all A C T C S with the property that bel(U, #) (A U (S - T)) and bel(U, #) (S - T) are finite numbers, the equality bel(U,#)(AIT) =~bel(UI,P1)(AIT ) - flbel(Uz,P2)(AIT )
(25)
holds. Proof. Applying (9) to the sets A U (S - T) C S and S - T C S we obtain, by an easy calculation, that bel(U,#)(AIT) =bel(U,#)(A U(S - T ) ) - b e l ( U , # ) ( S - T) =[e bel(U1,P1 ) (A U (S - T)) - [~bel( Uz, P2) (A U ( S - r ) )]
- [ ~ bel(U1 ,PI ) (S - T ) - f l bel(U2, P2) (S - T)] =~[bel(U1,P1 ) (A U (S - T))-beI(UI,P1 ) (S - T)] -fl[bel(Uz,Pz)(A U (S - T)) - b e l ( U 2 , P 2 ) ( S - T)] =~bel(U1,P1)(AIT) -flbel(U2,Pz)(AIT).
[]
(26)
Hence, Jordan decomposition of a conditioned signed belief function can be obtained as the linear combination of the corresponding conditioned classical probabilistic belief functions with the same coefficients as in the unconditioned case. The last fact is quite intuitive, as in the case when T = S, Eq. (25) immediately converts into Eq. (9).
165
words, there is a number of other notions and results developed within the framework of the theory of classical probabilistic belief functions the generalization of which to the case of signed belief functions could and should be suggested and investigated, however, because of the limited extent of this contribution we have to terminate our effort now. Just as an inspiration and orientation for further research effort let us pick up the three most important (at least in the author's opinion) open problems: (i) To define, at least in the case of a finite set S, a "signed" analogy to the notion of signed probability assignment and to investigate the relations between those signed belief assignments and the signed belief functions. Is every signed belief function over (the power-set of) a finite set S definable by a uniquely defined signed basic assignment as it is the case for classical belief functions? (ii) Given a finite set of classical belief functions (or their corresponding basic probability assignments) over a finite set S, it is always possible to define a common probability space (I2, ~ , P) so that every belieffunction in question is defined by a corresponding set-valued random variable taking f2 into ~(S). Is this also the case for measurable spaces with signed measures and for signed belief functions? (iii) How to define the Dempster combination rule for signed belief functions and which are the algebraic properties of the space of signed belief functions over a fixed (and perhaps finite) set S with respect to the binary operation ® defined by the Dempster rule. Is it possible to define an inverse element and an analogy of the operation of substraction in this space? This problem stands in close relation to that of "deconditialization" which is, in general, unsolvable within the framework of the classical probability theory. Let us hope that at least some of these questions will be answered in a future paper dealing with signed belief functions.
Acknowledgements 5. Conclusions It is a matter of an obvious fact that the analogies between classical and signed belief functions can be drawn much further than we have done above. In other
This work has been partially sponsored by the COPERNICUS project MUM No. 10053 from the European Union. Another support has been given by the grant No. 201/93/0781 of the Grant Agency of
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the Czech Republic and by the grant No. A1030504 of the Grant Agency of the Academy of Sciences of the Czech Republic. References [1] A.P. Dempster, Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Statist. 38 (1967) 325-339. [2] P.R. Halmos, Measure Theory, D. van Nostrand, New York, 1950.
[3] J. Kohlas, The reliability of reasoning with unreliable arguments, Ann. Oper, Res. 32 (1991) 67-113. [4] I. Kramosil, Believeability and plausibility functions over infinite sets, Internat. J. General Systems 23 (1994) 173-198. [5] I. Kramosil, Approximations of believeability functions under incomplete identification of sets of compatible states, Kybernetika 31 (1995) 425-450. [6] Ph. Smets, The representation of quantified belief by belief functions, an axiomatic justification, Technical Report No. TR/IRIDIA/94-3.3, Institut de R6cherches Interdisciplinaires et de D6veloppements en Intelligence Artificielle, Universit6 Libre de Bruxelles, Belgium, 1994.
sets and systems ELSEVIER
Fuzzy Sets and Systems 92 (1997) 157-166
Belief functions generated by signed measures Ivan Kramosil* Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod voddrenskou yogi 2, 18207 Praha 8, Czech Republic
Received April 1997
Abstract It is a well-known fact that the usual and already classical combinatorial definition of belief function over (the powerset of) a finite set can be generalized in such a way that belief function is defined by the quantile function of a set-valued (generalized) random variable defined over an abstract probability space. In this contribution we shall investigate a further stage of generalization resulting when the probability space in question is replaced by a measurable space equipped by a signed measure; signed measure is a a-additive set function which can take values also outside the unit interval, including the negative and infinite ones. An assertion analogous to the Jordan decomposition theorem for signed measures is stated and proved, according to which each signed belief function restricted to its finite values can be defined by a linear combination of two classical probabilistic belief functions, supposing that the basic set is finite. © 1997 Elsevier Science B.V. Keywords: Dempster-Shafer theory; Belief function; Signed measure; Signed belief function; Hahn decomposition theorem
I. Introduction Dempster-Shafer ( D - S ) approach to uncertainty quantification and processing or, as it is often called, D - S theory, has been developed for more than twenty years now (the basic Dempster paper [1] dates back to 1967). It represents, in our days, an interesting and promising mathematical model which can be seen, from the purely formal point o f view, as a non-traditional application o f probability theory, but which offers interpretations going out o f the frameworks o f the usual interpretations o f the probability theory. The belief function has been introduced and investigated in the D - S theory as a numerical characteristic o f uncertainty ascribing to each set of possible answers to a question (or set o f possible states o f * Fax: +42-2 85 85 789; e-mail: [email protected]. 0165-0114/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved PH S01 65-01 14(97)00167-X
an investigated system, under another interpretation) the value which can be taken as the probability with which the obtained random empirical data (observations) are such that the true answer (or the actual state o f the system) can be proved to belong to the set in question. In other words, this value is defined by the probability that the set o f all answers or states compatible with the randomly obtained data forms a subset o f the set the (degree of) belief of which is to be defined. As the aim o f this paper is to present some generalizations of the classical belief functions and to investigate some purely mathematical issues connected with these generalizations, we omit here discussions concerning the philosophical and methodological problems as well as the motivation and we begin with the most simple and the most common definition of belief functions. We shall present this definition at an
L KramosillFuzzy Sets and Systems 92 (1997) 157-166
158
informal level, as it will be covered, as a special case, by a more general definition below. Let S be a finite nonempty set of possible answers or states and let us adopt the assumption of the closed world according to which the true answer to the given question or the true state of the investigated system is in S. Let m :~(S)---~ (0, 1) be the so called basicprobability assignment, i.e., a probability distribution on the power-set ~ ( S ) of all subsets of S. Hence, 0 ~
bel(m'A)=(1-m(O))-I Z
m(B). ¢48 c A
(1)
Taking the value m(A) as the probability that just A C S is the subset of all answers (states) compatible with the random data at hand, the interpretation of the value bel(m,A) agrees with that one outlined above. The relation (I) can be easily rewritten in a way which is equivalent to the just presented one but which admits a generalization to the case when S is infinite. Let (~,~d, Pl be a probability space, i.e. ~d is a nonempty ~-field of subsets of a set ~ and P is a probability measure defined on ~d. Let U be a measurable mapping taking I~,~d, Pl into (~(S), ~ ( ~ ( S ) ) ) , i.e., U : f2 --+ ~ ( S ) is such that for all ~' C ~ ( S ) , {co E f2 : U(~o) E ..~} E ~d holds. As S is finite, the condition {o9 E f2 : U(co) =A} E ~ ' for all A C S obviously suffices. Moreover, let U be such that the equality m(A) =P({co C f2: U(co) = A } ) holds for each A C S. Then (1') obviously converts into
bel(m,A) P({co: ~o E (2, 0 # U(og) CA})
P({~: ~En, 0-¢ u(o~)}) = B({co E 12: U(co) C A}l{co E 12: U(co) ¢ 0}) (2) using the elementary definition of conditional probability P(A/B) = P(A fqB)/P(B), if P(B) > O. In this approach we do not avoid the case when m(0) is positive, i.e., when the random event U(co) = 0 can occur with a positive probability. This case means that no state or answer from S is compatible with the observed data and, under the assumption of the closed
world, the only interpretation is that the data are logically inconsistent and that the original results of observations and measurements have been deformed in a confusing way in an information channel: D-S theory solves this problem by a simple re-normalization of the values of the belief function to the set of cases when the data are consistent. This solution can be discussed and criticized from different points of view and the problem of inconsistent data processing in D--S theory deserves a more detailed investigation, but it is not our goal in this paper. Abandoning the assumption of the closed world, the event U(og)= 0 can be interpreted in such a way that the true answer to the question or the actual state of the system is beyond the set S. Very often the simplifying assumption that m(0) = P({o9 E f2: U(co) = 0}) = 0 holds is accepted; in this case obviously
bel(m,A) =P({co E f2: U(og) cA})--- Z
m(B). (3)
BCA
In what follows, we shall also often use this simplification. There are at least three following ways how to generalize the simple and already classical model just described. (i) We can suppose that the set S is infinite, so that the set ~ ( S ) is uncountable. In this case, however, it is not too reasonable to take the whole power-set ~ ( ~ ( S ) ) as the G-field ~ , as in this case the class of possible probability measures on d is rather restricted (cf. [3], e.g.). So, a better solution is to choose a afield SP C ~ ( ~ ( S ) ) as a new free parameter in our constructions, and to define bel(m,A) by (2) only for those A C S for which ~(A) E 5~ holds (and supposing that {0} E 5 a and P({coEY2: U ( c o ) = 0 } ) < l hold). This approach is developed, in more detail, in [4]. (ii) Another generalization arises when we admit that the set-theoretic inclusion U(og) C A, the probability of validity which defines the value bel(m,A), needs not be always effectively decidable and that we can be sure that U(co) c A holds only if it holds for all the class of subsets of S equivalent with (or indistinguishable from) the set A. This model leads to some reasonable, "pessimistic", or "by the worst-case analysis motivated" approximations of belief functions. Some more detailed results in this direction can be found in [5].
I. Kramosil/ Fuzzy Sets and Systems 92 (1997) 157-166
159
(iii) In this paper we shall investigate the case when the probability measure P defined on the measurable space (f2, ~¢) is replaced by a more general measure, namely, by a measure keeping the properties of a-additivity, but taking also values outside the unit interval of real numbers. We shall define appropriate generalizations of the notions of belief functions and their elementary properties as defined in the classical D - S theory over finite spaces S (cf., e.g. [6]).
( - 1 ) . cx~= - cxD, ( - 1 ) . ( - c x ~ ) = oo. We shall also suppose that a/cx~ = a / ( - c ~ ) = 0 for all a E ( - o c , cx~). No conventions concerning the expressions like 0/0, 0. (-4-oc), ( ± o e ) / ( ± o o ) , etc. are adopted at this instant; in the case of necessity we shall do so later. We denote by R + = (0, e~) and by R - = (-cx~,O) the corresponding subsets of (-cx~,o~). All these conventions can be easily seen to be consistent with the elementary properties of the corresponding operations on ( - o o , ~ ) .
2. From probability measure to signed measures and to the corresponding belief functions. Hahn decomposition theorem
Definition 2. Let ( f 2 , J ) be a measurable space. A mapping # : ~ ¢ ~ (0, e~) is called a (a-additive) measure, if #(0) = 0 (0 is the empty subset off2 which evidently is in z¢), and if #([_J~= 1Ai) = ~ o i= ~ 11"t(Ai ) for each sequence Al, A2 .... of mutually disjoint sets from ~¢.
Let us start with the well-known definition of probability measure. Definition 1. Let f2 be a nonempty set, let d be a nonempty a-field of subsets of f2, so that, for each A, A 1, A2 .... E d , f 2 - A E s,¢ and [.Ji~ l Ai E ~ holds as well. A mapping # : ,~¢ ~ (0, 1) is called probability measure on the measurable space (f2, d ) , if it is nonnegative, normalized and a-additive, i.e., if (i) # ( f 2 ) = 1 (obviously, f 2 E z ¢ holds for each nonempty a field ~¢), (ii) for each sequence A1, A2,... of mutually disjoint oo o(3 sets from ~¢, #(Ui=l A~) = ~-~i=1 #(Ai). During the following generalizations we shall work with the extended real line ( - o o , oo). Elementary arithmetical operations and relations are extended from R = ( - e o , oo) to R* according to the usual conventions. Namely, Y'~7= 1Xi = 0(3, if xi = oo for at least one i and xi = - o o for no index i, dually, n ~ i = x xi = - e ~ , if xi = - ~ for at least one i and OQ xi = ~ for no index i, ~ i = 1xi = c~, if xi = oo for at least one i, xi = - c x z for no i, and if there exists i0 such that xi>~O holds for all i>~io. Dually, 0<3 E i = 1Xi = - - 0 0 , if xi = - - o o f o r at l e a s t o n e i, X i = (DO for no i, and if there exists i0 such that xi ~<0 holds for all i >~i0. If there are i, j such that xi = ~ and xj = - e ~ , ~7(=°~l)x i is not defined. The inequality < is extended from ( - o o , oo) to { - o o , oe) in such a way that - c x D < x < o o holds for each x CR. Moreover, a • ~ = o o for all 0 < a ~ < o ~ , a • ~ = - o o for all - o o ~ < a < 0 , a • ( - e ~ ) = - o o for all 0 < a ~ < o o , and a - ( - c x ~ ) = o o for all - o o ~ < a < 0 . Namely,
Definition 3. Let ( f 2 , d ) be a measurable space. A mapping # : d - - + ( - o c , oo) is called a (aadditive) signed measure, if # ( 0 ) - - 0 , if for each sequence A1, A2,... of mutually disjoint sets from d the series ~ i ~ t l d ( A i ) is defined and the equaloo o~ ity #(Ui=lAi)= ~-~i=l~(Ai) holds, and if there are no sets A1,A2 in d such that / ~ ( A 1 ) = ~ and # ( A 2 ) = - c ~ (i.e., # assumes at most one of the infinite values to avoid expressions like oo - e~z). The two definitions above copy the definitions of the corresponding notions from [2] up to the difference that in [2] both the measures are supposed to be defined on a-rings of subsets of Q, not on a-fields. A system d of subsets off2 is a a-ring, if for each A1, A2,... E ~¢ also the sets A1 - A 2 and Ui= l Ai are in ze'. As each afield is, due to the de Morgan laws, obviously closed with respect to finite and countable intersections, and for each A1, A2 E d , Al - A 2 -----AtA ( f 2 - A 2 ) E d as well, each a-field is also a a-ring, so that the definitions in [2] are more general than our ones. On the other side, it is generally accepted that the complement of a random event, consisting in the fact that this random event has not occurred, is also a random event, so that the closedness of the system of random events with respect to the operation of complement seems to be quite natural. As it is not our goal, in this paper, to generalize the structure of the system of random events, or to propose an alternative structure, but rather to extend the system of possible numerical
160
I. Kramosil/Fuzzy Sets and Systems 92 (1997) 15~166
quantifications ascribed to the random events, we shall suppose, throughout all this paper, that random events form a a-field and we shall adapt all our definitions, here and below, to this case. It is almost obvious that each probability is also a measure, the only we have to prove is that #(0) = 0 for each probability measure/~. But, (A1,A2.... ) = (12, 0, ~, ~3.... ) is a sequence of disjoint sets from ~¢, so that OO 1 = # ( ( 2 ) = ~--~i=1 #(Ai)>f#( A1 ) + #(A2) = # ( ( 2 ) + #((~)~>#(O) = 1, hence, #(~) = 0 . An important, and very useful for our further purpose, property of signed measures is that they can be defined by differences of two (non-negative) measures. This property is formally stated and proved by the so called Hahn decomposition theorem which can be found, e.g., in [2]. Theorem 1 (Hahn decomposition theorem). Let (g2,
~¢) be a measurable space, let # be a signed measure defined on (f2, ~¢). Then there exist disjoint subsets (2+, f2- of f2 such that f2+ W (2- = O, (2+E ~¢, hence, O - E s~, and both the mappings #1, #z :du¢~ ( - ~ , oo) definedby #I(A) = #(A M f2 +) and # 2 ( A ) = - # ( A M f 2 - ) for each AC~¢, are measures in the sense of Definition 2, so that #i(A) E (0, cxD)for both i = 1,2. So, for each signed measure # defined on (f2, ~¢) and for each A E su¢ the value # ( A ) = #1(A) - # 2 ( A ) is the difference of the values ascribed to A by two (non-negative) measures defined on (f2, ~¢). The pair (f2+, f2-) is called the Hahn decomposition of f2 and it need not be defined uniquely, the pair (#1,#2) is called Jordan decomposition of the signed measure # and it is defined uniquely. Proof. Cf. [2] for more general case of signed measures defined on a-rings of subsets of 12. [] Let us close this chapter by the definition of the basic notion of this paper, namely, of the three variants of belief functions corresponding to the three types of measures defined on the basic measurable space. Definition 4. Let (f2, ~¢) be a measurable space, let S be a nonemp~ set, let 5a C ~ ( ~ ( S ) ) be a nonempty a-field of systems of subsets of S, let U : f2 --~ ~ ( S ) be a measurable mapping from the measurable space
(O, d ) into the measurable space (~(S), Sf), hence, let the inclusion {{co E f2: U(co) E N}: N E 5 °} C ~¢ hold. Let {co E 12: U(co) = ~} = ~. (i) Let # be a probability measure defined on (f2, ~¢). Then the (partial) mapping bel(U,#): ~(S)--* (0, 1), ascribing to each A C S such that ~(A) E 5e the value
bel(U, #) (A) = #({co E (2: U(co) E ~(A)}) = #({co E f2: U(co) c A } )
(4)
is called the (classical or probabilistic) belief function defined by U and # in ~ ( S ) (if ~ ( A ) ~ 5 a, bel(U, #)(A) is not defined). (ii) Let # be a measure defined on (f2, d ) . Then the (partial) mapping bel(U, #) : ~ ( S ) --+ (0, ~ ) defined by (4) is called the generalized belief function defined by U and # in ~(S). (iii) Let # be a soned measure defined on ((2, d ) . Then the (partial) mapping bel(U, #) : ~ ( S ) --~ (-cxD, ~ ) defined by (4) is called the signed belief function defined by U and # in ~(S). As can be easily seen, if the set S is finite and S P = ~ ( ~ ( S ) ) , (4) reduces to (3), hence, Definition 4 corresponds to the simplified version of the usual definition of belief function when m ( ~ ) = 0 is supposed to hold, in fact, our assumption that {co ~ O: U(og) = 0} = 0 is still stronger. The reason for our restriction to the case when the random empirical data are supposed to be logically consistent is that in the more general case corresponding to (2) we have to define: first of all, conditional measures and conditional signed measures, and to investigate the conditions under which they are defined. Such an investigation can be interesting and useful, but it would bring us beyond the scope to which this paper should be oriented. Moreover, such an approach would charge our further reasoning by many technical difficulties and this is why we have decided to follow, at least now, the most simple way suggested by Definition 4. In the next chapter we shall try to arrive at some decompositions of signed belief functions into generalized or even into classical probabilistic belief functions inspired by, and similar to, the Jordan decompositions of signed measures. We shall also prove that generalizations of basic probability assignments, generated
L KramosillFuzzy Sets and Systems 92 (1997) 157-166
on ~ ( S ) with a finite S by signed belief functions, are also signed measures on the measurable space (~(S), ~ ( ~ ( S ) ) ) .
161
the same relation - ~ o < m * ( ~ ) = ~-'m(A)<~oo
(6)
AE2#
3. Decompositionof signed belief functions Theorem 2. Let (0, ag) be a measurable space, let # be a signed measure on (fLag), let S be a finite nonempty set, let U be a measurable mapping which takes the measurable space (f2, ag) into the measurable space ( ~ ( S ) , ~ ( ~ ( S ) ) ) . Set, for each A c S , m(A) = #({o9 E f2: U(co) =A}) (i.e., #({~o E f2: U(o9) E {A} E ~a(~(S))})) and set, for each ~ c ~ a ( S ) , m * ( ~ ) = }--~aE~m(A) with the convention that m*(O)=0 for the empty subset of ~a(~(S)). Then m* : ~(~(S))--+ (-oo, oo) is a signed measure on (~(S), ~a(~(S))}. Set ~ ( S ) = {A C S: m(A) > 0}, ~a2(S) = {A C S: m(A) < 0}. Let ~+(S) C ~(S), ~ - ( S ) C ~ ( S ) be such that ~ ( S ) C .~+(S), °o~2(S)C~.~-(S), ~ + ( S ) O ~ - ( S ) = ~ , ~+(S) U ~ - ( S ) = ~ ( S ) . Then (~+(S), ~ - ( S ) ) is a Hahn decomposition of ~ ( S ) with respect to the signed measure m*. Remark. Obviously, this Hahn decomposition is not uniquely defined, as m(O)= 0, so that we can take either ~ ~ ~+(S) or ~ E ~ - ( S ) . Proof. The equality m*(O)=0 follows immediately from the definition of m*. Due to the finiteness of S and, consequently, of ~a(S), a-additivity reduces to finite additivity and this property obviously holds for m*. Or, if ~1, ~2 are disjoint subsets of ~a(S), then m*(°~lU~2) -----
Z
re(A)= Zm(A)+
AE g~'lU.~82
AE.~I
Zm(A) AE°~2
holds for each ~3 C ~(S). Hence, m* is a signed measure on the measurable space (~(S), ~(~(S))}. Let ( ~ + ( S ) , ~ - ( S ) } be a decomposition of ~ ( S ) such that, for all A E ~+(S) (A E ~a-(S), resp.), the relation m(A) >10 (m(A) <~O, resp.) holds. Then, obviously, the inclusions ~ (S) = {A C S: re(A) > 0} C ~+(S) and ~2(S) = {A C S: m(A) <0} C ~ - ( S ) are valid. Let B C ~a(S). Then the inequalities m*(~' M/~+(S)) =
Z
m(A)>~O
(7)
m(A)<,O
(8)
AE~A,~+(S)
and m*(~ n ~a-(S)) =
Z AE.~rq~-(S)
obviously hold, as m(A)>~O for each A E ~ ( S ) and m(A)~O for each A E ~ - ( S ) . Hence, (~+(S), ~ - ( S ) ) is a Hahn decomposition of ~ ( S ) with respect to the signed measure m* and the (nonnegative and a-additive) measures m]'(-)= m*(. N ~a+(S)) and rn~(.) = - m * ( - N ~ - ( S ) ) on ~a(~a(S)) represent the (obviously uniquely defined) Jordan decomposition of m*. [] Obviously, each Hahn decomposition (~+(S), ~a-(S)) o f ~ ( S ) i s such that ~l(S) C ~+(S) C {A C S: m(A)>~0} and ~a2(S) c ~ - ( S ) c {A C S: m(A)~<0} hold. Or, if there exists A C S such that m(A)<0 and A E ~ + ( S ) hold simultaneously, then for ~ = {A} C ~ ( S ) we obtain that m * ( M N ~ + ( S ) ) = m*({A})=m(A)
(5)
Theorem 3. Let the notations and conditions of
Let the signed measure # be such that - o o < #(E) ~
Theorem 2 hold. Let ago C ag be the minimal a-field of subsets of f2 containing all the sets { co E f2: U ( ag) = A } , ag C S. Let x be an object different from all subsets of S. Then there exist probability measures P1,P2, defined on the measurable space (f2, do), two random variables U1, U2 taking
= m*(~l ) + m*(~2).
I. Kramosil/Fuzzy Sets and Systems 92 (1997) 157-166
162
(O,S~¢o) into (.~(S) U {x},,~(~@(S) U {x})), and two finite nonnegative real numbers ~, fl such that, for all A c S with -oo
(9)
Remark. The values c~ and fl are independent of A. Hence, the value ascribed to A C S by the signed belief function bel(U, 12) can be obtained as a linear combination of the values ascribed to the same set A by two (classical probabilistic) belief functions bel(Ut,P~) and bel(U2, P2). The relation (9) can be taken as something like a Jordan decomposition of signed belief functions. If bel(U, 12)(A) is infinite, (9) obviously cannot hold for finite ~, fl, as bel(Ui, Pi)(A), i = 1,2, are probability values, hence, values embedded within the unit interval of reals. Proof of Theorem 3. Let (N+(S), ~ - ( S ) ) be a Hahn decomposition of ~ ( S ) with respect to m*. Define the mappings Ui:g2~(S)t_J{x}, i= 1,2, in this way: UI(CO)=AcS iff U(co)=A and AE~+(S), Ul(co)=x otherwise, U2(co)=A iff U(co)=A and A E ~ - ( S ) , U2(co) = x otherwise. Obviously, both U1,U2 are measurable mappings, i.e., (generalized, non-numerical) random variables, as {co E (2: Ui(CO)=A} E ~¢0 holds for both i = 1,2 and for aliA C S orA = x . ForA C S it is clear, forA = x we obtain that {co E f2: Ul(co) = x } = UA e ~-(s){co E £2: U(co)=A} and {co E £2:U2(co)=x}=UA~+(s){co EO:U(CO)=A}, 'and both these sets are in ~¢0 due to the fact that ~ + ( S ) and ~ - ( S ) are finite systems of sets. Moreover, if Ul(co)=A for some A c S , then m(A)=12({coEO: U(co)=A})/>0, and if U2(CO)=ACS, then rn(A)<~O, due to the definitions of U1 and (-/2. As {co E £2: U(co)= 13} = ~, also {co E f2: Ui(co) = ~} = 13for b o t h / = 1,2. Let us suppose that there exist A, B C S such that 0 < m ( A ) < e c and 0 > m ( B ) > - e ~ hold, in other words, suppose that ~l(S)~Fin+(m,S)¢13 and ~2(S) ~ Fin- (m, S) ¢ ~, where Fin+(m, S) = {A C S: re(A) < oo} and fin-(m, S) = {A C S: re(A) > - ec}; let us recall that ~ ( S ) = { A C S : m ( A ) > O } and ~2(S) = {A C S: re(A)<0}. Set
o¢=
~ A~+(S)AFin+(m,S)
m(A ),
(10)
re(A). AE~-(S)MFin-(m,S)
By assumptions ( ~ ( S ) ) finite, ~1(S) and ~2(S) nonempty) we obtain that 0 < ~ < oc and 0 < fl < oc hold. Define P/: s¢0 ~ (0, 1), i = 1,2, in this way:
P ({co c 8: u(co) =A})
= (~m(A)/a,
i
ifA E ~ + ( S ) , m(A)
0,
P2({co c o: g(co) =A})
S -m(A)/fl,
L0,
(11)
i f A E ~ - ( S ) , m(A)> - oc, for other A C S.
Both the mappings PI,P2 can be uniquely extended to a-additive probability measures on the a-field d 0 , as nonnegativity is clear and ~-~Ac s/~({co E £2: U(co) = A } ) = 2AEc~+(S)NFin+(m,S ) m(A)= 1 (similarly for P2). Hence, for both i = 1,2, (£2,~¢0,P~.) is a probability space and Ui: (2 ~ ~(S) U {x} is a random variable (measurable mapping with respect to the a-field ~ ( N ( S ) U {x})). Consequently, we can define the classical probabilistic belief functions bel(U1,Pl) and bel(U2,P:) on ~(St_J{x}), setting bel(Ui,Pi) (A) =/],.({co E £2: Ui(co) CA}) for each A C S U {x}. In particular, for A C S, an easy calculation yields that
bel(U1,Pl ) (A) = P1 ({co E (2:U1 (co) CA}) =
e ({coE BCA
=
((coE
U (co) =B})
B C A,BE~+(S), re(B) < oo
=
Z
m(B)/~
B C A,BE~+(S),m(B) <
E Q:
N~+ (S)NFin+ (m, S)}),
(12)
so that #({co E f2: U(CO) E ~(A) M~ + ( S ) [-1Fin+(m, S)})
=ocbel(U~,P1) (A).
(13)
I. Kramosil/ Fuzzy Sets and Systems 92 (1997) 157-166
Analogously,
bel( U2, P2) (A) =
Z
( m(B )//3 )
= - (1//3) #({co • K2: U(co) • ~(A) f-q~ - ( S ) (14)
and #({co • f2: U(co) • ~(A) Yl~ - ( S ) MFin-(m, S)})
= - fibel(U2,Pz)(A).
(15)
Let A C S be such that
bel(U, #)(A) = #({co • f2: U(co) • A}) = Z
m(B)
Let us consider, now, the case when ~1 (S) or ~2(S) is empty. If ~1 ( S ) = ~ 2 ( S ) = 0, then m ( A ) : 0 for all A CS, so that bel(U,#)(A)=O for all A CS, so that
bel(U, #) (A) = O. bel(U~, Pt ) (A) - O. bel(U2, P2) (A) for no matter which probability distributions P1,P2
B C A,BE~-(S),m(B)< oc)
A Fin-(m, S)}),
163
(16)
BCA
on d 0 . Let ~ l ( S ) # 0 , ~2(S)~---0, so that m (A)>~0 holds for each A C S. Then (~(S),0) is also a Hahn decomposition of ~@(S) with respect to m* (here 0 is the empty subset of ~(S), i.e. the empty system of subsets of S). As the Jordan decomposition of m* is independent of the Hahn decomposition of ~(S), we can replace ( ~ + ( S ) , ~ - ( S ) ) by (~(S), (~) and apply the same way of reasoning as above. Let there exist at least one A C S such that 0
bel(UI,P1)(A) = (1/c~) #({co E ~2: U(co)
If there exists B CA such that m ( B ) = o c , then it cannot exist B ' C S such that m ( B ' ) = - o o , as in this case m * ( { B } ) = o c and m * ( { B ' } ) = - o o , what contradicts the proved fact that m* is a signed measure on ~ ( ~ ( S ) ) . Hence, m ( B ) = e c for some B CA implies that re(A)= o~ as well, in other words, bel(U,#)(A)
bel(U, #) (A) =#({co • ~: u(co) cA}) =#({co • f2: U(~o) • ~(A)})
E~(A)NFin+(m, S)}).
(18)
In the same way as in (11) we prove that if bel(U, #) (A) < oc, then ~(A) C Fin+(m, S), so that
bel(U,p)(A) = #({co E I2: U(co) CA}) =#({co E f2: U(co) E ~(A) N Fin+(m, S)})
=~bel(U1,P1)(A) - O. bel(U2,P2)(A),
(19)
so that (9) holds again with f l = 0. The case when ~1(S) = 0 and ~2(S) MFin-(m, S) # 0 is processed analogously. Finally, if re(A)= ~ for all A c S (or if re(A) = - ~ for all A C S), then there is no A C S with finite and non-zero value bel( U, #) (A) so that (9) holds either trivially or with ~ = fl = 0. The theorem is proved. H
=#({co • f2: U(co) • ~(A)
A ~+(S) fq Fin+(m, S)})
4. Conditioned belief functions
+#({co • o: u(co) • ~(A)
fq ~@-(S) fq Fin- (m, S)}) =abel(UbP1)(A) - fl bel(Uz,Pz)(A) by (13) and (15), so that (9) holds.
(17)
First of all, let us extend the definition of conditioned belief functions also to the generalized versions of belief functions defined above. The assertion following this definition proves that if the basic space S is finite, the well-known combinatorial expressions for the conditioned belief functions are valid in the
1. Kramosil/Fuzzy Sets and Systems 92 (1997) 157-166
164
generalized case as well. As elsewhere in this paper, we shall limit ourselves to the purely formalized and mathematical aspects of conditioning operations for belief functions, leaving aside the motivation and interpretation (the reader is kindly invited to consult [6] or other sources for this sake).
As S and, consequently, ~ ( S ) are finite sets, the tr-additivity of # immediately yields that, for A C T,
bel(A[T)
c
=
u( o) = B}).
(22)
B C S: O~BATCANT
Definition 5. Let (~, .~¢) be a measurable space, let # be a probability measure (a generalized measure, a signed measure, resp.) defined on (f2,~¢), let S be a nonempty finite set, let U : (~, ~¢)---~(~(S), ~ ( ~ ( S ) ) ) be a measurable mapping which takes (2 into ~(S), let T be a nonempty subset of S. The conditioned classical probabilistic (generalized, signed, resp.) belief function bel(U,#)(.IT ) is the mapping of ~ ( S ) into (0, 1) defined, for each A C S, by the relation: bel(U, #) (AIT) = 0, ifA g~ T, otherwise
Each B c S such that 0 ¢ B N T C A N T can be uniquely decomposed into disjoint subsets 0 ~ B1 = B n T and B2 = B n (S - T), conversely, for each 0 CBl C A N T and B2 C S - T, the set B = B1 U B2 belongs to the class of sets over which the summation in (22) is defined. As the mapping B ~ (B1,B2) is one-to-one, (22) immediately yields that
bel(U,#)(A[T)
and the equality between the first and the third item in (21 ) is proved. In fact, it is just this equality which serves, as a rule, as the definition of conditioned belief function in the works dealing with belief functions over finite spaces S. By the definition of the (unconditioned) signed belief function we obtain that
(20)
Obviously, bel(U, #) (.) = bel(U, #) (. IS) for all the three kinds of belief functions.
Theorem 4. Let the notations and conditions of Definition 5 hold. Then for the conditioned classical probabilistic (generalized, signed, resp.) belief function bel(U, #) (.IT) the relation bel(U,/2) (A[ T)
m(B, UB2),
(23)
bel(A U (S - T)) - bel(S - T) =
Z
m(B)-
BcAU(S-- T)
=
~ m(BUX)
~
O~Bi CANT B2CS-- T
Z
m(B)
BCS-- T
Z
re(B)
BCAU(S--T),B~S-T
=bel(U,#)(A U(S - T)) - bel(U,#)(S - T)
= Z
~
bel(AIT)=
(21)
=
Z
m(B)
BE { CcS: CAANT} ~O, CC(AAT)U(S-- T)
05~BCANT X C S-- T
Z
holds for each A C T, here m(C) = #({co c ~2: U(co) = C } ) , C c S , as above. Proof. Because of the evident fact that probability measures are a special case of (generalized) measures and the later ones are a special case of signed measures, it is sufficient to prove the assertion for the case when # is a signed measure. For the sake of simplicity we omit the parameters U and p throughout this proof.
m(B1 U B2)
B=BI UB2, O~Bi CANT, B2 C S-- T
=
~
Z
m(B1 UB2).
(24)
O~BI CANT B2cS-- T
The assertion is proved.
[]
Corollary 1. Let the notations and conditions of Theorem 2 hold, let ~o and x be as in Theorem 3, let
bel(U,p)(AIT ) be defined as in Definition 5. Then
I. Kramosil/Fuzzy Sets and Systems 92 (1997) 157-166 there exist probability measures P1, P2, random variables Ul, U2, and finite nonnegative real numbers c~,t~ with the properties claimed in Theorem 3 and such that, for all A C T C S with the property that bel(U, #) (A U (S - T)) and bel(U, #) (S - T) are finite numbers, the equality bel(U,#)(AIT) =~bel(UI,P1)(AIT ) - flbel(Uz,P2)(AIT )
(25)
holds. Proof. Applying (9) to the sets A U (S - T) C S and S - T C S we obtain, by an easy calculation, that bel(U,#)(AIT) =bel(U,#)(A U(S - T ) ) - b e l ( U , # ) ( S - T) =[e bel(U1,P1 ) (A U (S - T)) - [~bel( Uz, P2) (A U ( S - r ) )]
- [ ~ bel(U1 ,PI ) (S - T ) - f l bel(U2, P2) (S - T)] =~[bel(U1,P1 ) (A U (S - T))-beI(UI,P1 ) (S - T)] -fl[bel(Uz,Pz)(A U (S - T)) - b e l ( U 2 , P 2 ) ( S - T)] =~bel(U1,P1)(AIT) -flbel(U2,Pz)(AIT).
[]
(26)
Hence, Jordan decomposition of a conditioned signed belief function can be obtained as the linear combination of the corresponding conditioned classical probabilistic belief functions with the same coefficients as in the unconditioned case. The last fact is quite intuitive, as in the case when T = S, Eq. (25) immediately converts into Eq. (9).
165
words, there is a number of other notions and results developed within the framework of the theory of classical probabilistic belief functions the generalization of which to the case of signed belief functions could and should be suggested and investigated, however, because of the limited extent of this contribution we have to terminate our effort now. Just as an inspiration and orientation for further research effort let us pick up the three most important (at least in the author's opinion) open problems: (i) To define, at least in the case of a finite set S, a "signed" analogy to the notion of signed probability assignment and to investigate the relations between those signed belief assignments and the signed belief functions. Is every signed belief function over (the power-set of) a finite set S definable by a uniquely defined signed basic assignment as it is the case for classical belief functions? (ii) Given a finite set of classical belief functions (or their corresponding basic probability assignments) over a finite set S, it is always possible to define a common probability space (I2, ~ , P) so that every belieffunction in question is defined by a corresponding set-valued random variable taking f2 into ~(S). Is this also the case for measurable spaces with signed measures and for signed belief functions? (iii) How to define the Dempster combination rule for signed belief functions and which are the algebraic properties of the space of signed belief functions over a fixed (and perhaps finite) set S with respect to the binary operation ® defined by the Dempster rule. Is it possible to define an inverse element and an analogy of the operation of substraction in this space? This problem stands in close relation to that of "deconditialization" which is, in general, unsolvable within the framework of the classical probability theory. Let us hope that at least some of these questions will be answered in a future paper dealing with signed belief functions.
Acknowledgements 5. Conclusions It is a matter of an obvious fact that the analogies between classical and signed belief functions can be drawn much further than we have done above. In other
This work has been partially sponsored by the COPERNICUS project MUM No. 10053 from the European Union. Another support has been given by the grant No. 201/93/0781 of the Grant Agency of
166
I. Kramosil/Fuzzy Sets and Systems 92 (1997) 157-166
the Czech Republic and by the grant No. A1030504 of the Grant Agency of the Academy of Sciences of the Czech Republic. References [1] A.P. Dempster, Upper and lower probabilities induced by a multivalued mapping, Ann. Math. Statist. 38 (1967) 325-339. [2] P.R. Halmos, Measure Theory, D. van Nostrand, New York, 1950.
[3] J. Kohlas, The reliability of reasoning with unreliable arguments, Ann. Oper, Res. 32 (1991) 67-113. [4] I. Kramosil, Believeability and plausibility functions over infinite sets, Internat. J. General Systems 23 (1994) 173-198. [5] I. Kramosil, Approximations of believeability functions under incomplete identification of sets of compatible states, Kybernetika 31 (1995) 425-450. [6] Ph. Smets, The representation of quantified belief by belief functions, an axiomatic justification, Technical Report No. TR/IRIDIA/94-3.3, Institut de R6cherches Interdisciplinaires et de D6veloppements en Intelligence Artificielle, Universit6 Libre de Bruxelles, Belgium, 1994.