284

Conditional Independence in Uncertainty Theories

Prakash P. Shenoy School of Business University of Kansas Summerfield Hall Lawrence, KS 66045-2003USA [email protected]

Abstract Anabstract framework that unifies variousuncertainty calculi is that of valuation-based systems [Shenoy 1989, Thispaper introduces the notions of independence 199Ia]. In VBS, knowledge about a setof variables is rep­ and conditionalindependence in valuation-based resentedby a valuation for that set of variables. Thereare systems (VBS). VBS is an axiomatic framework threeoperators in VBS that areused to makeinferences. capable of representingmany differentuncertainty These are called combination,marginalization, and re­ calculi.We definein dependenceand conditional moval. Combination representsaggregation of knowledge. independencein terms of factorization of the joint Marginalizationrepresents coarseningof knowledge. And valuation.The definitions of independence and removal represents disaggregation of knowledge. conditional independencein VBS generalizethe The framework of VBS is able to uniformly represent corresponding defmitions in probability theory. probability theory, Dempster-Shafer's belief-function the­ Our definitions apply not only to probability ory, Spohn'sepistemic-belie ftheory,and Zadeh'spossi­ theory, but also to Dempster-Shafer's belief-func­ bility theory. In this paper, we will develop thenotion of tion theory, Spohn's epistemic-belief theory, and independenceand conditional independence for variables in Zadeh's possibility theory. In fact, they apply to the framework of VBS. One advantage of this generalityis any uncertaintycalculi that fitin the framework that all resultsdeveloped here will apply uniformly to of valuation-based systems. all uncertaintycalculi that fit in the framework of VBS. Thus the results describedin this paper apply to, for example, 1 INTRODUCTION probabilitytheory, Dempster-Shafer's belief-function the­ The concept of conditional independencebetween two sub­ ory, Spohn's epistemic-belief theory,and Zadeh's possi­ sets of variables given a third has been extensively studied bility theory. in probability theory [Dawid 1979, Spohn 1980, What does it meanfor two disjoint subsetsof variables to Lauritzen 1989, Pearl1988, Smith 1989, Geiger 1990]. bein dependent? Intuitively,independence can definedbe in Theconcept of conditional independencein probability terms of factorization of the joint valuation. If t is a valu­ theory has been interpreted interms of relevance. If r, s ation for rus,then we say thatr and s are independent and t are disjoint subsets of variables, then to say that r with respectto t iff t factors into two valuations, one and s conditionallyindependent given means that the are t, whose domainonly involves r, and the other whose do­ conditional distributionof r, given values of sand t, are main only involves s. One implication of this is that if governed by the value of t alone-further information we are interestedin constructinga valuation for rvs, then about the value of s is irrelevant. independence of r and s allows us to constructthis valua­ The concept of conditionalindependence forvariables has tion by, frrst, constructing two valuations-one whose also been studied in Spohn's theory of epistemic beliefs domain involving only r and the other whose domain in­ [Spohn 1988, Hunter 1991]. However, the concept of in­ volving only s-and second, by simply combining the dependence for variables has not been studied inDempster­ two valuations to get the result. Shafer' s theory of belief functions [Dempster 1967, Shafer What does it mean for two disjoint subsets of variables to 1976] or in Zadeh's possibility theory [Zadeh 1979, beconditionally independentgiven a third disjoint subset? Dubois and Prade 1988].1 Conditional independencecan alsobe describedin terms of factorization of the joint valuation. Suppose't is a valua­ tion for rusul We say rand s are conditionally indepen­ 1 Dempster [1967], Shafer [1976, 1982, 1984, 1987, 1990], dent given t with tot iff factors and Smets [ 986] have defmed independence for belief respect the valuation t 1 into whosedomain functions, but not for variables on which belief functions are two valuations, one involves variables defined. Shafer [1976] has defmed independence for frames of in rut, and the other whose domain involves only vari­ discenunent, a concept further studied by Shafer, Shenoy and ables in sut. Mellouli [1987]. Belief functions in belief-function theory are analogs of probability functions in probability theory. Conditional Independence in Uncertainty Theories 285

bility theory,for example, a valuation for s is a function An outline of this paperis as follows. In section 2, we s�IR.where is the set of all numbers. describethe frameworlc of valuation-basedsystems (VBS). cr:'UI' R real TheVBS frameworkwas describedearlier in [Shenoy Zero Valuations For each s�X. there is at most one 1989, 1991a]. Herewe extendthe frameworkby defining valuation �sE '\J5 called the zero valuation for s. Let Z three new classes of valuations callednormal, proper nor­ denote { �s I S!;;;X}, the setof all zero valuations. Note mal, and positiveproper normal. Also,we introduce a that we arenot assuming zero valuations always exist. If new operatorcall ed removal,and defme some new axioms zero valuations do notexist, Z = 0. We call valuationsin for the removal operator. Shenoy [1991b]shows how '\J- Z nonzero valuations. Intuitively, a zero valuation probability Dempster-Shafer'stheory, belief-function the­ representsknowledge thatis internallyinconsistent. In ory, Spohn's epistemic-belief theory, and Zadeh's possi­ probability theory,for example, a zero valuation for sis bility theory fitin the framework of VBS. the valuation �s such that�5(x) = 0 for all xe 'UI' 5• In section3, we defmeindependence and conditional inde­ Proper Valuations For each subset s of$, there is a pendence for sets of variables. We show that thesedefmi­ subset P5 of V 5-{�5). We call the elementsof P5proper tions satisfy some well known propenies that have been valuations for s. Let P denote u{ P s I s�X ), the set of stated by Dawid [1979], Spohn [1980], Lauritzen [1989], all proper valuations. Intuitively, a propervaluation repre­ Pearl [1988], and Smith [1989] in the contextof probabil­ sentsknowledge that is partially coherent In probability ity theory. Using Pearl's terminology, the conditional in­ theory, for example,a propervaluation is a nonzerovalua­ dependence relation inVBS is a graphoid. Finally, in sec­ tion cr such that cr(x) � 0 for all xe cur5• tion 4, we make some concluding remarks. Proofs of all Normal Valuations For h s�X, there is another resultscan be found in [Shenoy 1991b]. eac subset'JI. s of V 5-{ �s}. We call the elements of 'JI.s nor­ mal valuations for s. Let 'J1. denoteu { 'JI.5 I �$ ) , the set 2 VALUATION-BASED SYSTEMS of all normal valuations. Intuitively, a normalvaluation In this section, we describe the framework of valuation­ representsknowledge that is partially coherent in a sense based systems (VBS). In aVBS, we representknowledge different from propervaluations. In probability theory,for by entities calledvariables andvaluations. We infer inde­ example, a normalvaluation is a nonzero valuation cr pendence relations using three operators calledcombina­ such that :E[cr(x) I XE curs l = 1. tion, marginalization, andremoval. We use these opera­ Proper Normal Valuations For each s�$, let� 5 tors on valuations. denote P 8n'J\.8• We call the elementsof 1\5proper nor­ Theframework ofVBS is describedin [Shenoy 1989, mal valuations for s. Let 1\ denote u(l\5I s�X) , the set 1991a]. The motivation there was to describea localcom­ of all proper normal valuations.Intuitively, a propernor­ putational methodfor computing marginals of the joint mal valuation representsknowledge that is completely co­ valuation. In this paper, we embellishthe VBS framework herent. by introducing threenew classesof valuations called nor­ Positive Proper Normal Valuations For each mal, proper normal, and positive proper normal, and by s�$, there is a subset 1\s + of 1\5• We call theelements introducinga new operatorcalled removal. Our motivation here is todefine independence and describe itsproperties. of 1\5+positive proper normal valuations for s. Let 1\ + denote u( 1\s + I sr;;;;$), the set of all positive proper Mr­ Variables We assume there is a fmite set X whose ele­ mal valuations. As we will la r, positive propernor­ ments called variables.Variables willbe denoted by see te are mal valuationsare proper normal valuations that have upper-case letters, etc. Subsets of X will be de­ X, Y, Z, unique identities. In probability theory, for example, a noted by lower-case letters, r, s, t, etc. positive propernormal valuationfor s is a propernormal Valuations For each s�$, there is a set 'If5• We call valuation crsuch that cr(x)> 0 for all xe cur5• the elements of '\15 valuations for s. Let V denote , Figure 1 shows the relations betweenthe different typesof u { V 5 I s�X } the set of all valuations. If cr is a valua­ valuations. As our defmitions, Z�'lf, ('V- ), tion for s, then we say that sis the domain of cr. per P� Z Valuations will bedenoted by lower-case Greek alphabets, 'J\.QV -Z), 1\ = Pn'n.,and 1\+!:;1\. p, t, cr, etc. Combination We assume there is a mapping Valuations are primitives inour abstractframework and, (£);'\Jx'\1 � 'JI. uZ, called combination, such that the as such, require no definition. But as we shall see shortly, following axioms are satisfied: they are objectswhich can be combined, marginalized,and (Cl) (Domain) If p anda arevaluations for r and s, re­ removed. A valuation for s represents some knowledge spectively, then p€£lcr is a valuation for rus. about variables in s. (C2) (Associative) p€£l(cr€£lt)= (pea)et. In probability theory, with eachvariable X, we associate a (C3) (Commutative) pea= aep. set '\Ifx called the framefor X. Also for each �X, we as­ sociate the set '\If5 = x { '\Ifx I Xe s) called theframe for s. (C4) (Zero)Suppose zero valuations exist, supposecr is a Elements of '\If5 are called configurationsof s. In proba- valuation for s, and suppose pis a valuation for r. Then cr€£lCr= �s€£lP= �rus· 286 Shenoy

that for eachS!:: X. (f\.5+, ffi) is a commutative subsemigroup.

Identity Valuations We will assume that foreach s�$, and for each cre 11.8u { �sl. there exists at least one identity for it, i.e., there ex­ ists &,€11.su{�5) such that

(Ml) (Domain) If a is a valuation for s, then sa.l.( -{X}) is The following lemmastates some easy implications of a valuationfor s-(X). Axiom CM1.3 2 (M ) (Order of Deletion) Suppose a is a valuation for s, Lemma 2.1 Suppose Axioms Cl-C7, Ml-M6, and 1 hold. Then the following statements hold. and suppose E s. Then ( X ) CM X1, X2 (CJJ. s-{ l} )J.(s-{Xt,X2)) (i). If is a normal valuation for s, and r�s. then = cr!

2 Axiom M2 is equivalent to the "consonance of 4 The statement of Lemma 2.2 was fuststated as an axiom in marginalization" axiom in [Shenoy andShafer 1990], which [Shenoy and Shafer 1990]. Shenoy (199la] statedaxiom CM2, which is stronger than the statement of 2.2. is stated as follows: If a is a valuation for s, andq 1: r 1:s, Lerruna The added strengthof axiom CM2 has advantagesin the then (a..l..r)J.q::::a!q. computation of marginals-see Theorem 2.1. 2t\S Shenoy

describesthe fusion algorithm, a methodfor computing (ii). If cr is nonnal, then {(crEBp)®p)J.s"' cr. (crtEB... EBcrm)J..{X} using only local computations. (iii). ((crEBp)®p).l.sep = crEBp. [Shenoy 1991a] Suppose (0'1, ..• , Theorem 2.1 (iv). ((crEBp)®p)ep= csep. O"m) is a collectionof valuations such thatO"i is a valuation for Si· Suppose Axioms Cl, C2, C3, Ml, (v). If p is positive proper normal, then p®p = tr. M2, and CM2 hold. Let$ denote S}V ••• VSm. (vi). If pis positive proper normal, then (crEBp)®p = Suppose Xe$, and suppose XtX2···Xn�l is a se­ crEBtr. quence of variablesin ��{X). Then (vii). If cr is normal, and p is positive propernormal, cr EB . EBom)J.{X = { u { ... ( i .. ) E9 F sx n�l then (O"EBp)®p=cr.

Fusx2{Fusx1 {crt • ..., O'm)}} } . (viii). If cr is normal, and r�s. then(cr®cr.J.I)ecr.l.r = 0". Removal We assume there is a mapping ®:VxV � probability distribution. T make this specificationtask suppose pis a normal valuation for r. Then the fol­ simpler, we make assertionsof independence that are nec­ lowing statements hold. essarily subjective. However, once we have a specification of a joint probability distribution (obtained either objec- (i). ((crEBp)®p).l.s = crffi'-0. Conditional Independence in Uncertainty Theories 289

tively or subjectively), all further statements of indepen­ Defmition 3.2 is a generalization of Definition 3.1. Note denceare necessarily objective with respectto the joint thatr ..i s iff .l{ r, s}. We knowfrom probability theory probability distribution. that functions of independent random variablesare inde­ Lett be a proper nonnal valuation for �- We will hence­ pendenL If X 1 andX2 are independentrandom variables, forth assume thatt represents theglobal knowledge re­ then f(X1) andg(X2) are also ind ependentrandom vari­ garding all variables in the VBS. For example, in proba­ ables. More generally, if Xt • ... , Xn are independent,{Nt. bility theory,t would represent the joint probability dis­ ... , Nk} is a partition of theset {XJ, .•. , Xn}. and Yj is a tribution for all variables in �. function of theX i in Nj. then Y 1· ••• , Y k are independenL The following lemma makes an analogous statemenL6 Definition 3.1 (Independence)Suppose tis a propernormal valuation for$, and suppose r, s�$, Lemma 3.1Suppose tis a proper nonnal valuation rns = 0. We sayra nds are independelllwith respect for$, and suppose r1, ..., r0 are disjoint subsets of r •... , Nk} is to f, written as r ..i't s, ifft.l.(rus)= pEBo, where p $.Suppose ..i{q, ... , 0}.Suppose {Nt a partition of { 1, ... , n}, i.e., Nir.Nj= 0 if i * j, and and o are valuations for r and s, respectively. NtU... UNk= (1, ... , n}.Suppose Sj�(u{rjliENj)). When it is clearthat all independence statements arewith for j = 1, .•. , k. Then ..i(st .... , Sk}. respect tot, we will simply say'r and s are independent' The statement in the following corollary toLemma 3.2 is instead of 'r ands are independent withrespect to t,' and called decomposition [Pearl 1988]. It is a special case of use the simpler notation r ..i s instead of r � s. Lemma3.2. Theorem 3.1 (Symmetry) Suppose tis a proper Corollary (Decomposition)Suppose tis a proper normal valuation for$, and suppose r, s �$, rr.s= normal valuation for$, suppose r, s, tare disjoint 0. If r .l s, then s .l r. subsets of$, and suppose r ..i (sut) . Then r .l s. The following lemma gives alternative characterizations of The following lemma gives four alternative characteriza­ theindependence relation.5 tions of joint independence.7 Lemma 3.1Suppose tis a proper normal valuation Lemma 3.3Suppose t is a proper nonnal valuation for $, and suppose r, s � X, rr.s= 0. The following for$, and suppose ft, ... , r0 are disjoint subsets of statementsare equivalent $. Then the following statements areequivalent. (i). r .l s. (i) . .l(q, ... , rn} (ii). t.l.(rus)= where panda are valuations for pEBa, "" !(rl u ...ur n) m m . (n.) t h r r and s, respectively. = Ptw ... wpn. w e e Pi lS a v a1ua- tion for rj, i = 1, ... , n. us ! (r ) r . ur _ (iii). tJ.. = t et.!.s. ("'") .!.(rlu .. n) .l.rlm m J..rn 111 • t - t w ...wt

(iv). There exists an identity 5'tJ.r for tJ..r such that (iv) . .i{C}, ... , rn-tJ and (r}U ... Urn-1) .i rn. !( u t r s)®t.!.r = aEB5'tJ.r, where a is a valuation for s. (v). ri ..i u {rj I j= 1, ... , n, j :F- i} for i= 1, •.., n.

(vi). rj .l (Tt u ... Urj-1) for j= 2, •.. , n. r (v). There exists an identity 5'tJ.rfor t.!. such that Defmition 3.3 defmes conditional independence for two .!.(rus)® .!.r .!. t t = t se5'tJ.r . subsets given a third.

Defmition 3. 2 generalizes Defmition3.1 for any number Definition 3.3 (Conditional Independence) of subsets of variables. Suppose t is a proper nonnal valuation for�.and (Joint Independence) suppose r, s, and tare disjoint subsets of$. We say r Definition 3.2 Supposet is and s are conditionally independent given t with re- a proper normal valuation for$, and suppose f1, ... , sut spect to f, written as r ..i't s I t, ifft !(ru )= rn are disjoint subsets of X. We say rJ, ...• rn are O:rutEB where CXru

5 The statements of Lemma 3.1 are analogs of corresponding 6 An analogous statement is stated and proved in [Shafer, statements in [Dawid 1979] in the context of probability Shenoy, and Mellouli 1987] in the context of qualitative theory. Our contribution here is in showing that these independence. statements hold in our more general framework of VBS. Thus they hold not only in probability theory (as shown by Dawid 1 The statements in Lemma 3.3 are analogs of corresponding [1979]) but also in other uncertainty calculi that fit in the statements in Shafer, Shenoy and Mellouli [1987] in the framework of VBS. context of qualitative independence. 290 Shenoy

conttaction,and intersection a graphoid. From Theorems Lemma 3.4 Suppose tis a proper nonnal valuation 3.1-3.4 and the corollary to Lemma 3.2, it follows that for�. and supposer, s, andtar e disjointsubsets of thedefmition of conditional independence in Definition �. Thefollowing statements are equivalent. 3.3 is a graphoid. (i). r .l s I t.

vt) = CONCLUSION (ii). tJ.(rvs r:xrv1EDasvt• where arvt and asvt 4 are valuations forrut and sut, respectively. Themain objective of this paperis to defme independence rvsvt) J.t andconditional independence in theframework of (iii). tJ.( ®t E9 o where f3rv and = �vt i3svt t valuation-based systems. Although these conceptshave valuations for rut and sut, respectively. i3svt are beendefmed and extensively studiedin probabilitytheory, they have not been extensively studiedin non-probabilis­ (iv). tJ.(rvsvt) = tic uncertainty theories. tJ.lEB(t.l.(rvt)®t.l.l)ED(t,l..(svt)®tJ.� . Drawing upon the literature on independence in probabil­ ity theory [Dawid 1979, Spohn 1980, Lauritzen 1989, (v). t.!.(rvsvt)®tJ,t= Pearl1988, Smith 1989], we defme independence andcon­ (t.l.(rvt)®t.l.t) (t.l.(svt)®t.l.t). E9 ditional independence in VBS. Theframework of VBS was (vi). tJ.(rvsvt) = (tJ..(rvt)®tJ.l)EDt.l.(sut) defmed earlierby Shenoy [1989, 1991aJ. However, the VBS frameworkdefmed there isinadequate the for pur­ (vii). There exists an identitys't!.(IIVt) fort.l.( svt) such posesof studying properties of independence. In this pa­ per, we embellish the frameworkby includingthree new t t.!.(rvsvt)®t.l.(svt) = (t.!.(rvt)®t.l.nea tha -, 't"-(•ut)•, classes of valuations called proper, normal, and positive propernormal, and by including a new operatorcalled re­ sv ) (viii). There exists an identity S'tJ..(sut) for tJ..( t moval. The new defmitions are stated in the fonn of ax­

= ioms. Shenoy [199lb] shows that these axioms are gen­ such that tJ.(rvsvt)®tJ..(svt) arvtEDS't!.(sVl)· eralenough to include probability theory, Dempster­ Theorem 3.2 states another property of independence.This Shafer's belief-function theory,Spohn's epistemic belief property is called weak union [Pearl1988]. theory,and Zadeh's possibility theory. Theorem 3.2 (Weak Union) Suppose tis a proper Theframework of VBS as describedin this paper enables nonnal valuation for �. and suppose r, s, and tare us to defme independence andconditional independence, disjoint subsets of $. If r .l sut. then r l.. s It. and enables us to derive all major properties of conditional independence that have been derived in probability theory. Theorem 3.3 states another property of conditional inde­ Independence and conditiooal independence aredefmed in pendence.This property is called contraction [Pearl1988]. tenns of factorization of the joint valuation. Thus, not Theorem 3.3 (Contraction) Suppose tis a proper only dowe have a deeperunderstanding of independence in normalvaluation for$, and supposer, s and tare probability theory, we also understand what independence disjoint subsets of $. If r .l s, and r .l t1 s, then r l.. means in various non-probabilistic uncertainty theories. sut. This should deflect some criticism thatnon-probabilistic uncertaintytheories are not as well developed asprobabil­ The next theorem states a property of conditional indepen­ ity theory. dence thatho lds only ifthe joint valuation t is positive proper nonnal. Acknowledgments Theorem 3.4 (Intersection) Suppose tis a positive This work was supported in part by theNational Science propernormal valuation for �. and suppose r, s, and Foundationunder grantIRI-8902444. I am grateful to t disjoint subset of $ . If r s I t, and r .l t I s, are .l Pierre Ndilikili.kesha, Glenn Shafer, Philippe Smets, then r .lsvt. Leen-Kiat Soh, and an anonymous refereefor comments Defmition 3.4 generalizes Defmition 3.3 from two sub­ and discussions. sets to any number of subsets. References Definition 3.4 (Joint Conditional Independence) Suppose t is a proper nonnal valuation for $, and Cano, J. E., M. Delgado, and S. Moral (1991), "An ax­ suppose rlt ..., rn. t are disjoint subsets of$. We iomatic for the propagation of uncertainty indirected

say 'J, ... , rn are conditionally independent given t acyclic graphs,"unpublished manuscript,Departamento de Ciencias de Computaci6ne I. A., Universidad de with respect to t', written as .l't { f1, ... , q} I t, iff Ia Granada.Spain. J.(r1v... vr0vt) ll:lo E9 . t = . . here IS Clr1vtw . arnVt• W ClriVt Dawid, A. P. (1979), "Conditional independence in statis­ a valuation for riut, i = 1, ... , n. tical theory (with discussion),"Journal of the Royal Pearland Paz [1987] calla conditionalindependence rela­ Statistical Society, Series B, 41(1), 1-31. tion that satisfies symmetry, decomposition, weak: union, Conditional Independence in Uncertainty Theories 291

Dempster, A.P. (1967), "Upper and lower probabilities Shenoy, P. P. {1989), "A valuation-based language for induced by a multivalued mapping," Annals of expert systems," InternationalJournal of Approximate Mathematical Statistics, 38, 325-339. Reasoning, 3(5), 383-411.

Dubois, D. and H.Prade (1988), Possibility Theory:An Shenoy, P. P. (I991a), "Valuation-based systems: A Approach to ComputerizedProcessing of Uncertainty, framework for managing uncertaintyin expert systems," Plenum Press, New York, NY. in Zadeh, L.A. andJ. Kacprzyk (eds.),Fuzzy Logic for the Management of Uncertainty, John Wiley & Sons, Geiger, D. (1990),"Graphoids: A qualitative framework New York, NY. for probabilistic inference," Ph.D.dissertation, Department of Computer Science, University of Shenoy,P. P. (1991b), "Independence in Valuation-Based California at Los Angeles, CA. Systems," Working PaperNo. 236, School of Business, University of Kansas,Lawrence, KS. Hunter, D. (1991), "Graphoids and natural conditional functions,"International Journal of Approximate Shenoy, P. P.and G. Shafer (1990),"Axioms for proba­ Reasoning, 5(6), 489-504. bility and belief-function propagation," in Shachter, R. D., T. S. Levitt, F. Lemmer and L. N. Kanal (eds.), Lauritzen, S.L. (1989), "Lectures on contingency tables," J. Uncertainty in ArtificialIntelligence, 169-198, 3rd edition, Technical ReportNo. R-89-24, Institute for 4, North-Holland, Amsterdam. Reprinted in Shafer, G. and Electronic Systems, University of Aalborg, Denmark. J. Pearl, eds. (1990), Readings in Uncertain Reasoning, Pearl, J. (1988), Probabilistic Reasoning in Intelligent 575-610, Morgan·Kaufmann, San Mateo, CA. Systems: Networks of Plausible Inference, Morgan Smets, P. (1986), "Combining non distinct evidences," Kaufmann, San Mateo, CA. Proceedings of the North American FuzzyInformation Pearl, J. and A. Paz (1987), Graphoids: Graph-based logic Processing Society Conference (NAFIPS'86), New for reasoningabout relevance relations, in Boulay, B. Orleans, LA, 544-548. D., D. Hogg, and L. Steele (eds.),Advances in Smith, J. Q. (1989), "Influence diagrams for statistical Artificial Intelligence - II, 357-363, North-Holland, modelling," The Annals of Statistics, 17(2), 654-672. Amsterdam. Spohn, W. (1980), "Stochastic independence, de­ Petrich, M. (1973), /ntroduction to Semigroups, Charles causal in pendence, andshieldability," Journal of Philosophical E. Merrill Publishing Co., Columbus, OH. Logic, 9, 73-99. Shafer, G. (1976), A Mathematical Theory of Evidence, Spohn, W.(19 88), "Ordinal conditional functions: A dy­ Princeton University Press, Princeton, NJ. namic theory of epistemic states," in Harper, W. L. and Shafer, G. (1982), "Belief functions andparametric models B. Skynns (eds.), Causation in Decision, Belief (with discussion)," Journal of the Royal Statistical Change, and Statistics, 2, 105-134, D. Reidel, Society, Series B, 44(3), 322-352. Dordrecht, Holland.

Shafer, G. (1984),''The problem of dependent evidence," Zadeh, L. A. (1979), "A theory of approximate reason­ Working Paper No. 164, School of Business, ing," in Ayes, J. E., D. Mitchie, and L. I. Mikulich University of Kansas, Lawrence, KS. (eds.), Machine Intelligence, 9, 149-194, Ellis Horwood, Chichester, U.K. Shafer, G. (1987), "Belief functions and possibility mea­ " sures, in Bezdek, J. (ed.), The Analysis of Fuzzy Information, 1, 51-84, CRC Press, Boca Raton, FL. Shafer, G. (1990), "Perspectives on the theory andpractice of belief functions,"International Journal of Approximate Reasoning, 4(5-6), 323-362.

Shafer, G. (1991), "An axiomatic study of computation in hypertrees," Working PaperNo. 232, School of Business, University of Kansas, Lawrence, KS. Shafer, G. and P. P. Shenoy (1990), "Probability propa­ gation,"Annals of Mathematics and Artificial Intelligence, 2, 327-352.

Shafer, G., P. P. Shenoy, and K.Mellouli (1987), "Propagating belief functions in qualitative Markov trees,"International Journal of ApproximateReasoning, 1{4), 349-400.