Representation of Evidence by Hints

This paper introduces a mathematical model of a hint as a body of imprecise and uncertain information. Hints are used to judge hypotheses: the degree to which a hint supports a hypothesis and the degree to which a hypothesis appears as plausible in the light of a hint are de ned. This leads in turn to supportand plausibility functions. Those functions are characterized as set functions which are normalized and monotone or alternating of order 1. This relates the present work to G. Shafer's mathematical theory of evidence. However, whereas Shafer starts out with an axiomatic de nition of belief functions, the notion of a hint is considered here as the basic element of the theory. It is shown that a hint contains more information than is conveyed by its support function alone. Also hints allow for a straightforward and logical derivation of Dempster's rule for combining independent and dependent bodies of information. This paper presents the mathematical theory of evidence for general, in nite frames of discernment from the point of view of a theory of hints. KEYWORDS Hints, evidence, support functions, plausibility functions, Dempster's rule 1. HINTS | AN INTUITIVE INTRODUCTION Intuitively, a hint is a body of information relative to some question 2 J. KOHLAS, P.A. MONNEY which is in general imprecise in that it does not point to a precise answer but rather to a range of possible answers. It is also often uncertain in the sense that the information allows for several possible interpretations and it is not entirely sure which is the correct one. There may be internal con ict within a hint because di erent interpretations may lead to contradictory answers. Also there can be external contradictions between distinct and di erent hints relative to the same question. The goal of this paper is to develop a mathematical model of this intuitive notion of a hint and to study some of its basic properties. It takes as its starting point A. Dempster's (1967) multivalued mapping and develops into similar lines as G. Shafer's (1976) mathematical theory of evidence. The theory will however be developed for the most general case and not be limited to the case of nite frames as in Shafer's book. For an introduction and as a motivation the simpler case of nite hints will rst be discussed. Let be an arbitrary nite set whose elements represent the possible answers to a given question which has to be considered. One of the elements of represents the true, but unknown answer. is called the frame of discernment. The subsets of represent possible propositions about the answer to the question considered. Let denote the nite set of possible interpretations of the information contained in the hint to be represented. One of the elements ! 2 must be the correct interpretation, but it is unknown which one. However, not all possible interpretations are equally likely. Thus, a probability p(!) for the interpretations ! 2 is introduced. Each possible interpretation ! restricts the possible answers within somehow. If ! is the correct interpretation, then the correct answer is known to be within some nonempty subset (!) of , the focal set of the interpretation. Alternatively, for any possible interpretation !, the family S of the propositions (subsets of ) implied by the interpretation ! can be considered. S is simply the family of supersets of the focal set (!). It has thus trivially the following properties: (1) H 2 S and H H 0 imply H 0 2 S (2) H1 2 S; H2 2 S imply H1 \H2 2 S: (3) belongs to S, ; does not belong to S. In addition, the intersection of all implied sets of an interpretation equals (!). Furthermore, for any possible interpretation, one can also look at the family P of propositions which are possible under the interpretation. A subset H is possible, when H intersects the focal set (!) of the interpretation. Equivalently, H is possible, i its complement is not implied, Hc 62 S. P has the following properties: REPRESENTATION OF EVIDENCE BY HINTS 3 (1') H 2 P and H H 0 imply H 0 2 P (2') H1 2 P; H2 2 P imply H1 [H2 2 P: (3') belongs to P, ; does not belong to P. Furthermore, if H 2 S, then Hc 62 S and thus S P. A hint is thus de ned by a frame of discernment to which it refers, a set of possible interpretations together with a probability p(!) and nally a multivalued mapping from the set of interpretations into the frame . If the interpretation ! happens to be the correct one, then the answer to the question considered is restricted to the set (!). So far, any hint H is a quadruple ( ; p; ; ). If a proposition H is xed as a hypothesis about the correct answer, then it will be interesting to judge this hypothesis in the light of a hint H. Let S(!) and P(!) denote the families of implied and possible propositions of an interpretation !. Then one can look at the subsets of interpretations under which H is implied, u(H), or possible, v(H) u(H) = f! 2 : H 2 S(!)g v(H) = f! 2 : H 2 P(!)g (1) A hypothesis H, which is implied or supported by many possible interpretations, or more important, by very probable interpretations, is very credible in the light of the hint. Also, if the hypothesis is possible under many interpretations, or under very probable interpretations, then the hypothesis is very plausible in the light of the hint. Thus, in order to measure the degree of credibility or support sp(H) and the degree of plausibility pl(H), the probabilities of u(H) and v(H) can be considered: sp(H) = P (u(H)) pl(H) = P (v(H)): (2) The values sp(H) and pl(H) are de ned for all subsets of . sp is called a support (or belief) function and pl a plausibility function (or upper probability). These concepts were introduced by A. Dempster (1967) and extensively studied by Shafer (1976) for nite frames of discernment. The goal of this contribution is to study hints with respect to arbitrary, especially in nite frames. To the best of our knowledge, only very few papers study evidence theory in this general case (Goodman, Nguyen, 4 J. KOHLAS, P.A. MONNEY 1985; Nguyen, 1978; Shafer, 1979; Strat, 1984) The case of belief functions on in nite frames of discernment was in particular studied by Shafer (1979). In this paper belief functions are axiomatically de ned as Choquet capacities, monotone of order1. Using an integral representation theorem of Choquet (1953, 1969) an allocation of probability for belief functions is derived. This concept provides for an interpretation of the meaning of belief. However, with this interpretation, the de nition of Dempster's rule for the combination of belief functions is less straightforward. In an unpublished paper G. Shafer (1978) de nes rst product belief functions on a product space and then Dempster's rule as a conditioning of the product belief function to the diagonal of . This seems somehow to be a detour. Hints on the other hand allow for a straightforward and logical derivation of Dempster's rule for combining independent and also dependent bodies of information. Furthermore and more importantly, it will be seen that in the general case a hint contains more information than is conveyed by its support function alone. Therefore, hints cannot be combined on the base of their support functions alone as proposed in Shafer's paper (1978)! This would result in a loss of information. This will be one of the main results of this paper. Another main result is that supportand plausibility functions as de ned by (2) can be characterized as Choquet capacities, monotone of order 1. The proof of this result rejoins Shafer's (1979) development and will only be sketched here. Finally, a new inclusion relation between hints will be introduced in this paper which generalizes a similar relation between support functions introduced by Yager (1985, see also Dubois, Prade, 1986). In section 2 the general mathematical concept of a hint will be dened. In section 3 supportand plausibility functions will be introduced. A process of re ning hints is presented in section 4. It leads to a relation of inclusion between hints. Section 5 studies inclusion relations between hints which are equivalent in the sense that they de ne partially the same supportand plausibility functions. Finally, in section 6, the combination of hints will be discussed and Dempster's rule derived. In particular, it will be shown that inclusion of hints is maintained under Dempster's rule. The results of this section show that Dempster's rule cannot be de ned in terms of support functions only. 2. THE MATHEMATICAL MODEL OF HINTS The frame of discernment is now an arbitrary set and in particular it can be in nite. The set of possible interpretations can then also be REPRESENTATION OF EVIDENCE BY HINTS 5 arbitrary. However, will be a probability space ( ;A; P ) with a -algebra A and a probability measure P on it. As before (section 1) any possible interpretation ! 2 restricts the possible answers in somehow. It will be assumed here that to any ! 2 a family S(!) of implied propositions H , satisfying conditions (1) to (3) of section 1, is assigned. A family of subsets satisfying conditions (1) to (3) of section 1 is called a lter. The family P(!) = fH : Hc 62 S(!)g of possible propositions satis es conditions (1') to (3') of section 1 above. A pair of such dual families R = (S;P) will be called a restriction. A restriction R is called vacuous, if S contains only (and P all subsets of except the empty set). A vacuous restriction does not restrict at all the possible answers. It is used to represent the situation that, under some interpretations, a hint contains possibly no information at all concerning the question considered. The set R = \fH : H 2 Sg is called the base of the restrictionR. One might wonder, whether S should not be closed under arbitrary intersections and thus R 2 S. This will not be assumed here | for reasons which become clear later. However, a restriction R with R 2 S will be called set-based, because in this case S = fH : R Hg and P = fH : R\H 6= ;g. For a set-based restriction we write R = R. Similarly, if (2) and (2') of section 1 hold for countable families, the restriction will be called a restriction. To go back to the model of a hint, it will thus be assumed, that every possible interpretation ! 2 has assigned a nonempty restriction (!) = (S(!);P(!)) describing its implied and possible propositions. is a mapping from into the set R( ) of restrictions on . This is a generalization of the multivalued mappings considered by A. Dempster (1967). A hint H is thus nally a quintuple H = ( ;A; P; ; ) of elements as described above. A hint H = ( ;A; P; ; ) is called set-focussed, i its restrictions (!) are set-based for all ! 2 . The bases of (!) are then called focal sets. If is a nite set, then all restrictions and thus all hints are setbased. But even in the general case many important classes of hints are set-focussed. For all ! 2 , if (!) is either a xed set-based restriction R or the vacuous restriction, then the hint is called simple. If (!) equals the vacuous restriction for all ! 2 , then the hint is called vacuous; it represents full ignorance about the question at hand. If H is a set-focussed hint whose focal sets (!) all contain only one single point (!) of , then the hint is called precise. A precise hint corresponds essentially to a random variable (under reserve of the appropriate measurability condition). 6 J. KOHLAS, P.A. MONNEY Restrictions are fundamental to the theory. In many respects they behave like ordinary subsets of . Especially the operation of intersection or conjunction can be de ned: If R1 and R2 are two restrictions known to hold on , then their conjunction forms a new restriction R = R1 \R2 de ned by S = fH1 \H2 : H1 2 S1; H2 2 S2g. It is easily veri ed, that S is a lter if ; does not belong to S. If ; 2 S, then R1 and R2 are called contradictory. If R1 = R1 and R2 = R2, then R1\R2 = R1\R2. In the same way, the intersection is de ned for arbitrary families of restrictions, not only for nite ones. In order to judge hypotheses H in the light of a hint H, the subset u(H) of interpretations which imply H and the subset v(H) of interpretations under which H is possible are de ned as in (1). u and v are mappings from the power set P( ) to the power set P( ). The following theorem lists some of their elementary properties: Theorem 1. (1) u(;) = v(;) = ; (2) u( ) = v( ) = (3) u(H) = v(Hc)c (4) v(H) = u(Hc)c (5) u(\fHi : i 2 Cg) = \fu(Hi) : i 2 Cg, where C is nite in general, countable for hints with -restrictions (!), ! 2 , and arbitrary for set-focussed hints. (6) u([fHi : i 2 Cg) [fu(Hi) : i 2 Cg for an arbitrary C. (7) v([fHi : i 2 Cg) = [fv(Hi) : i 2 Cg, where C is nite in general, countable for hints with -restrictions (!), ! 2 , and arbitrary for set-focussed hints. (8) v(\fHi : i 2 Cg) \fv(Hi) : i 2 Cg for an arbitrary C. (9) u(H 0) u(H 00) if H 0 H 00. (10) v(H 0) v(H 00) if H 0 H 00. Proof. (1) and (2) are trivial. By de nition, v(H)c = f! 2 : Hc 2 S(!)g = u(Hc) and (4) is proved. (3) follows by applying (4) to Hc. (5): If ! 2 u(Hi) for all i 2 C, then Hi 2 S(!), thus \fHi : i 2 Cg 2 S(!) and therefore ! 2 u(\fHi : i 2 Cg). Inversely, ! 2 u(\fHi : i 2 Cg) implies \fHi : i 2 Cg 2 S(!), hence Hi 2 S(!) and ! 2 u(Hi) for all i 2 C. (6): If ! 2 u(Hi) for some i 2 C, then Hi 2 S(!), thus [fHi : i 2 Cg 2 S(!) and ! 2 u([fHi : i 2 Cg). (7) and (8) are proved using (3),(4),(5) and (6) together with de Morgan laws. (9) and (10) follow immediately from the de nitions of u and v. Q.E.D. REPRESENTATION OF EVIDENCE BY HINTS 7 In view of (5) u is called a \ homomorphism and in view of (7) v is called a [ homomorphism. 3. SUPPORT AND PLAUSIBILITY FUNCTIONS For a hint H = ( ;A; P; ; ) the degree of support sp(H) and the degree of plausibility pl(H) are de ned by (2) for any subset H of for which u(H) 2 A and v(H) 2 A respectively. Let Es be the class of all subsets H of for which u(H) 2 A, i.e. for which the degree of support is de ned. The sets of Es are called s-measurable and Es is the domain of the set-function sp. Similarly let Ep be the class of all subsets H for which v(H) 2 A, i.e. for which the degree of plausibility is de ned. The sets of Ep are called p-measurable and Ep is the domain of the set-function pl. Note that there is a strong link between the supportand the plausibility function. In fact, according to theorem 1 (4) and (3) pl(H) = P (v(H)) = P (u(Hc)c) = 1 sp(Hc) sp(H) = P (u(H)) = P (v(Hc)c) = 1 pl(Hc) (3) whenever the corresponding probabilities are de ned. Theorem 2. (1) Es is a multiplicative class (i.e. closed under nite intersections) or a -multiplicative class (closed under countable intersections) depending on whether (!); ! 2 are general restrictions or -restrictions. (2) Ep is an additive class (i.e. closed under nite unions) or a -additive class (closed under countable unions) depending on whether (!); ! 2 are general restrictions or -restrictions. (3) Ep = fH : Hc 2 Esg, Es = fH : Hc 2 Epg and ;; belong to both Es and Ep: Proof. (1) and (2) are direct consequences of theorem 1 (5) and (7) and the fact that A is a -algebra. (3): H 2 Es is equivalent to u(H) 2 A, which is equivalent to v(Hc) 2 A (theorem 1 (3) and (4)) which nally is equivalent to Hc 2 Ep. ;; belong to Es and Ep because of theorem 1 (1) and (2). Q.E.D. 8 J. KOHLAS, P.A. MONNEY Es and Ep are called dual classes of sand p-measurable sets. If is a nite set, then all subsets of are sand p-measurable. However, in general Es and Ep are strict subclasses of the power set of . Let's illustrate theorem 2 by a simple, albeit somewhat pathological example: If ( ;A; P ) is a probability space and B a subset which does not belong to A; (!) = F for all ! 2 B; (!) = otherwise, then Es contains all subsets of which do not contain F plus the set . We have u(H) = ; for all H 2 Es; H 6= and thus sp(H) = 0, unless H = . Ep contains all subsets of which are not contained in F c plus ;. Theorem 3. The supportand plausibility functionns of a hint H = ( ;A; P; ; ), sp : Es ! [0; 1] and pl : Ep ! [0; 1] respectively, satisfy the following conditions: (1) sp(;) = pl(;) = 0 and sp( ) = pl( ) = 1: (2) sp is monotone of order 1, i.e. sp(E) Xf( 1)jIj+1sp(\i2IEi) : ; 6 = I f1; . . . ; ngg (4) for all n 1 and sets E;Ei 2 Es, such that E Ei; and pl is alternating of order 1, i.e. pl(E) Xf( 1)jIj+1pl([i2IEi) : ; 6 = I f1; . . . ; ngg (5) for all n 1 and sets E;Ei 2 Ep, such that E Ei: Furthermore, if all (!); ! 2 are -restrictions, then the following conditions hold: (3) sp and pl are continuous, i.e. if E1 E2 . . . is a monotone decreasing sequence of sets of Es, then sp(\1i=1Ei) = lim i!1 sp(Ei) (6) and if E1 E2 . . . is a monotone increasing sequence of sets of Ep, then pl([1i=1Ei) = lim i!1 pl(Ei): (7) REPRESENTATION OF EVIDENCE BY HINTS 9 Proof. (1) follows from theorem 1 (1) and (2). In order to prove (2) for the support function, the well-known inclusion-exclusion formula of probability theory, together with theorem 1 (5), (6) and (9) is used: sp(E) = P (u(E)) P (u([fEi : i = 1; 2; . . . ; ng)) P ([fu(Ei) : i = 1; 2; . . . ; ng) =Xf( 1)jIj+1P (\i2Iu(Ei)) : ; 6 = I f1; . . . ; ngg =Xf( 1)jIj+1P (u(\i2IEi)) : ; 6 = I f1; . . . ; ngg =Xf( 1)jIj+1sp(\i2IEi) : ; 6 = I f1; . . . ; ngg: Condition (2) for the plausibility function is proved in the same way or by using (4) together with (3). E1 E2 . . . implies u(E1) u(E2) . . . (theorem 1 (9)) and \1i=1Ei 2 Es (theorem 2 (1)). By the continuity of probabilities and theorem 1 (5) sp(\1i=1Ei) = P (u(\1i=1Ei)) = P (\1i=1u(Ei)) = lim i!1P (u(Ei)) = lim i!1 sp(Ei) and condition (3) is proved. Q.E.D. Note that in particular set-focussed hints have continuous supportand plausibility functions. Does it make sense to de ne the degree of support for a hypothesis H outside the class Es of s-measurable subsets? If u(H) is not measurable, the model of the hint H does not contain the necessary information to determine the probability of the set of interpretations supporting H. But any measurable set of interpetations A which is contained in u(H) is a support for H. Hence one may say that the unknown support for H must be at least P (A), for any A u(H) and A 2 A. Thus, in the absence of further information the support of H could be de ned as spe(H) = sup fP (A) : A u(H); A 2 Ag = P (u(H)) (8) 10 J. KOHLAS, P.A. MONNEY where P is the inner probability to P . This is an extension of the support function sp onto the whole power set P( ) because the restriction of spe to Es equals sp. We call spe the vacuous extension of sp to underline that no information not contained in the hint ( ;A; P; ; ) has been added. By duality, we may also extend the plausibility functions pl from Ep to P( ): ple(H) = 1 spe(Hc): (9) This is similarly called the vacuous extension of pl. This name is justi ed by the following proposition: Theorem 4. The equality ple(H) = inf fP (A) : A v(H); A 2 Ag = P (v(H)) (10) holds. P is the outer probability to P . Proof. From the de nitions (8) and (9) and theorem 1 (4) it follows that ple(H) = 1 spe(Hc) = 1 sup fP (A) : A 2 A; A u(Hc)g = 1 sup fP (A) : A 2 A; u(Hc)c Acg = inf fP (Ac) : A 2 A; v(H) Acg = inf fP (A) : A 2 A; v(H) Ag: Q.E.D. Furthermore, it turns out that spe and ple satisfy also the conditions of theorem 3. Theorem 5. Let spe and ple be the extended supportand plausibility functions of a hint H = ( ;A; P; ; ). Then (1) spe and ple are monotone and alternating of order1 respectively on P( ). (2) If (!) is a -restriction for all !, then spe and ple are also continuous. REPRESENTATION OF EVIDENCE BY HINTS 11 The proof of this theorem will not be given here. It seems to be surprisingly di cult and relies on the notion of an allocation of probability (Shafer, 1979). See Kohlas (1990) for a proof of this theorem. The connection between inner probability measures and support or belief functions have also been noted by Ruspini (1987) and Fagin, Halpern (1989), see also Shafer (1990). 4. REFINING HINTS A hint H = ( ;A; P; ; ) can be re ned in several respects by adding supplementary information to it: (1) The restrictions (!) associated with the interpretations ! may become more precise: A restrictions (S 0;P 0) is said to be more precise than (or included in) a restriction (S;P) i S 0 S (or equivalently P 0 P), i.e. if it implies more propositions and if less propositions are possible. We write then (S 0;P 0) (S;P). (2) Some interpretations which originally are considered as possible may become known as impossible: The new set of possible interpretations 0 becomes a subset of . This implies also that the original probability P must be conditionned on 0. This leads to a new probability space ( 0;A0; P 0) of possible interpretations, where A0 = A \ 0 and P 0(A) = P (A \ 0)=P ( 0), provided that P ( 0) > 0. Note that 0 is not necessarily measurable; P 0 is still a probability measure on A0 (Neveu, 1964). (3) The probability measure P 0 on the set of possible interpretations 0 may be extended from the -algebra A0 to a probability measure P 00 on a larger -algebra A00 containing A0. Let's note that in this case P 0 (A) P 00(A) P 0 (A) (11) for all A 2 A00. Thus, combining all three re ning steps in the above sequence, a new, re ned hint H00 = ( 00;A00; P 00; 00; ) may be obtained, such that 00 ;A00 A \ 00; P 00 is an extension to A00 of the probability measure P 0(A) = P (A \ 00)=P ( 00) on A\ 00 and 00(!) (!) for all ! 2 00. In this case we write H00 H and say that H00 is included in or is ner than H (and H is coarser then H00). Of course, many times not all three re ning steps are present; in particular often only step (1) or steps (1) and 12 J. KOHLAS, P.A. MONNEY (3) are considered. These particular cases correspond to Yager's (1985) de nition of inclusion. This notion of inclusion of hints leads to the following comparison of the corresponding supportand plausibility functions: Theorem 6. Let H00 = ( 00;A00; P 00; 00; ) and H = ( ;A; P; ; ) be two hints such that H00 H and with sp00 e ; pl00 e and spe; ple as their respective extended supportand plausibility functions. If k = P ( 00), then (1) spe(H) k sp00 e (H) + (1 k) for all H (2) ple(H) k pl00 e (H) for all H : Proof. Let v00(H) and v(H) be the subsets of interpretations of 00 and respectively under which H is possible. Then clearly v00(H) v(H)\ 00 by the re ning step (1). Now, for any H , ple(H) = P (v(H)) P (v(H) \ 00) P (v00(H)) = P (v00(H) \ 00): Let P 0(A) = P (A \ 00)=P ( 00) for A 2 A \ 00 and P 0 (A) denote the outer probability measure with respect to P 0. Then it follows easily that P 0 (v00(H) \ 00) = P (v00(H) \ 00)=P ( 00) and hence ple(H) P 0 (v00(H) \ 00)P ( 00): If P 00 (A) is the outer measure with respect to the probability measure P 00 on A00, then clearly P 0 (A) P 00 (A) for any A 00. Thus ple(H) P 00 (v00(H) \ 00)P ( 00) = P 00 (v00(H))P ( 00) = pl00 e (H)P ( 00) = k pl00 e (H): This proves (2). By (9) we have spe(H) = 1 ple(Hc) 1 k pl00 e (Hc) = 1 k (1 sp00 e (H)) = k sp00 e (H) + (1 k): REPRESENTATION OF EVIDENCE BY HINTS 13 This proves (1). Q.E.D. If only re ning steps (1) and possibly (3) are present, then k = 1 and [sp00 e (H); pl00 e (H)] [spe(H); ple(H)]: To any hint H = ( ;A; P; ; ) a vacuous hint V = ( ;A; P; vac; ) can be associated, where vac(!) is the vacuous restriction for all !. Clearly (!) vac(!) for all ! and therefore we have always H V. 5. FAMILIES OF HINTS RELATED TO A SUPPORT FUNCTION A hint generates a support function sp on some multiplicative class Es. This function has the properties (1) and (2), possibly (3) as stated in theorem 3. If now sp is a function on a multiplicative class Es, satisfying conditions (1) and (2) of theorem 3, is there always a hint which generates this support function? The answer is a rmative. This is a consequence of an integral theorem of Choquet (1953) as was noted by Shafer (1979). But it can easily be seen that di erent hints may generate the same support function sp on Es, but with di erent extensions spe to P( ). In fact, let ( ;A; P ) be a probability space and let B1; B2 be two di erent non-measurable subsets of which have di erent inner probabilities. Furthermore, let be a frame of discernment and F a strict subset of . This allows to de ne two distinct hints Hi = ( ;A; P; i; ); i = 1; 2, where i(!) = nF if ! 2 Bi otherwise. For both hints, the class Es equals all subsets of which do not contain F plus the set and the support functions of H1 and H2 coincide. But if sp1e and sp2e denote their respective extended support functions, then sp1e(F ) = P (B1) 6= P (B2) = sp2e(F ): Thus there exists a whole family of hints related to a support function sp on Es. The goal of this section is to study this family of hints. In a similar vain, Shafer (1979) studied various extensions of support (or belief) functions. This section puts some of his results into the perspective of hints. In the context of the theory of hints Choquet's theorem can be stated as follows: 14 J. KOHLAS, P.A. MONNEY Theorem 7. Let Es be a multiplicative class and sp : Es ! [0; 1] a function satisfying conditions (1) and (2) of theorem 3. Then there exists a hint whose support function is sp. If furthermore Es is a -multiplicative class and sp satis es condition (3) of theorem 3 (continuity), then there exists a hint whose restrictions are all -restrictions and whose support function is sp. For a formal proof we refer to Choquet (1953) (see also Shafer, 1978 and Kohlas, 1990). Let's only describe the hint constructed in this proof: As set of possible interpretations the set R(Es) of all lters on the multiplicative class Es is selected. Note that to any restriction R = (S;P) in R( ) can be associated a lter '(R) = S \ Es on Es. The maping ' from R( ) to R(Es) is onto because for any lter F 2 R(Es) the restriction Rc(F) 2 R( ) de ned by its class of implied propositions S = fH : there is an E 2 F such that E Hg is in ' 1(F). This shows that f' 1(F) : F 2 R(Es)g is a partition of R( ). Moreover, Rc(F) is the coarsest restriction in ' 1(F): if R0 2 ' 1(F), then R0 Rc(F). De ne 00(F) = Rc(F) for any F 2 R(Es). Then there is according to Choquet (1953) a -algebra A00 in R(Es) and a probability measure P 00 de ned on it such that the hint (R(Es);A00; P 00; 00; ) has sp as support function. Note that using ' the probability space (R(Es);A00; P 00) induces a probability space (R( );A0; P 0). If we de ne c(R) = Rc('(R)), then the hint (R( );A0; P 0; c; ) generates clearly also the support function sp on Es. Let uc(H); vc(H) be the functions (1) de ned with respect to c and let uc(Es); vc(Ep) (where Ep is the dual class to Es) be the images of Es and Ep with respect to uc and vc respectively. By theorem 1 (5) and (7), uc(Es) is a multiplicative class and vc(Ep) an additive class. Both uc(Es) and vc(Ep) are contained in A0. Now, let Ac be the smallest -algebra containing uc(Es) and vc(Ep); Ac is a subalgebra of A0. Let nally Pc be the restriction of P 0 to Ac. Then the hint Hc = (R( );Ac; Pc; c; ) still has sp on Es as support function. This hint is called the canonical hint of the support function sp on Es. We shall see that Hc is in some sense the coarsest hint which generates sp on Es: among all hints generating sp, it contains the least information. This will be formulated more precisely using the inclusion relation between hints introduced in the previous section. Thus, let H = ( ;A; P; ; ) be any hint, which de nes the support function sp on Es. More precisely, suppose that the class of s-measurable sets of H contains Es and that on Es its support function equals sp. Hints which de ne in this sense identical support functions on Es are called equivalent. In order to compare equivalent hints among themselves and REPRESENTATION OF EVIDENCE BY HINTS 15 in particular with the canonical hint, they must be represented with respect to an identical set of possible interpretations. By the mapping , the -algebra A and the probability measure P can be transported to the set R( ) in the usual way: Consider the -algebra A0 of all subsets B R( ) for which 1(B) 2 A and de ne a probability P 0 on A0 by P 0(B) = P ( 1(B)). This leads to an equivalent hint (R( );A0; P 0; id; ) where id stands for the identical mapping id(R) = R. This is called the canonical representation of H. In particular, note that this new hint de nes the same extended support function sp0e as H. In this sense H and its canonical representation Hcr contain exactly the same information. The following theorem states now that the canonical hint is the coarsest hint among all equivalent hints with respect to a support function sp on Es. Theorem 8. Let Hc be the canonical hint with respect to a support function sp on a multiplicative class Es. If H is any equivalent hint with respect to this support function and Hcr its canonical representation, then Hcr Hc. Proof. Both Hcr and Hc have the same set of possible interpretations R( ). Moreover, clearly id(R) Rc('(R)); A0 Ac and the restriction of P 0 to Ac equals Pc. Q.E.D. As a consequence of this theorem, it follows that [spe(H); ple(H)] [spce(H); plce(H)] for all H , if spce; plce denote the extended support and plausibility functions of the canonical hint and spe; ple the extended support and plausibility functions of the hint H. Shafer (1979) studied extensions of support functions and identi ed among others the minimal extension of a support function sp on Es. It turns out that this minimal extension is in fact as one expects the extension of the canonical hint with respect to sp on Es. Theorem 9. If spce; plce are the extended support and plausibility functions of the canonical hint Hc with respect to a support and plausibility function sp and pl on a multiplicative class Es and its dual additive class Ep, then spce(H) = sup fX f( 1)jIj+1sp(\i2IEi) : ; 6 = I f1; . . . ; ngg : Ei H;Ei 2 Es; i = 1; . . . ; n;n = 1; 2; . . .g; (12) 16 J. KOHLAS, P.A. MONNEY plce(H) = inf fX f( 1)jIj+1pl([i2IEi) : ; 6 = I f1; . . . ; ngg : Ei H;Ei 2 Ep; i = 1; . . . ; n;n = 1; 2; . . .g; (13) Proof. Note that by theorem 1 (6) [ni=1uc(Ei) uc([ni=1Ei). Furthermore spce(H) = Pc (uc(H)) sup fPc([ni=1uc(Ei)) : Ei H;Ei 2 Es; i = 1; . . . ; n;n = 1; 2; . . .g = sup fX f( 1)jIj+1Pc(\i2Iuc(Ei)) : ; 6 = I f1; . . . ; ngg : Ei H;Ei 2 Es; i = 1; . . . ; n;n = 1; 2; . . .g = sup fX f( 1)jIj+1sp(\i2IEi) : ; 6 = I f1; . . . ; ngg : Ei H;Ei 2 Es; i = 1; . . . ; n;n = 1; 2; . . .g: On the other hand, Shafer (1979) proves that the right hand side of (12) de nes indeed a support function spm on the power set P( ) satisfying the conditions of theorem 7. There exists therefore a hint H0 which generates this support function and letH0 cr its canonical representation. But theorem 8 implies that H0 cr Hc and by theorem 6 spce(H) spcre(H) = spm(H) since k = 1. Thus we obtain nally spce(H) = spm(H) which proves (12). (13) is deduced from (12) using (3) and theorem 1 (3) and (4) together with the de Morgan laws. Q.E.D. Theorem 9 together with theorems 6 and 8 show that spm is the smallest support function which extends sp from Es to all of P( ). If the support function sp on a -multiplicative class Es is continuous (satis es condition (3) of theorem 3), then a canonical hint associated to this support function can be constructed in a similar way with respect to the set of -restrictions R ( ) on . For any hint for which all restrictions are -restrictions, a canonical representation with respect to R ( ) can be de ned along similar lines as above. Then two further results corresponding to theorems 8 and 9 can be proved: Theorem 10. Let Hc be the canonical hint with respect to a continuous support function sp on a -multiplicative class Es. If H is any REPRESENTATION OF EVIDENCE BY HINTS 17 equivalent hint with respect to this support function and Hcr its canonical representation, then Hcr Hc. Theorem 11. If spce; plce are the extended support and plausibility functions of the canonical hint Hc with respect to continuous support and plausibility functions sp and pl on a -algebra Es = Ep, then spce(H) = sup f lim i!1 sp(Ei) : E1 E2 . . . ; Ei 2 Es;\Ei Hg (14) plce(H) = inf f lim i!1 pl(Ei) : E1 E2 . . . ; Ei 2 Ep;[Ei Hg: (15) These theorems will not be proved here. The proofs develop along similar lines as those of theorems 8 and 9. Note that for theorem 11 Shafer (1979) showed that the right hand side of (14) is indeed a continuous support function. This theorem shows that it is the smallest continuous support function which extends the continuous support function sp from Es to P( ). 6. COMBINING HINTS Let H1 and H2 be two hints relative to the same frame and de ned by ( 1;A1; P1; 1; ) and ( 2;A2; P2; 2; ). The basic idea for the combination of these hints into a combined body of information is that in each hint there must be exactly one correct interpretation !i; i = 1; 2 such that | looking at both hints together | !1 and !2 must be simultaneously correct interpretations. Hence (!1; !2) must be the correct combined correct interpretation. Therefore, in order to combine the two hints H1 and H2 into one new combined hint, we form rst the product space of the combined interpretations from the two hints ( 1 2;A1 A2; P 0) where P 0 is any probability measure on A1 A2 re ecting the common likelihood of combined interpretations. The two hints are called independent, if the interpretations of the two hints are stochastically independent. Then P 0 is the product measure of P1 and P2. This is the case which will be pursued here although other cases would be equally possible. If the combined interpretation (!1; !2) is the correct one, then the restriction (!1; !2) = 1(!1) \ 2(!2) (16) 18 J. KOHLAS, P.A. MONNEY must necessarily hold. Note that it is possible that 1(!1) and 2(!2) are contradictory. Then !1 and !2 are called contradictory interpretations. De ne now u0(H) = f(!1; !2) 2 1 2 : H is implied by (!1; !2)g v0(H) = f(!1; !2) 2 1 2 : H is possible under (!1; !2)g: (17) Theorem 1 | except (1) and (2) | clearly applies to u0 and v0; (1) is replaced by v0(;) = ; and (2) by u0( ) = 1 2. u0(;) represents the set of contradictory interpretation pairs. Such a pair can never be the correct one because contradictions are not possible. Therefore contradictory interpretations must be eliminated and the probability must be conditioned on the event that there is no contradiction. Provided that u0(;) is measurable, i.e. u0(;) 2 A1 A2 and P 0(u0(;)) < 1, the new combined hint H1 H2 = ( ;A; P; ; ) can be formed, where = u0(;)c = v0( ); A = u0(;)c \ A1 A2; P (A) = P 0(A)=P 0(u0(;)c) and is de ned by (16) (and restricted to ). This way to combine hints is called Dempster's rule (A. Dempster (1967)). Let u and v be de ned by (1) relative to the hint H1 H2. Then u(H) = u0(H) \ = u0(H) u0(;) and v(H) = v0(H). Dempster's rule may be extended even to the case where u0(;) is not measurable. In this case the conditional probability space ( ;A; P ) can be considered, where ( ;A) is de ned as above and P (A) = P 0 (A \ u0(;)c)=P 0 (u0(;)c), provided that P 0 (u0(;)c) > 0. This leads to the combined hint H1 H2 = ( ;A; P; ; ). As before, we have u(H) = u0(H) \ = u0(H) u0(;) and v(H) = v0(H). Let Es and Ep be the classes of sand p-measurable sets relative to the hintH1 H2. Denote by E 0 s and E 0 p the classes of sets H such that u0(H) and v0(H) are measurable with respect to A1 A2. From u0(H) 2 A1 A2 it follows that u(H) 2 \ A1 A2 and thus E 0 s Es. Similarly, because v0(H) ; v0(H) 2 A1 A2 implies v(H) 2 \ A1 A2 or E 0 p Ep. If u0(;) is measurable, then E 0 s = Es and E 0 p = Ep. The next theorem states that inclusion of hints is maintained under Dempster's rule: REPRESENTATION OF EVIDENCE BY HINTS 19 Theorem 12. Let H1;H2;H00 1 ;H00 2 be four hints such that H00 1 H1 and H00 2 H2. Then H00 1 H00 2 H1 H2. Proof. 00 1(!1) 1(!1) and 00 2(!2) 2(!2) imply 00 1(!1)\ 00 2(!2) 1(!1) \ 2(!2). This, together with 00 1 1 and 00 2 2 implies 00 . Also A00 1 A1 \ 00 1 and A00 2 A2 \ 00 2 imply that A00 = A00 1 A00 2 \ 00 (A1 \ 00 1) (A2 \ 00 2) \ 00 = (A1 A2) \ ( 00 1 00 2) \ 00 = A1 A2 \ 00 = (A1 A2 \ ) \ 00 = A \ 00: It remains to show that P 00(A) = P (A)=P ( 00) for any A 2 A \ 00. Let Q00; Q denote the product measures of P 00 1 and P 00 2 and P1 and P2 on the product spaces ( 00 1 00 2 ;A00 1 A00 2) and ( 1 2;A1 A2) respectively. Then by de nition P 00(A) = Q00 (A)=Q00 ( 00) for any A 2 A \ 00. It is thus su cient to show that Q00 (A) = k P (A) for some constant k independent of A. To begin with, let's suppose that the sets 00 1 ; 00 2 ; 00 and are measurable with respect to A1;A2;A and A1 A2 respectively. Then P 00(A) = Q00(A)=Q00( 00) and P (A)=P ( 00) = P (A)=P ( 00) for A 2 A\ 00 and we must prove that Q (A) = k P (A). Let XA denote the indicator function of A. Then Q00(A) = Z P 00 1 (d!1)P 00 2 (d!2)XA: Because XA is a measurable function with respect to A, it is su cient to take the restrictions of the probability measures P 00 1 and P 00 2 to A1 and A2. But there these probabilities are conditional probabilities such that Q00(A) = Z P1(d!1)P2(d!2)XA=P1( 00 1)P2( 00 2) = Q(A)=P1( 00 1)P2( 00 2) = P (A)(Q( )=P1( 00 1)P2( 00 2)): 20J. KOHLAS, P.A. MONNEYThis proves the theorem in the case of measurable sets 001 ; 002 ; 00 and. If is not measurable, then there exists a measurable set , containing, such that Q ( ) = Q( ). If A 2 A\ , then A = A \ is measurable,contains A, and Q (A) = Q( A).Thus P ( A) = P (A) for all A 2 A\ and may be replaced by andA \ by A \ without changing the relevant probability values. In thisway the case where some or all sets 001 ; 002 ; 00 and are not measurablecan be reduced to the former case. This proves the theorem. Q.E.D.In the case of theorem 12, the constant k appearing in theorem 6 equalsP ( 00), where 00 contains all combined interpretations (!1; !2) which arenot contradictory under H001 H002 . Some combined interpretations, whichare not contradictory under H1 H2 may however be contradictory underH001 H002 . This accounts for the possible di erence between and 00.If the situation is such that 00 = , then k = 1 and [sp00e (H); pl00e (H)][spe(H); ple(H)].Let V be the vacuous hint associated with H2. Then theorem 12implies that H1 H2 H1 V. Similarily H1 H2 V H2. 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