An Axiomatic Framework for Propagating Uncertainty in Directed Acyclic Networks

An Axioma ic F amewo k o P opaga ing Unce ain y in Di ec ed Acyclic Ne wo ks Jos6 Cano, Miguel Delgado, and Se a in Mo al Uni e sidad de G anada, G anada, Spain ABSTRACT This pape p esen s an axioma ic sys em o p opaga ing unce ain y in Pea l's causal ne wo ks, (P obabilis ic Reasoning in In elligen Sys ems: Ne wo ks o Plausible In e - ence, 1988 [7]). The main objec i e is o s udy all aspec s o knowledge ep esen a ion and easoning in causal ne wo ks om an abs ac poin o iew, independen o he pa icula heo y being used o ep esen in o ma ion (p obabili ies, belie unc ions o uppe and lowe p obabili ies). This is achie ed by exp essing concep s and algo i hms in e ms o alua ions, an abs ac ma hema ical concep ep esen ing a piece o in o ma ion, in oduced by Shenoy and Sha e [1, 2]. Th ee new axioms a e added o Shenoy and Sha e 's axioma ic amewo k [1, 2], o he p opaga ion o gene al alua ions in hype ees. These axioms allow us o add ess om an abs ac poin o iew concep s such as condi ional in o ma ion (a gene aliza ion o condi ional p obabil- i ies) and gi e ules ela ing he decomposi ion o global in o ma ion wi h he concep o independence (a gene aliza ion o p obabili y ules allowing he decomposi ion o a bidimensional dis ibu ion wi h independen ma ginals in he p oduc o i s wo ma ginals). Finally, Pea l's p opaga ion algo i hms a e also de eloped and exp essed in e ms o ope a ions wi h alua ions. KEYWORDS: causal ne wo k, unce ain y, hype ees, PULCINELLA sys em, ma ginaliza ion, combina ion, condi ional in o ma ion 1. INTRODUCTION Shenoy and Sha e [1, 2] ha e gi en an axioma ic amewo k o he p opaga ion o unce ain y in hype g aphs. In his wo k he p opaga ion Add ess co espondence o Se a in Mo al, Depa men o A i icial In elligence, Campus de Fuen e Nueca, Uni e si y o G anada 18071, G anada, Spain. Recei ed Janua y 1, 1992; accep ed Sep embe 9, 1992. In e na ional Jou nal o App oxima e Reasoning 1993; 8:253-280 © 1993 Else ie Science Publishing Co., Inc. 655 A enue o he Ame icas, New Yo k, NY 10010 0888-613X/93/$6.00 253 254 Jos6 Cano, Miguel Delgado, and Se a in Mo al algo i hms a e abs ac ed om he pa icula heo y being used o ep e- sen in o ma ion. They in oduce he p imi i e concep o alua ion, which can be conside ed as he ma hema ical ep esen a ion o a piece o in o ma ion. A alua ion may be pa icula ized o a possibili y dis ibu ion, a p obabili y dis ibu ion, a belie unc ion, e c. Then hey de elop and exp ess p opaga ion algo i hms in e ms o ope a ions wi h alua ions. These algo i hms may be pa icula ized o any conc e e heo y by ansla - ing alua ions and ope a ions o hei special in e p e a ion in his heo y. These gene al algo i hms ha e been implemen ed in he PULCINELLA sys em [3]. The no a ion used in his pape , and examples o wha is a alua ion in P obabili y Theo y and Theo y o Belie Func ions, a e as ollows: NOTATION Assume ha we ha e an n-dimensional a iable, (X~ ..... Xn), each dimension, X i, aking alues on a ini e se U~. The ollowing con en- ions will be ollowed: • I I ___ {1 .... , n}, we shall deno e by X I he I/L-dimensional a iable (lI[ is he numbe o elemen s o se I), (Xi)i~ 1, and by UI he ca esian p oduc l--[ i E 1 Ui, ha is he se in which X i akes i s alues. • I u ~ U~, hen we shall deno e by u i he i h coo dina e o u, ha is he elemen om U/. • I u ~ U 1 and J ___ I, we shall deno e by u + g he elemen om Uj ob ained om u by d opping he ex a coo dina es; ha is, he elemen gi en by u) J = u j, Vj ~ J. • I A ___ U 1 and J __. I, we shall deno e by A + J he subse o Uj, gi en by A *J = ( E UjI = u +J,u cal. EXAMPLE 1 In P obabili y Theo y a alua ion is he ep esen a ion o a p obabilis ic piece o in o ma ion abou some o he a iables, X 1, I {1,..., n}. Mo e conc e ely, i we ha e h ee a iables (X1, X2, X 3) aking alues on U 1 × U 2, x U3, whe e U/= {uil, ui2}, i = 1, 2, 3, hen a alua ion may be a p obabili y dis ibu ion abou X1, p(u11 ) = 0.8 p(u12 ) = 0.2. I may also be a condi ional p obabili y dis ibu ion abou X3 gi en 2(2, p(U31[U21 ) = 0.9 P(U32]U21 ) = 0.1 p(U31[U22 ) = 0.6 P(U32[U22 ) = 0.4. Di ec ed Acyclic Ne wo ks 255 F om a ma hema ical poin o iew, a p obabilis ic alua ion abou a iables X~ is a non-nega i e mapping, p : Ul --, ,~¢ ~, whe e ~ deno es he non-nega i e eals. These mappings a e no conside ed no malized, bu a e conside ed equi alen upon mul iplica ion by a posi i e cons an ; ha is, wo alua- ions pl, P2 de ined on he same ame /3/ a e conside ed equi alen i he e exis s a cons an a > 0, such ha Vu ~ U~, p~(u) = ,~.p2(u). F om s ic ma hema ical poin o iew, a alua ion should be conside ed an equi alence class on he se o non-nega i e mappings om U I on ,~ 0-, unde he abo e equi alence ela ion; howe e , o simpli y he language and no a ion, we shall conside ha a alua ion is a mapping, bu ha wo mappings a e conside ed iden ical i hey a e equi alen . EXAMPLE 2 Fo Belie Func ions [4-6], a alua ion abou X is a non- necessa ily no malized mass assignmen on U~, ha is, a mapping m: 9(U ) ~ ,~c~, whe e ~(U I) is he se o all he subse s o U1, and m(Q) = 0. Two basic ope a ions a e assumed o be de ined among alua ions: combina ion and ma ginaliza ion. Combina ion is an ope a ion o summa- ize in a single alua ion he in o ma ion o wo alua ions. I he wo alua ions o be combined a e V 1 and V 2 de ined on U and Uj, espec- i ely, hei combina ion will be deno ed V ® V2 and will be de ined on UIuj" Ma ginaliza ion is an ope a ion o calcula e he in o ma ion induced by a alua ion de ined on a ame U/, on a less ine ame: Uj, whe e J c I. I V is he alua ion de ined on U~, i s ma ginaliza ion o Uj is deno ed by V +g EXAMPLE 3 In he pa icula case o p obabilis ic alua ions, combina ion is de ined by poin -wise mul iplica ion. I p~ and P2 a e non-nega i e unc ions de ined on U I and Uj, espec i ely, hen p~ ® P2 is a mapping de ined on U/g g o ~- gi en by, Pl @P2(u) =Pl(us')'p2(ulg), Vu E UIu J. This ope a ion is used in p obabili y o combine a ma ginal dis ibu ion wi h a condi ional one o p oduce a bidimensional dis ibu ion, o used o 256 Jos6 Cano, Miguel Delgado, and Se a in Mo al calcula e condi ional in o ma ion. Remembe ha as we a e no conce ned abou no maliza ion, condi ioning o a se A may be conside ed as he mul iplica ion wi h he likelihood associa ed o A (i s cha ac e is ic unc- ion: lA(u) -- 1, i u ~ A; lA(u) = 0, o he wise). Manginaliza ion is de ined in he usual way: I p is a alua ion de ined on U and J c_ I, hen p J( ) = Y'. p(u), ,,c u~J=l) EXAMPLE 4 In Belie Func ions [4, 5] combina ion is ca ied ou by means o Demps e 's ule: m I ® m2(A ) = y' ml(B1) "m2(B2). BIOB2=A We do no no malize because wo alua ions a e conside ed as equi alen i one is ob ained om he o he by mul iplying by a posi i e cons an . I m is de ined on U and J c_ I, hen he ma ginaliza ion o m o Uj is gi en by, m~J(A) = • m(B) B~J=A Shenoy and Sha e [1, 2] show ha i hese wo ope a ions e i y a sys em o h ee axioms, hen he calculus wi h alua ions may be done by means o p opaga ion algo i hms. Mo e speci ically, hey show ha i we ha e n alua ions, V~,..., V n, and we wan o calcula e (V 1 ® V 2 ® -.. ® Vn) ~ oil o each a iable Xi, hen we can do so wi hou explici ly calcula - ing he global alua ion V 1 ® V 2 ... ® V~, bu by doing local compu a ions among he ini ial alua ions, a anged in an app op ia e way. The ad an- age o a oiding he calcula ion o V~ ® V 2 -.. ® V, is ha , in gene al, his calcula ion is e y ine icien . Fo example, in he case o p obabili ies, i each a iable X i appea s a leas on a alua ion ~ and we ha e m a iables, hen I/1 ® V 2 ... ® V~ will be de ined on U 1 × ... × U m. I each U~ has k i elemen s, we will need l-I m,=~ k i alues o speci y his alua ion. In he bes case (all he k i equal o 2) his numbe is 2 m. Al hough Shenoy and Sha e , [1, 2], ocus hei wo k on he p oblem o calculus, he e a e o he impo an aspec s in he p ocess o p oblem modeling and esolu ion ha ha e no been conside ed. I we ha e wo pieces o in o ma ion ep esen ed by alua ions V 1 and V2, hen hei combina ion, V a ® V 2 does no always gi e ise o a meaning ul o alid in o ma ion o he p oblem. Conside , o example, wo p obabilis ic alua ions abou a iables X 1 and X2: p, a bidimensional p obabili y abou (X1, X2), and Pl, a p obabili y abou X 1. The combina ion p ® p~ Di ec ed Acyclic Ne wo ks 257 makes no sense om a p obabilis ic iewpoin . I is no a alid p obabili y o he wo a iables; howe e , i p is a condi ional p obabili y abou X 2 gi en X 1, hen Pl ® P is alid p obabili y dis ibu ion abou (X 1, )(2). I is a bidimensional p obabili y. The concep o independence plays an impo - an ole in his kind o ules: I Pl and P2 a e p obabili y dis ibu ions o X 1 and X2, espec i ely, and hese a iables a e independen , hen, Pl ® P2 is a alid p obabili y o (X~, X2). I X I and X 2 a e no independen , hen his is no ue. Pea l [7] uses hese ules in P obabili y Theo y o show ha in a causal ne wo k (di ec ed acyclic g aph), gi ing a condi ional p obabili y o each node gi en i s pa en s de e mines one and only one global p obabili y dis ibu ion o all he a iables. In o he wo ds, he ini ial pieces o in o ma ion a e comple e and cohe en . These impo an issues a e he main opics o he p esen pape . Shenoy, [8], also s udies condi ional independence o alua ions in e ms o ac o s o he join alua ion. In his pape , i is no assumed he exis ency o a join alua ion. We gi e ules o build mo e complex alua ions om elemen al ones using he gi en independence ela ionships. We also gi e condi ions o de e mine one and only one join alua ion. The g aphical s uc u es used o ep esen ela ionships among a iables in ou wo k a e Pea l's causal ne wo ks, no Shenoy and Sha e 's hype - g aphs, because he o me a e mo e app op ia e o ep esen indepen- dence ela ionships among a iables in a di ec way. In he second sec ion we in oduce h ee new axioms o ope a ions wi h alua ions, and de ine, in an abs ac way, he concep s o condi ional alua ion, obse ~'a ion, and 'a pos e io i' in o ma ion. We hen in oduce ules o build new alua ions om ini ial ones. In he hi d sec ion we show ha in a di ec ed acyclic g aph, ha ing a alua ion o each node, gi en i s pa en s, de e mines one and only one global alua ion alid o all he a iables. In he ou h sec ion, we ob ain gene al p opaga ion algo- i hms in di ec ed acyclic g aphs. These a e a gene aliza ion o Pea l's algo i hms, [7], bu now a e exp essed in e ms o ope a ions wi h alua- ions, and, he e o e, applicable o di e en unce ain y heo ies. Finally, in he las sec ion we ela e p opaga ion algo i hms wi h he independence ela ionships associa ed wi h he g aph. 2. VALUATION-BASED SYSTEMS In his sec ion we desc ibe how o ep esen in o ma ion wi h alua ions and how o do calcula ions wi h hem, gi ing a me hod analogous o Bayes Theo em. Le X = (X1,..., X n) be an n-dimensional a iable such ha each X i 258 Jos6 Cano, Miguel Delgado, and Se a in Mo al akes i s alues on a ini e se U A alua ion is a p imi i e concep meaning he ma hema ical ep esen a ion o a piece o in o ma ion in a gi en unce ain y heo y. We will assume ha o each I ___ {1 ..... n} he e is a se V/o alua ions de ined on he ca esian p oduc , U 1. V will be he se o all alua ions V = U i c{l ...... }VI. Two basic ope a ions a e necessa y (see Zadeh [9]; Shenoy, Sha e [1, 2]): • Ma ginaliza ion. I J c_ I and V 1 e VI, hen he ma ginaliza ion o V l o J is a alua ion Vl + 1 in Vj. • Combina ion. I V 1 ~ V/ and V 2 ~ Vj, hen hei combina ion is a alua ion V l ® V 2 in V l u ]- Shenoy and Sha e [1, 2], conside he ollowing h ee axioms o hese ope a ions on alua ions: Axiom l Vl ® V2 = V2 ® V1, (VI ~ V2) ~ V3 = VI ~ (V2 ~ V3). Axiom 2 l l c J c K, and V ~ VK , hen ( V + g ) +1= V ~ I Axiom 3 I V 1 ~ V/, V 2 ~ Vj, hen (V 1 ® 1/2) ;1= V l ® V2 "L(J•I). We assume h ee mo e axioms, Axiom 4 Neu al Elemen . The e exis s one and only one alua ion V 0 de ined on U 1 x ... x U, such ha VV ~ V~, ~¢J c_ I, Vo ~ ] ® V=V. Axiom 5 Con adic ion. The e exis s one and only one alua ion, V c, de ined on U 1 x -.. x U n, such ha VV E V, V~ ® V -- V~. Axiom 6 VV E Vo, i V ~ Vc ;¢, hen V = V0 ~°. The h ee i s axioms p o ide he necessa y condi ions o deduce p opaga ion algo i hms. The hi d axiom is o pa icula impo ance o he de elopmen o p opaga ion algo i hms, as i allows us o calcula e (V 1 ® 1/'2) ~1 wi hou explici ly calcula ing (V 1 ® V2), a alua ion de ined on U u ]. This can be done by calcula ing I/2 ~ (g n 1) and combining he esul wi h V 1. In his las case we need only handle alua ions on Uj, U I n ], and UI, which is much mo e e icien . Remembe ha , in gene al i is ine i- cien o handle alua ions de ined o a la ge numbe o a iables. The ou h axiom deals wi h he exis ence o he neu al elemen . This neu al elemen is conside ed by Sha e and Shenoy [1], bu i s exis ence is no pos ula ed by an axiom. This axiom is essen ial in he p esen s udy o de ine he concep o condi ional in o ma ion in an abs ac way. In he case o P obabili y Theo y, he neu al elemen is a cons an (non-ze o) alua ion: po(U) = 1, Vu ~ U1 X "" x U, The cons an alue is no impo an because p obabilis ic alua ions a e equi alen upon mul iplica ion by a posi i e cons an . In he case o Belie Func ions, he neu al elemen is he so-called acuous belie , [5, 6], gi en Di ec ed Acyclic Ne wo ks 259 by i A =U~ ×..'× Un. o he wise In he Axiom 4, he exis ence o he neu al elemen is pos ula ed on he lame co esponding o all he a iables, U × ... × U n. In smalle ames, he neu al elemen is ob ained by ma ginaliza ion. The e o e, V~j ~K will be called he neu al elemen o V~, and when he e is no chance o con usion, i will be deno ed simply by V 0. The con adic ion is cha ac e ized in Axiom 5 as a alua ion such ha i i is combined wi h any o he alua ion, i p oduces he con adic ion. In P obabili y Theo y i is gi en by he ze o- alued unc ion p,(u) = O, Vu ~ U~ x ... × U,, In Belie Func ions Theo y, he con adic ion is he ze o alued mass assignmen , m~,(A) = O, VA c U~ x ... x U,, These alua ions a e ob ained by combining wo con adic o y alua- ions. Fo example, in he case o Belie Func ions, i can esul om combining he masses, m~ and m 2, de ined on (uE, u2, u d and gi en by nl({ul} ) = 1; ml(A ) = 0, o he wise nz({U2} ) = 0.5; m2((u2,u3} ) = 0.5; m2(A ) = 0, o he wise. In Shenoy and Sha e [2], he con adic ion is conside ed om a di e - en s andpoin : They de ine a subse o he se o alua ions called he amily o p ope ualua ions. The elemen s o his se a e he alua ions di e en om he con adic ion. As in he case o he neu al elemen , Vc ~ ~ will be called he con adic- ion o V K, and will some imes be deno ed by V,.. Acco ding o Axiom 6, in he ame co esponding o he emp y se o a iables, U~, all alua ions a e he con adic ion o he neu al elemen . To be e app ecia e his, le us show i s meaning in P obabili y Theo y. The ca esian p oduc U~ has one elemen : {e}. A alua ion, p, in his se ~, + is, he e o e, a mapping om {e} on ,R0, ha is, a numbe , p(e). I his numbe is ze o, hen we ha e he con adic ion; i i is di e en om ze o, hen we ha e he neu al elemen : combina ion wi h his alua ion p o- duces an equi alen alua ion. I s meaning and consequences will be discussed below when we conside he ela ionships o he calculus wi h alua ions and he concep o independence. 260 Jos6 Cano, Miguel Delgado, and Se a ln Mo al The ollowing p oposi ions illus a e some addi ional p ope ies o alu- a ions. We assume ha Axioms 1-6 a e e i ied. PROPOSITION 1 VV E VK, V ® Vc ~ K = Vc~ K P oo On he basis o Axioms 5 and 3, ~. ~K = (V~ ® V) ~ = ® Vc *K Q.E.D. • PROPOSITION 2 VV ~ V / being I _ K, we ha e V ® Vc+ K = V+ P oo On he basis o Axioms 4, 1, and P oposi ion 1, ® ~ ~K = ® (K. ~K ® ~ ~K) = ( ® Vc ~) ® V~ ~ = V~ ~ P oposi ion 1 has been applied in he las equali y. • The ollowing p oposi ion is qui e na u al: i a alua ion is ma ginalized on he a iables in which i is de ined, we ob ain he same alua ion; howe e , p oo o his equi es in oca ions o Axiom 4. PROPOSITION 3 VV ~ V/, V ~ l = V P oo By applying Axioms 4, 3, 2, and 4, V ~1 = (V® Vo~) ~I = V® (Vo~) ~ = V® V0 ~= V. The ollowing is a echnical p oposi ion ha will be used in la e p opaga ion algo i hms. PROPOSITION 4 I 1/1,V 2~V,I_{1,2 ..... n}, VI E V],V z ~ V K wi h I__c_J u K, J n K c_ I, hen (1/1 ~ ~) ~' = , .1~] ~ 2 ~'n~ P oo On he basis o Axiom 2, (V, ® V2) $' = ((V 1 ® V2)$Ju') j'l As (V 1 ® V 2) is a alua ion on J u K and we know ha I c J u K, hen by using Axiom 4, we ob ain (V 1 ® V2) +1 = (((V, ~ V2) ® Vo$1)$JUl) $1 On he basis o Axioms 1 and 3, (V 1 ® g2) $I = ((V 1 ® V 2 ~ Vo$l)J'Jul) $1 = (((V1 ® goJ, I) ® V2)J'J )l) j'' = ((V 1 * Vo ,[,I) ® V2J~(jUl)nK) $1 = (1/1 ® ® +I * V~+'~K) z~ Di ec ed Acyclic Ne wo ks 261 The las equali y is de i ed om he ac ha i (J n K)_ I, hen (JUI) AK=IAK. Now, using Axioms 1, 3, and 4 we ge , (V 1 ~ e) "LI = ((Vo ,L1 ~ V2 "LINK) ~ Vi) ~'I ~__ (Vo ,~1 (~ V2 SINK ~ Vl"~lnJ). As he alua ion I/2 * nK® V1 ~l~J is de ined on (I N K) U (I N J) = I n (K U J), which--because I ___ J u K--is equal o I, his alua ion is de ined on I. The e o e, applying Axiom 4 we ge (V 1 ~ V2) " l = (V2 $1oK ~ Vl"/oJ), Q.E.D. [] No e ha in his p oo , mul iplica ion by he neu al alua ion on V , V0 ~* is used, in gene al, o ex end a alua ion V ~ V~ o he se V K u I. This me hod will be used se e al imes h oughou he a icle. An immedia e consequence o abo e p oposi ion is he ollowing one, which we gi e wi hou p oo . PROPOSITION 5 I V 1,V 2 ~ V,I___{1,2 ..... n},V 1 ~ Vj, V 2 ~ V K J n K = I, hen (7 I ~ V2) *l = pl $1 ~ V2 $1 The ollowing p oposi ion can be deduced using Axiom 6. wi h PROPOSITION 6 I V 1, V 2 ~ 1,1, V 1 ~ V~, V 2 ~ V K wi h J • K = • and V2 ~ 4= V~ +e hen (V 1 ~ V2) SJ ~- Vl " Acco ding o his p oposi ion, i we ha e wo alua ions, V 1 and V 2, gi en o disjoin se s o a iables, J and K, and we combine hem-- ob aining a alua ion o he se J u K--and i we hen ma ginalize on he i s se , hen we ob ain he same alua ion V]. This is no necessa ily e i ied i Axiom 6 is no pos ula ed as ue. In ha case, he combina ion o a alua ion wi h a disjoin alua ion, ollowed by ma ginaliza ion, may a ec his alua ion. This may p oduce some incohe ence be ween he concep o independence and he calculus wi h alua ions o be gi en below. We shall conside his p oblem in mo e de ail in sec ion 5. The main de ini ions ela i e o he calculus wi h alua ions a e gi en below. DEFINITION 1 A alua ion V ~ V/ is said o be abso ben i and only i i is no he con adic ion in V/ and (VV' ~ VIX(V ® V' = V) o (V ® ' = ~)). 268 Jos6 Cano, Miguel Delgado, and Se a in Mo al Then he ollowing ac s a e e i ied. a. (H', h') is ac ually a sys em o in o ma ion. b. V~ ~ H', Vi ~ {1 .... , n}, wi h h'(V i) = (i, e(i)). c. I o a sys em o in o ma ion (H", h") wi h he same dependence s uc u e, (b) is e i ied, hen M , R1, i, SI. ] belong o H". d. The only global alua ion, V, in H' wi h h(V) = ({1, 2 ..... n}, O) is Vl® 2 ® ... ® .. I hese poin s a e p o ed, hen om (a) and (b) we conclude ha H _c H', H being he sys em o in o ma ion gene a ed by alua ions {V/}i~ l,2 ..... n~. Taking in o accoun (c), we ha e he equali y H = H', because H is a sys em o in o ma ion ha e i ies (b). Acco ding o (d), he e is one and only one global alua ion de ined o all he a iables, {1,..., n}: V 1 ® V 2 ® ... ® V~. Now all we need is o p o e hese poin s. To show (a), we need o p o e he ou p ope ies o sys ems o in o ma ion: 1. I is immedia e by he way alua ions MI, R z, S/, ] a e de ined. 2. P ope y 2 may be applied only o alua ions Rl, i, S~,] combined wi h alua ions M . I is e i ied because he combina ion R~, i ® M u e~0 = MI u P(i) u {i}, and he combina ion S , ] ® M] = M 1 u ], ha is, ele- men s om H'. 3. The combina ion o a alua ion M/wi h V0 ~ ], wi h D(I, O, J) = 0, is p ecisely S/, ]. Applying his ule o alua ions in he o m o R , ~ and S , ] yields elemen s o he same ype. 4. Ma ginaliza ion o elemen s M and S ,] p oduces elemen s o he same ype. The only p ope ma ginaliza ion ope a ions applicable o elemen s R , i a e o ma ginalize o U/ e~n and o U~ u e(i)u {i}. In he i s case, as RI, i is a alua ion condi ioned on a iables (Xj)j ~ i u e~o, yields he neu al alua ion on U/u e(i) which is equal o S~, e<0" The second is no eally a ma ginaliza ion. We ge he same alua ion Rl, i. The p oo o (b), (c), and (d) is s aigh o wa d and will no be desc ibed in de ail. • 4. PROPAGATION OF VALUATIONS IN POLYTREES Assume ha (X1,... , gn) is an n-dimensional a iable and (H, h) is a sys em o in o ma ion de ined on he g aph (T, E) and gi en by alua ions {V~}i ~ l ..... n~. Unde hese condi ions, he only possible global alua ion o all he a iables is V 1 ® V z ® .-. ® V~. Conside also a amily o obse a- ions {Oi}i~i, and ha ou objec i e is o calcula e he 'a pos e io i' alua ions o each single a iable, ha is, o calcula e he ollowing Di ec ed Acyclic Ne wo ks 269 alua ions: PSi = ® 0 i ~ V ieI o each j e {1 ..... n}, whe e V = 1/1 ® .-" ® Vn __ is he only exis ing global in o ma ion. The main p oblem, as indica ed in he in oduc ion, is o calcula e, 1/1 ®... ® V~. P opaga ion algo i hms p oposed by Pea l, [7], a oid he calcula ion o V] ® .-- ® V n i he g aph does no ha e undi ec ed cycles (also called loops). These g aphs a e called poly ees, [7]. In his sec ion we gene alize hese algo i hms o he unce ain y exp essed by means o alua ions. The no a ion will be based on he ollowing alua ions, Hk = Vk,i k ~ I Hk = V~ ® Ok,i k ~ l 0~= V o~ V{j.},i j~I O; = Oj, i j ~ l and he ollowing se s o each j c {1 ..... n}, • I , he se o indexes i such ha X i is a descenden o Xj ( he e is a di ec ed pa h om Xj o Xi). • I[ = {1,...,n} - (17 U {j}). • P(j) is he se o pa en s o node Xj. • Fo each k ~ P(j), I~ is he se o indexes o nodes X i such ha he e is an undi ec ed pa h om X i o X k o om X~ o X i no con aining Xj. • C(j), is he se o child en o node Xj. • Fo each k ~ C(j), Ij~, he se o indexes o nodes connec ed wi h X k ia an undi ec ed pa h no con aining )(i.. • I~ is he se o indexes o nodes no connec ed wi h node Xj. The ollowing p oposi ions o m he basis o p opaga ion algo i hms. In all o hem, i is assumed ha he g aph has no loops. PROPOSITION 8 Vje {1 ..... n}, PSi P oo = 7 j ® ; i, whe e ] '~ {j} ,~ (j} Taking in o accoun he exp ession o PSi and he de ini ion o PSi = (H, ® H 2 ® "'" ® Hn) ~{j} 270 Joss Can®, Miguel Delgado, and Se a in Mo al Taking in o accoun Axioms 1 and 2, PSj = ( H 1 ® H 2 ® ''' ® H~) ~ {j) = ((iE~lj+Hi) ® Hj ® (iE~llj Hi)) J'{j} On he basis o P oposi ion 5, and aking in o accoun ha (I S u {j}) n (I~ u {j}) = {j}, ® ) ~ {j) ,' '[ {j} ) Q.E.D. • PROPOSITION 9 I (H 1 ® H 2 ® ... ®/4,) ~° is di e en om he con a- dic ion, hen Vj~ {1 ..... n}, z ) = Vj® ( @ 7 ' ; A i = ~ h~ ®q k~P(j) "/ I I k~C(j) whe e P oo = l~j = (iE~iijkHi) i ljk The exp ession o 7 j is Taking in o accoun ha he g aph has no loops, we can show ha Di ec ed Acyclic Ne wo ks I = LI~ ~ p(j) I)~ wi h Ij~, n I)] 2 = ~ i k, 4: k 2. The e o e, and on he basis o Axiom 3, ® ~] ~ ({Jlue ))] klj} o k~P(j) i~ljk 271 Now, by a epea ed applica ion o P oposi ion 4, Taking in o accoun he exp ession o ~ [, we ob ain he desi ed equali y o 7 j, k~P(j) The alua ion Aj was de ined as As he g aph has no loops, I~ = U k ~ c(j)u (01 I~, wi h Ijk ' n I~ = Q, i k I 4= k 2. The e o e, Aj = ® ® ® o~. By a epea ed applica ion o P oposi ion 5, ~= ® /4, ® ®o~. • kGC(j) l Ilk i~ljo Taking in o accoun he exp ession o A~, k~C(j) i o 272 Jos6 Cano, Miguel Delgado, and Se a in Mo al As (H 1 ® ..-® H,) ~ is di e en om he con adic ion hen (®i~ 6o Hi)~° is also di e en om he con adic ion, and aking in o accoun Axiom 6, Thus, a = ® a: ®o:= ® aj ®o:. k~C(j) k~C(j) Q.E.D. • We now ha e he alua ions we wan o calcula e, PSi, in e ms o k and alua ions 7 j and A:, which a e exp essed in e ms o alua ions h: • . In he nex p oposi ion, we show how we calcula e hese alua ions in a node, assuming ha he co esponding alua ions in all he neighbo ing nodes a e known. PROPOSITION 10 I j ~ {1,..., n}, hen Vk ~ P(j), i~C(k), i¢j And Vk~ C(j), k Ak® ' ®V~® ® ~- Aj = O k i~P(k), icj P oo I k ~ P(j), hen [ ~{k} i ®/-/,/ i~ C(k), iej [ ( .)] = %®0~,® ® ,~}, . i~ C(k), i=/=j Di ec ed Acyclic Ne wo ks 273 Analogously, i k ~ C(j), • lj~ = h k ® O k ® V k ® ® 7 k icP(k), i4-j Q.E.D. • The con adic ion aises he ollowing p oblem. I we ha e a g aph wi h wo disconnec ed pa s, o example one wi h e ices {1, 2,..., i} and o he wi h e ices {i + 1 .... , n} ( ha is, he e is no a c om e ices o one se o e ices o he o he se ), hen i H l ® ... ® H i is he con adic ion in U~I ..... i~, i can be shown ha H 1 ® ... ® H n is he global con adic ion and PSi is he con adic ion o each a iable Xj, j ~ {1,2 ..... n}. How- e e , i we use he abo e o mulas, hen he con adic ion is no p opa- ga ed be ween non-connec ed pa s o he g aph and wha we ob ain o PSi, j ~ {i + 1 ..... n} is (Hi+ 1 ® ... ® [In) ~{j}, which is no PSi i he e is no con adic ion in a iables Xi+ 1 .... , X n. Wha o do om a p ac ical poin o iew? The answe is simple. I he g aph is connec ed, hen he p opaga ion o mulas a e co ec . We ha e used he lack o con adic ion o show ha and in his case Ij0 is emp y. I he g aph is no connec ed, hen o a global con adic ion o exis , some o he connec ed pa s ha e o be con adic o y. In such a case, om a ma hema ical poin o iew all alues o PSi will be con adic o y. Bu he e is an al e na i e: no o conside he in o ma ion om a iables gi ing ise o he con adic ion, emo ing such con adic o y in o ma ion om he sys em and keeping only non-con adic o y alua ions. Shenoy [12] desc ibes an e icien way o isola e a maximal consis en se o alua ions. Wi h his es ic ed sys em, he o mulas o p opaga ion can be used. Ou objec i e is now o calcula e he alues ~j, hi, 7 ~ k (k E C(j)), h~ (k ~ P(j)), o each j E {1 ..... n}. I he se o obse a ions, {Oi}i~ 1 is emp y (I = ~3), hen his can be done e y easily. Fi s , we shall p o e ha all alua ions h i a e he neu al alua ion. 274 Jos6 Cano, Miguel Delgado, and Se a in Mo al PROPOSITION 11 I I = Q and V c ~ (V 1 ® ... ® Vn) *e hen Aj = V o KU}, Vj ~ {1 .... , n}. P oo Fo e e y lea a iable in (T, E), we ha e Aj = (®k~c(j) A~) ® O~=V o. I Xj is a lea (a node wi h no child en) hen C(j)=Q, he e o e, O~ = V o. Now le us p o e ha i A k = Vo ~ k} ~ V k ~ o e e y k ~ C(j) hen Aj = Vo ~j~ + +. k~C(j) k~C(j) and ,~= ,~,®o~® , ® ~ i~P(k), i.,~j i~P(k), i#:j [< ]"J' = ® -,,-~ ® /p<k) i~P(k), i~j Now, aking in o accoun ha V k ~ V~k}le(k), [ < ,)1 A~ = Vo ~'<') ® ® ~-~ i~P(k), iV:j By a epea ed applica ion o P oposi ion 4 we ge k VoCJ}® ( @ -a'~' ~) = Vo cj} Aj= i~P(k), imj Tha is, ; ~ = V0 * u} ~ V m hence a.j=( ® a.~)=Vo~U}~ u,. k~C(j) Q.E.D. • Unde hese condi ions, i I = 0, we can ind ~ j o e e y j ~ {1 ..... n} wi h he ollowing algo i hm, whe e i is assumed ha i ] > i, hen Xj is Di ec ed Acyclic Ne wo ks 275 no a pa en o X i ( ha is, he j alue o a node will be calcula ed when co esponding 7 ~, j ~ P(k) a e known). The s eps a e as ollows: • Fo e e y j = 1 .... , n calcula e aj I~,, ~j PS i= j ® Aj. Vk ~ C(j), 7 ~ = k Vi ~ P(j), aj = Vo+ ~1 Assume now ha we wan o calcula e PSi wi h a se I 4= 0. We shall gi e, like Pea l, [7], an algo i hm ha calcula es hese alues (PSi, 7 j, Aj, A~, ~-~) o a se I' = 1 U {@}, assuming ha we know hese alues o he se o obse a ions I; ha is, he algo i hm upda es he alues o alula ions in he ligh o a new obse a ion. Fo k ~ P(j), he 7 alues a e said o be messages om he pa en s o Xj o i , o incoming messages om i s pa en s. Analogously, o k ~ C(j), he A~ alues a e he messages ha his node ecei es om i s child en. On in oducing a new obse a ion, O , all incoming messages o a node do no change, unless he e is an undi ec ed pa h be ween his node and X . In his case, only he message a i ing om his pa h changes; ha is, A~ o 71"j k change only when he e is an undi ec ed pa h be ween Xj and X ia X k. As he g aphs we a e wo king wi h do no ha e loops, he e is, a mos , only one undi ec ed pa h om each node o X . All incoming alua ions o Xj wi h he se o obse a ions I' a e, he e o e, he same as he alua ions o se o obse a ions I, excep pe haps o he message coming om one o i s child en o pa en s. The ou going message om Xj o his node does no change, bu all he o he ou going messages a e di e en (Figu e 1). Fo node X l, he a iable o which we ha e in oduced he new obse a ion, he si ua ion is as ollows: all incoming messages a e he same and all ou going messages a e new (Figu e 1). In he ligh o hese conside a ions we can design an algo i hm o pe o m he upda ing. The algo i hm is based on he ac ha when we a i e a a node, all he incoming messages a e calcula ed, and we hen calcula e 7 /, A/, PS/, and he ou going messages. Le J be he se o pai s (Jl, J2) whe e Jl is a node o be upda ed and J2 he incoming node. Then he algo i hm is as ollows: • J = {(1, -1)} • While J =~ Q Choose (Jl,J2) ~ J, J "- J - {(JJ,J2)} calcula e • j~ *- [~ ® (®~ ~ k) l~u'} e.g(j~) jl ] k • Ajl ~ (®*¢<c<j,) Ajl) ® O;I • PSjj ,-- jl ® Ajl 276 Jos6 Cano, Miguel Delgado, and Se a in Mo al Figu e 1. Changing messages wi h a new obse a ion 09. Fo e e y k e C(jl) , k 4:j2 calcula e jl * 77 k ~ [77jl ~ O~l ~ (~i~:C(jl)i.k /~'1 )] , J ~ J u {(k, Jl)} Fo e e y k E P(Jl), k 4= Jl calcula e * A~ 1 ~ [~jl ~ Oil * Vii ~ (®iEP(j,),.k ;1)] "~(k} * J (- J U ((k, jx)} I is immedia ely clea ha his algo i hm upda es all he messages, and he alues o PSi co esponding o I'. 5. PROPAGATION ALGORITHMS AND CONDITIONAL INDEPENDENCE In his sec ion we p o e ha i all he axioms hold, hen he p opaga ion algo i hms a e cohe en wi h he ini ial dependence s uc u e associa ed wi h a g aph. To p o e his cohe ence, we need Axiom 6. In a heo y o unce ain y ep esen a ion in which his axiom does no hold, we could eso o use p opaga ion o mulas, bu hen we would ha e o admi ha we a e iola ing he cohe ence wi h he dependence s uc u e associa ed wi h he g aph. PROPOSITION 12 Le (T, E) be a DAG wi hou loops associa ed wi h n-dimensional a iable (X 1 ..... X,), (H, h) a sys em o in o ma ion de ined on i , and I,J,K_c{1 ..... n} in such a way ha D(I,J,K)=O. Unde Di ec ed Acyclic Ne wo ks 277 hese condi ions, i we know he alues o a iables X j, ha is, i we ha e he obse a ions {Oj}j ~ j and he con adic ion is ne e ob ained, hen he in oduc ion o a new obse a ion o a a iable in {Xi} i ~ ~ does no change he 'a pos e io i' alua ion in any o he a iables in {X~} k ~ s.. P oo Assume ha we in oduce a new obse a ion, Oi~ ,, whe e i~b E 1, and assume a node Xk, whe e k ~ K. As he g aph has no loops he e is only one undi ec ed pa h going om Xi,, o X k. Along his undi ec ed pa h a el he messages wi h he in luence o Oi,, on Xk. I a a gi en momen , one o hese messages does no change a e in oducing obse a- ion O~,, hen his obse a ion does no ha e any e ec on X k. As we assume ha D(1, J, K) = 0, hen by he d-sepa a ion c i e ion (see De ini ion 13) we ha e ha o his pa h one o he wo ollowing condi ions holds • The e exis s a node wi h con e ging a ows in he pa h ha does no belong o {Xj}j ~j and i s descenden s do no belong o {Xj}j~ j. • The e exis s a node wi hou con e ging a ows in he pa h ha belongs o {Xj}j~ j. In he i s case, le his node be X I. None o i s descenden s is in {X~}j~ j. By applying a p oo simila o ha in P oposi ion 11, i can be shown ha be o e and a e in oducing obse a ion Oi,,, all messages ha his node sends o i s pa en s a e he neu al alua ion. As he chain o messages ca ying he e ec o Oi, ' o X k goes h ough X 1 om one o i s pa en s o a di e en pa en , he ou going message does no change. I is he neu al elemen , wi h no e ec on X k. In he second case, we ha e a node, Xj, j ~ J, wi hou con e ging a ows in he pa h om Xi, ' o X k. As j is in he se o obse a ions, we ha e an obse a ion Oj o his node, which is an abso ben alua ion. The messages his node sends o i s child en. X m, m ~ C(j) a e i~ C(j) i ~ m As Oj is abso ben and he con adic ion is ne e ob ained, ~ = O i . Thus, he messages om Xj o i s child en ne e change, and i he pa h om Xi, ' o X k passes h ough Xj om a pa en o a child, o om a child o a di e en child, hen i does no change in X k and has no e ec on S k .