d-separation: Strong Completeness of Semantics in Bayesian Network Inference

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1 -Separation: Strong Completeness of Semantis in Bayesian Network Inferene Cory J. Butz 1,WenYan 1, an Aners L. Masen 2,3 1 Department of Computer Siene, University of Regina, Canaa {butz,yanwe111}@s.uregina.a 2 HUGIN EXPERT A/S, Aalborg, Denmark aners@hugin.om 3 Aalborg University, Department of Computer Siene, Denmark Abstrat. It is known that -separation an etermine the minimum amount of information neee to proess a query uring exat inferene in isrete Bayesian networks. Unfortunately, no pratial metho is known for etermining the semantis of the intermeiate fators onstrute uring inferene. Instea, all inferene algorithms are relegate to enoting the inferene proess in terms of potentials. In this theoretial paper, we give an algorithm, alle Semantis in Inferene (SI), that uses -separation to enote the semantis of every potential onstrute uring inferene. We show that SI possesses four salient features: polynomial time omplexity, sounness, ompleteness, an strong ompleteness. SI provies a better unerstaning of the theoretial founation of Bayesian networks an an be use for improve larity, as shown via an examination of Bayesian network literature. 1 Introution In [12], Pearl avoate the restoration of probabilisti methos in artifiial intelligene systems an explore the possibility of representing an manipulating probabilisti knowlege in graphial forms, latter alle Bayesian networks. When reounting the evelopment of Bayesian networks, Pearl [14] states that perhaps [12] mae its greatest immeiate impat through the notion of - separation. As a metho for eiing whih onitional inepenene relations are implie by the irete ayli graph of a Bayesian network, -separation provies the semantis neee for efining an haraterizing Bayesian networks. Observe that Pearl emphasizes the importane of -separation with respet to Bayesian network moeling. With respet to inferene, Pearl only states that - separation an etermine the minimum information neee for answering a query pose to a Bayesian network. No laim has ever been mae that -separation an also provie semantis uring Bayesian network inferene. Koller an Frieman [8] state that it is interesting to onsier the semantis of the potentials onstrute uring inferene. They mention that sometimes the probabilities are efine with respet to the joint istribution, but not at other times. As no pratial algorithm exists for eiing the semantis of inferene, all inferene algorithms enote the intermeiate fators onstrute uring inferene O. Zaïane an S. Zilles (Es.): Canaian AI 2013, LNAI 7884, pp , Springer-Verlag Berlin Heielberg 2013

2 14 C.J.Butz,W.Yan,anA.L.Masen as potentials. Potentials have no onstraints [8] meaning they o not have lear physial interpretation [4]. In this theoretial paper, we present Semantis in Inferene (SI), an algorithm for enoting semantis uring exat inferene in isrete Bayesian networks. SI works by introuing the notion of eviene normal form to organize how eah potential was onstrute. SI then eies semantis of the potential by performing one -separation test. Formal properties of the SI algorithm are obtaine, namely, polynomial time omplexity, sounness, ompleteness, an strong ompleteness. SI an be utilize for larity of exposition in Bayesian network literature, sine the semantis of potentials an now be artiulate. 2 Inferene Here we onsier only isrete Bayesian networks. U = {v 1,v 2,...,v n } is a finite set of ranom variables an eah v i U an take a value from a finite omain, om(v i ). Given X U, om(x) is the Cartesian prout of om(v i ), v i X. Apotential on om(x) is a funtion ψ on om(x) suh that ψ(x) 0 for eah x om(x), an at least one ψ(x) is positive. For brevity, we refer to ψ as a mapping on X rather than om(x). A potential p on U that sums to 1 is alle a joint probability istribution on U, enote p(u). A onitional probability table (CPT) for X given isjoint Y, enote ψ(x Y ), is a potential on XY that sums to 1, for eah onfiguration y om(y ). The unity-potential 1(v i )forv i is a funtion 1 mapping every element of om(v i ) to one. The unitypotential for a non-empty set X = {v 1,v 2,...,v k } of variables, enote 1(X), is efine as 1(X) =1(v 1 ) 1(v 2 ) 1(v k ). For simplifie notation, we may write {v 1,v 2,...,v k } as v 1,v 2,...,v k. A Bayesian network [13] is a pair (B,C). B enotes a irete ayli graph with vertex set U an C is a set of onitional probability tables (CPTs) {p(v i P (v i )) i =1, 2,...,n}, wherep (v i ) enotes the parents (immeiate preeessors) of v i B. The prout of CPTs in C is a joint probability istribution p(u). For example, the irete ayli graph in Figure 1 is alle the extene stuent Bayesian network (ESBN) [8]. We give CPTs in Table 1, where only binary variables are use in examples, an probabilities not shown an be obtaine by efinition. By the above, p(u) = p() p( ) p(i) p(g, i) p(h g, j). (1) We say X an Z are onitionally inepenent [16] given Y in p(u), enote I p (X, Y, Z), if given any x om(x), y om(y ), for all z om(z): p(x y, z) =p(x y), whenever p(y, z) > 0, where X, Y, Z U. Pearl [12] gave a metho, alle -separation, for etermining those inepenenies enoe in a irete ayli graph. The following is the efinition of -separation base on [8]. In a Bayesian network B, a trail (an unirete path) v 1,v 2,...,v n is ative given Y, if: (i) whenever we have a v-struture v i 1 v i v i+1,thenv i or one of its esenants are in Y ; (ii) no other noe along the trail is in Y.Notethatifv 1 or v n are in Y the trail is not ative.

3 Strong Completeness of Semantis in Bayesian Network Inferene 15 i g s h l j Fig. 1. The irete ayli graph of ESBN Table 1. CPTs for the ESBN in Figure 1 p() p( ) i g p(g, i) i p(i) g l p(l g) g j h p(h g,j) s l j p(j s, l) i s p(s i) We say that X an Z are -separate given Y in B, enote I B (X, Y, Z), if there is no ative trail between any variable v X an v Z given Y. In inferene, p(x E = e) is the most ommon query type, whih are useful for many reasoning patterns, inluing explanation, preition, interausal reasoning, an many more [8]. Here, X an E are isjoint subsets of U, ane is observe taking value e. We esribe a basi algorithm for omputing p(x E = e), alle variable elimination (VE), first put forth in [17]. We o not onsier alternative approahes to inferene suh as onitioning [6] an join tree propagation [1,2,10]. Inferene involves the elimination of variables. Algorithm 1, alle sum-out (SO), eliminates a single variable v from a set Φ of potentials [8], an returns the resulting set of potentials. The algorithm ollet-relevant simply returns those potentials in Φ involving variable v. Algorithm 1. SO(v,Φ) Ψ = ollet-relevant(v,φ) ψ = the prout of all potentials in Ψ τ = v ψ return (Φ Ψ) {τ}

4 16 C.J.Butz,W.Yan,anA.L.Masen SO uses Lemma 1, whih means that potentials not involving the variable being eliminate an be ignore. Lemma 1. [15] If ψ 1 is a potential on W an ψ 2 is a potential on Z, then the marginalization of ψ 1 ψ 2 onto W is the same as ψ 1 multiplie with the marginalization of ψ 2 onto W Z, wherew, Z U. The eviene potential for E = e, enote 1(E = e), assigns probability 1 to the single value e of E an probability 0 to all other values of E. Hene, for a variable v observe taking value λ an v {v i } P (v i ), the prout p(v i P (v i )) 1(v = λ) keeps only those onfigurations agreeing with v = λ. Algorithm 2, taken from [8], omputes p(x E = e) from a isrete Bayesian network B. VE alls SO to eliminate variables one by one. More speifially, in Algorithm 2, Φ is the set C of CPTs for B, X is a list of query variables, E is a list of observe variables, e is the orresponing list of observe values, an σ is an elimination orering for variables U XE,whereXE enotes X E. Algorithm 2. VE(Φ, X, E, e, σ) Multiply eviene potentials with appropriate CPTs While σ is not empty Remove the first variable v from σ Φ = sum-out(v, Φ) p(x, E = e) = the prout of all potentials ψ Φ return p(x, E = e)/ X p(x, E = e) As in [8], suppose the observe eviene for the ESBN is i =1anh =0an the query is p(j h = 0,i = 1). The weighte-min-fill algorithm [8] an yiel σ =(,, l, s, g). VE first inorporates the eviene: ψ(i =1) = p(i) 1(i =1), ψ(, g, i =1) = p(g, i) 1(i =1), ψ(i =1,s) = p(s i) 1(i =1), ψ(g, h =0,j) = p(h g, j) 1(h =0). To eliminate, the SO algorithm omputes ψ() = p() p( ). SO omputes the following to eliminate ψ(g, i =1) = ψ() ψ(, g, i =1). To eliminate l, ψ(g, j, s) = l p(l g) p(j l, s).

5 Strong Completeness of Semantis in Bayesian Network Inferene 17 SO omputes the following when eliminating s, ψ(g, i =1,j) = ψ(i =1,s) ψ(g, j, s). (2) s For g, SO an ompute: ψ(g, i =1,j) ψ(g, i =1) ψ(g, h =0,j) g = ψ(g, i =1,j) ψ(g, h =0,i=1,j) (3) g = ψ(h =0,i=1,j). Next, VE multiplies all remaining potentials as p(h =0,i=1,j) = ψ(i =1) ψ(h =0,i=1,j). Finally, VE answers the query by p(j h =0,i=1) = p(h =0,i=1,j) p(h =0,i=1,j). j 3 Unerstaning Semantis We review the urrent limite unerstaning of semantis in inferene. Kjaerulff an Masen [7] suggest that in working with probabilisti networks it is onvenient to enote istributions as potentials. In fat, the use of potentials is built into the stanar inferene algorithms (see the SO an VE algorithms, for instane). For example, suppose query p(j) is pose to the ESBN [8]. Even without eviene being onsiere, the initial step of VE is to regar CPTs as potentials, i.e., p(u) is fatorize as p(u) = ψ() ψ(, ) ψ(i) ψ(g, h, j). (4) By omparing (1) an (4), it is lear that semantis are estroye even before the CPTs in omputer memory are moifie. The notation use for potentials oes not onvey the semanti meaning of the probabilities omprising the potential. Darwihe [6] asribes meaning uring inferene by representing eah potential by what we will all eviene expane form, exept that prouts involving eviene potentials are taken. Let ψ be any potential onstrute by VE. The eviene expane form of ψ, enote F (ψ), is the unique expression efining how ψ was built using the multipliation an marginalization operators on the Bayesian network CPTs together with any appropriate eviene potentials.

6 18 C.J.Butz,W.Yan,anA.L.Masen For example, onsier potential ψ(g, i =1,j)in(2).F (ψ(g, i =1,j)), the eviene expane form, an be easily obtaine in a reursive manner as follows: ψ(i =1,s) ψ(g, j, s) s = s ψ(i =1,s) ( l (p(l g) p(j l, s))) (5) = s ((p(s i) 1(i =1)) ( l (p(l g) p(j l, s)))). Heneforth, parentheses are unerstoo an may not be shown. Unfortunately, the expane form by itself oes not iretly artiulate semantis. By semantis, wemeanthat a CPTψ(X Y ) onstrute by VE s manipulation of Bayesian network CPTs is not neessarily equal to the CPT p(x Y ) obtaine from the efine joint probability istribution p(u). For instane, it an be verifie that in the ESBN, p(h g, j) p(g, i) p() p( ) (6) proues the CPT ψ(g, h i, j) in Table 2 (left). In ontrast, the CPT p(g, h i, j) built from the joint istribution p(u) in(1)is shown in Table 2(right). Table 2. (left) CPT ψ(g,h i, j) built by (6). (right) CPT p(g,h i, j) built from p(u) in (1). i j g h ψ(g,h i, j) i j g h p(g,h i, j) Semantis in inferene are not well unerstoo. In their omprehensive an highly reommene text, Koller an Frieman [8] onsier the semantis of potential ψ(b,, ) built by eliminating variable a from the Bayesian network B in Figure 2 (left): ψ(b,, ) = a p(a) p(b a) p( a, ). (7)

7 Strong Completeness of Semantis in Bayesian Network Inferene 19 Koller an Frieman [8] inorretly state p(b, ) ψ(b,, ). (8) While this laim is almost always true, there are a few exeptions to refute it. For one ounter-example, eliminating variable a using the CPTs in Table 3 yiels: p(b, ) = ψ(b,, ). (9) Koller an Frieman [8] also state it must neessarily be the ase that p (b, ) = ψ(b,, ), (10) where p (U) is efine by a ifferent Bayesian network B - the one given in Figure 2 (right). Our objetive is to stipulate semantis in the urrent Bayesian network B - the one on whih inferene is being onute. a a b b Fig. 2. Bayesian networks B (left) an B (right) Table 3. Exeptional CPTs for B in Figure 2 (left) a p(a) a b p(b a) b p( b) a p( a, ) CPT Struture It is instrutive to review that, when eviene is not onsiere, eah potential built by VE is a CPT. A topologial orering [8] is an orering of the variables in a Bayesian network B so that for every ar (v i,v j )inb, v i preees v j in. For example, i g s l j h is a topologial orering of the irete ayli graph in Figure 1, but i g h l j s is not. Reall this feature of Bayesian networks, p(u) = p(v i P (v i )). v i U This an be establishe by showing 1 = p(v i P (v i )). U v i U More generally, we have the following two lemmas.

8 20 C.J.Butz,W.Yan,anA.L.Masen Lemma 2. [3] Consier a Bayesian network (B,C) on U. Given any nonempty subset X of U, v i X p(v i P (v i )) is a CPT ψ(x P (X)), wherep (X) = ( vi XP (v i )) X. Lemma 3. [3] When eviene is not onsiere, eah potential onstrute by VE is a CPT. Lemma 3 an be seen as first applying Lemma 1 on the eviene expane form of a potential built by VE, keeping in min E =, an then applying Lemma 2. For example, onsier the potential ψ built by (6), whih is alreay in eviene expane form. By applying Lemma 1, p(h g, j) p(g, i) p() p( ). (11) By Lemma 2, ψ(g, h i, j) = ψ(,, g, h i, j), Thus, the potential ψ built by (6) is, in fat, a CPT ψ(g, h i, j), in Table 2 (left). 5 Denoting Semantis The eviene expane form F (ψ) of any potential ψ onstrute by VE is in eviene normal form, if F (ψ) is written as γ N, where γ is the prout of 1 an all eviene potentials in F (ψ), an N is the same fatorization as F (ψ) exept without prouts involving eviene potentials. Reall ψ(g, h =0,i=1,j) in (3). The eviene expane form F (ψ) is p(h g, j) 1(h =0) p(g, i) 1(i =1) p() p( ), (12) an the eviene normal form γ N is 1(h =0,i=1) p(h g, j) p(g, i) p() p( ), (13) namely, γ =1(h =0,i=1)anN is (6). Lemma 4. The eviene expane form F (ψ) of any potential ψ onstrute by VE always an be equivalently written in normal form, i.e., F (ψ) =γ N. Proof. Sine eviene variables are never marginalize in VE, the laim follows from Lemma 1.

9 Strong Completeness of Semantis in Bayesian Network Inferene 21 Observe that, by Lemma 3, N in eviene normal form is a CPT. We may enote eviene normal form γ N simply as N with eviene γ unerstoo, sine γ only serves to selet onfigurations of N agreeing with the eviene. We now turn to enoting semantis. To unerstan when N = p(x Y ) in eviene normal form, some terminology is neee. A path from v 1 to v n is a sequene v 1,v 2,...,v n with ars (v i,v i+1 ) in B, i =1,...,n 1. With respet to a variable v i, we efine three sets: (i) the anestors of v i, enote A(v i ), are those variables having a path to v i ; (ii) the esenants of v i, enote D(v i ), are those variables to whih v i has a path; an, (iii) the hilren of v i are those variables v j suh that ar (v i,v j )isinb. The anestors of a set X U are efine as A(X) =( vi XA(v i )) X. The esenants D(X) are efine similarly. I B (X, Y, Z) means an inepenene statement I(X, Y, Z) [13] hols in B by -separation, where X, Y, Z U. We now give the Semantis in Inferene (SI) algorithm, whih uses -separation to enote the semantis of any potential ψ built by VE on B. Eah potential ψ onstrute by VE is represente in eviene normal form ψ(x Y ). If the semantis of B ensure the ψ(x Y )=p(x Y ), then ψ is enote as p B (X Y ); otherwise, it is enote as φ B (X Y ). S is the set of variables marginalize in F (ψ). A(XS)an D(XS) are ompute from the transitive losure, enote T,ofB [5]. Algorithm 3. SI(ψ) Compute the eviene expane form F (ψ) ofψ Compute the eviene normal form γ N of F (ψ) Compute the CPT struture ψ(x Y )of N Compute Z = A(XS) D(XS) Compute X 1 = X P (Z) if I B (X 1,,Y)holsinB by -separation return p B (X Y ) else return φ B (X Y ) Reall ψ(g, i =1,j) in (2). The eviene expane form is (6). Its eviene normal form γ N is γ =1(i =1)anN = ψ(j g, i). Now X = {j}, Y = {g, i} an S = {l, s}. By the transitive losure T of the ESBN, A(XS)={,, i, g} an D(XS)={h}. Hene, Z =, P (Z) =, anx 1 =. Trivially, I B (X 1,,Y) hols. Thus, SI enotes ψ(g, i =1,j) in (2) as p B (j g, i =1). Now onsier ψ(g, h = 0,i= 1,j) in (3). The eviene expane form is (12). The eviene normal form γ N is (13). Here N = ψ(g, h i, j), as seen in (6). With X = {g, h}, Y = {i, j} an S = {, }, fromt on the ESBN we have A({,, g, h}) ={i, j, l, s} an D({,, g, h}) ={j, l}. Thus,Z = {j, l}, giving P (Z) ={g, s} an X 1 = {g}. Now,I B (X 1,,Y) oes not hol. Thereby, SI enotes ψ(g, h =0,i=1,j) in (3) as φ B (g, h =0 i =1,j). In this example, there is a path from XS = {,, g, h} to XS through Z = {j, l}, starting at X 1 = {g}. With X 1 = {g} an Y = {i, j}, wefouson I B (g,,ij). Note that when eiing semantis of ψ(x Y ), the inepenene

10 22 C.J.Butz,W.Yan,anA.L.Masen to be teste is I B (X 1,,Y) an not I B (XS,Y,A(XSY)). In Figure 2 (left), I B (ab,, ) hols, but p(b, ) ψ(b, ) in (8) is possible. 6 Theoretial Founation We present four salient features of SI. Only the proofs of time omplexity an strong ompleteness are shown ue to spae onsierations. Theorem 1. Let ψ be any potential built by VE uring exat inferene in a isrete Bayesian network with n variables. Then the time omplexity of the SI algorithm to etermine the semantis of ψ is O(n 3 ). Proof. As ψ may require n 1 multipliations an n marginalizations, omputing F (ψ) takes2n steps. The normal form γ N an be eie in linear time, as an the CPT struture ψ(x Y )ofn. The transitive losure T of the irete ayli graph an be ompute in O(n 3 )[5].LetXS be a set of k variables, 1 k n. ThenA(XS)anD(XS) eah an be ompute in O(k n). Now Z an X 1 eah an be ompute in O(n 2 ). Testing I B (X 1,,Y) is linear in the size of B [6]. Thus, the semantis of ψ an be etermine by SI in O(n 3 ). Theorem 2. In a Bayesian network (B,C) efining a joint istribution p(u), suppose VE omputes a potential ψ whose eviene normal form is γ N. IfSI enotes the semantis of N as p B (X Y ), thenn = p(x Y ). Theorem 2 guarantees that if SI enotes the semantis of a VE potential ψ as γ p B (X Y ), then ψ = γ p(x Y ). Reall potential ψ(g, i =1,j) in (2). As illustrate in Table 4, Theorem 2 ensures that ψ(g, i =1,j)isequaltop(j g, i = 1), sine SI enotes it as p B (j g, i =1). Table 4. Potential ψ(g,i = 1,j)in(2)is p(j g,i = 1) i g j p B(j g,i =1) ψ(g,i = 1,j) = p(j g,i = 1) = With respet to inferene, the question of ompleteness is this. Can SI etermine the semantis of every VE potential efine with respet to the joint istribution? The answer is no. Theorem 3. In a Bayesian network B on U, suppose VE omputes a potential ψ whose eviene normal form is γ N. If SI enotes the semantis of N as φ B (X Y ), there exists a set C of CPTs for B efining a joint istribution p(u) suh that N p(x Y ).

11 Strong Completeness of Semantis in Bayesian Network Inferene 23 Theorem 3 states that whenever SI iniates that a potential is not efine with respet to the joint istribution, then this is true for at least one set of CPTs for the given Bayesian network. Reall one again ψ(g, h =0,i=1,j)in(3),whih SI enotes as φ B (g, h =0,l i =1,j). With respet to p(u) efine by the CPTs in Table 1, we have ψ(g, h =0,i=1,j) p(g, h =0 i =1,j). However, Theorem 3 an be mae signifiantly stronger. Lemma 5. [11] Exept for a measure zero set in the spae of all joint istributions p(u) efine by all isrete Bayesian networks (B,C), the inepenenies satisfie by p(u) are preisely those satisfie by -separation in B. 1 Lemma 5 says that for nearly all hoies C of CPTs for a Bayesian network B efining p(u), -separation perfetly haraterizes the inepenenies in p(u), i.e., for X, Y, Z U, I p (X, Y, Z) I B (X, Y, Z). Theorem 4. Exept for a measure zero set in the spae of all joint istributions p(u) efine by all isrete Bayesian networks (B,C), for any potential ψ built by VE, ψ = γ p(x Y ) SI enotes ψ as p B (X Y ), where γ N is the eviene normal form of ψ. Proof. ( ) Suppose VE onstruts a ψ whose eviene normal form is γ N an whose semantis are efine with respet to p(u). By ontraposition, suppose SI enotes N as φ B (X Y ). By SI, I B (X 1,,Y) oes not hol. Then, by Lemma 5, I p (X 1,,Y) oes not hol in essentially all possible p(u) efine over B. It follows that for eah suh p(u), γ p(x Y ) γ N. A ontraition to our initial assumption. Therefore, SI orretly enotes the potential ψ as p B (X Y ). ( ) Follows iretly from Theorem 2. Let B be any Bayesian network. Theorem 4 states that for nearly all hoies C of CPTs for B, the SI algorithm orretly enotes the semantis of potentials onstrute by VE uring exat inferene on B. 1 A set has measure zero if it is infinitesimally small relative to the overall spae [8].

12 24 C.J.Butz,W.Yan,anA.L.Masen 7 Conlusion We exten -separation s role from etermining the minimum amount of information neee to answer a query p(x E = e) [12] to also giving the semantis of the potentials onstrute when answering p(x E = e). Our results ontribute to a eeper unerstaning of Bayesian networks, sine semantis of VE s intermeiate fators are now artiulate with respet to the joint istribution. The main result (Theorem 4) showe that our SI algorithm orretly enotes the semantis of inferene in nearly all Bayesian networks. Future work will inlue applying the results here to ifferential semantis in Bayesian networks [6,9]. Referenes 1. Butz, C.J., Hua, S., Konkel, K., Yao, H.: Join Tree Propagation with Prioritize Messages. Networks 55(4), (2010) 2. Butz, C.J., Konkel, K., Lingras, P.: Join Tree Propagation Utilizing both Ar Reversal an Variable Elimination. Int. J. Approx. Reasoning 52(7), (2011) 3. Butz, C.J., Yan, W., Lingras, P., Yao, Y.Y.: The CPT Struture of Variable Elimination in Disrete Bayesian Networks. In: Ras, Z.W., Tsay, L.S. (es.) Avanes in Intelligent Information Systems. SCI, vol. 265, pp Springer, Heielberg (2010) 4. Castillo, E., Gutiérrez, J., Hai, A.: Expert Systems an Probabilisti Network Moels. Springer, New York (1997) 5. Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introution to Algorithms. MIT Press, Cambrige (2009) 6. Darwihe, A.: Moeling an Reasoning with Bayesian Networks. Cambrige University Press, New York (2009) 7. Kjaerulff, U.B., Masen, A.L.: Bayesian Networks an Influene Diagrams. Springer, New York (2008) 8. Koller, D., Frieman, N.: Probabilisti Graphial Moels: Priniples an Tehniques. MIT Press, Cambrige (2009) 9. Masen, A.L.: A Differential Semantis of Lazy AR Propagation. In: 21st Conferene on Unertainty in Artifiial Intelligene, pp Morgan Kaufmann, San Mateo (2005) 10. Masen, A.L.: Improvements to Message Computation in Lazy Propagation. Int. J. Approximate Reasoning 51(5), (2010) 11. Meek, C.: Strong Completeness an Faithfulness in Bayesian Networks. In: 11th Conferene on Unertainty in Artifiial Intelligene, pp Morgan Kaufmann, San Mateo (1995) 12. Pearl, J.: Fusion, Propagation an Struturing in Belief Networks. Artif. Intell. 29, (1986) 13. Pearl, J.: Probabilisti Reasoning in Intelligent Systems: Networks of Plausible Inferene. Morgan Kaufmann, San Franiso (1988) 14. Pearl, J.: Belief Networks Revisite. Artif. Intell. 59, (1993) 15. Shafer, G.: Probabilisti Expert Systems. SIAM, Philaelphia (1996) 16. Wong, S.K.M., Butz, C.J., Wu, D.: On the Impliation Problem for Probabilisti Conitional Inepeneny. IEEE Trans. Syst. Man Cybern. A 30(6), (2000) 17. Zhang, N.L., Poole, D.: A Simple Approah to Bayesian Network Computations. In: 7th Canaian Conferene on Artifiial Intelligene, pp Springer, New York (1994)

Bayesian Network Inference Using Marginal Trees

Bayesian Network Inference Using Marginal Trees Cory J. Butz 1?, Jhonatan de S. Oliveira 2??, and Anders L. Madsen 34 1 University of Regina, Department of Computer Science, Regina, S4S 0A2, Canada butz@cs.uregina.ca