Maximal Prime Subgraph Decomposition of Bayesian Networks: A Relational Database Perspective

Maximal Prime ubgraph ecomposition of ayesian Networks: Relational atabase Perspective an Wu chool of omputer cience University of Windsor Windsor Ontario anada N9 3P4 Michael Wong epartment of omputer cience University of Regina Regina askatchewan anada 4 02 bstract maximal prime subgraph decomposition junction tree (MP-J) is a useful computational structure that facilitates lazy propagation in ayesian networks (Ns). graphical method was proposed to construct an MP-J from a N. In this paper, we present a new method from a relational database (R) perspective which sheds light on the semantic meaning of the previously proposed graphical algorithm. 1. Introduction ayesian networks have been widely used for reasoning with uncertainty (Pearl 1988). Normally, a ayesian network is transformed into a junction tree on which probabilistic reasoning is conducted (auritzen & piegelhalter 1988; Huang & arwiche 1996). Various attempts have been made to improve the efficiency of probabilistic reasoning, for instance, the lazy propagation method (Madsen & Jensen 1998). Very recently, it was proposed to transform a ayesian network not into a junction tree, but into a maximal prime subgraph decomposition junction tree (MP-J) on which the lazy propagation can be greatly facilitated. graphical algorithm has been developed to construct the MP- J (Olesen & Madsen 2002). On the other hand, it has long been noted that there exists an intriguing relationship between Ns and Rs (Wen 1991) such that many problems in Ns can be considered as similar problems in Rs (Wong, Wu, & utz 2002; Wong & utz 2001). In this paper, we suggest a new algorithm for constructing an MP-J from a R perspective after carefully examining the graphical algorithm in (Olesen & Madsen 2002). y investigating the conditional independencies (Is) encoded in an MP-J, we show that constructing an MP-J from a N is equivalent to obtaining a conflict free set of Is encoded in the N. his new perspective makes it possible to apply a well-developed algorithm for constructing an acyclic database scheme in Rs to constructing an MP-J in Ns. his new method sheds light on the semantic meaning of the graphical method in (Olesen opyright c 2005, merican ssociation for rtificial Intelligence (www.aaai.org). ll rights reserved. & Madsen 2002) and further confirms the strong relationship between Ns and Rs. he paper is organized as follows. In ection 2, we review pertinent background. In ection 3, we discuss the relationship between Ns and Rs which serves as the basis for the new proposed algorithm. In ection 4, we present the proposed new method and discuss its complexity. We conclude the paper in ection 5. 2. ackground ayesian Networks ayesian network (N) defined over a set V = { i i = 1,..., n} of finite discrete variables is a directed acyclic graph (G) denoted, augmented with a set of conditional probability distributions (Ps). ach vertex in corresponds one-to-one to a variable in V (we thus use the terms vertex, node, and variable interchangeably). he parents of a node i in is denoted π i. ach variable i V is associated with a P p( i π i ) such that the joint probability distribution (JP) p(v ) = p( i V i π i ). he G of a N encodes Is satisfied by the JP p(v ). We use the notation Y Z, where, Y, and Z are disjoint subsets of V, to denote that given Y, and Z are conditionally independent (Pearl 1988). I is full if Y Z = V (by Y Z, we mean Y Z), otherwise, it is embedded. Junction rees he G of a N is normally transformed into a junction tree for probabilistic reasoning. he transformation consists of two graphical operations, namely, moralization and triangulation (auritzen & piegelhalter 1988). he moralized graph of a G, denoted M, is an undirected graph obtained by connecting every pair of nodes with a common child which are not already connected in and then dropping the directionality of all directed edges. M is then triangulated by adding a chord, called fill-in edge, to every cycle whose length is greater than three. triangulation is minimal if removal of any fill-in edge will result in an untriangulated graph. he resulting triangulated graph is denoted. junction tree, written as, is constructed by identifying all the cliques (i.e., maximal complete subgraphs) of and arranging them as nodes of a tree to sat-

(i) ( ii) igure 1: (i) he G of the sia N. (ii) he moralized graph M of the G. (iii) he triangulated graph of the G. (iv) he junction tree constructed from. isfy the running intersection property (auritzen & piegelhalter 1988), which requires that every clique on the path between two cliques and in contains. Note that multiple junction trees may be produced from on the arrangements of the cliques, satisfying the running intersection property. If cliques and are connected by an edge in, this edge is labelled with. he set = { there is an edge between and } is called the separator set of, each element in is called a separator of. xample 1 onsider the sia N (auritzen & piegelhalter 1988) whose G is in ig. 1 (i). Its moralized graph M and triangulated graph are shown in ig. 1 (ii) and (iii), respectively. he dotted edges (, ) 1 and (, ) in M in ig. 1 (ii) were added during moralization. he fill-in edge (, ), dotted in ig. 1 (iii), was added during triangulation. here are 6 cliques in the triangulated graph and they have been arranged as a junction tree in ig. 1 (iv). Note that in ig. 1 (iv), the oval represents a clique identified from, the square box represents a separator in. MP Junction rees Very recently, it was suggested to transform a N not into a junction tree, but into an MP-J to facilitate lazy propagation (Olesen & Madsen 2002). efore we introduce MP- J, the following pertinent graph terminologies are needed. We use G(V ) (or G) to denote an undirected graph consisting of a finite set V of vertices. efinition 1 et the triplet (V 1,, V 2 ) denote a partition of V where V 1 V 2 = V. If every path in G(V ) between i V 1 and j V 2 contains a vertex in, then is a separator of G(V ) with respect to (V 1,, V 2 ). urthermore, if the subgraph induced by the separator, i.e., G(), is complete, then is a complete separator of G(V ) with respect to (V 1,, V 2 ). efinition 2 partition (V 1,, V 2 ) is a decomposition of a graph G(V ) if is a complete separator. he complete separator is further called a minimal complete separator if for any, (V 1,, V 2 ( )) is not a decomposition anymore. ( iii) 1 We use ( i, j) to denote a directed edge in a G. (iv) Note that each separator in (i.e., the separator set of junction tree ) is a minimal complete separator of the triangulated graph from which the junction tree is constructed. efinition 3 n undirected graph is a prime graph if it has no complete separator. If G is a subgraph of G and G is also a prime graph, then G is a prime subgraph of G. G(V ) is a maximal prime subgraph (mp-subgraph) of G(V ) if G(V ) is a prime subgraph of G(V ) and for any other subgraph G(V ) of G(V ), where V V, G(V ) is not a prime subgraph. efinition 4 (Olesen & Madsen 2002) et G be an undirected graph, its mp- subgraph decomposition (MP) is the set of induced mp-subgraphs of G. MP junction tree (MP-J) of G is a junction tree whose nodes are the mpsubgraphs of G. efinition 5 onsider a N with its G, its MP is the set of induced mp-subgraphs of the moralized graph M (Olesen & Madsen 2002). MP-J of the N is a junction tree whose nodes are the mp-subgraphs of M. ( i) (ii) igure 2: (i) he moralized graph M of the sia N in ig. 1. (ii) he induced mp-subgraphs of M. (iii) he MP-J of the sia N. xample 2 onsider the moralized graph M of the sia N in ig. 2 (i). M induces 5 mp-subgraphs as shown in ig. 2 (ii). n MP-J of the sia N can be constructed by arranging these 5 mp-subgraphs to satisfy the running intersection property as shown in ig. 2 (iii). Junction rees and Hypertrees he notion of junction tree has a close tie with the notion of hypertree which is widely used in Rs (Maier 1983). efinition 6 hypergraph is a pair (N, H), where N is a finite set of vertices and H is a set of hyperedges which are arbitrary subsets of N (hafer 1991), that is, H = {h 1,..., h n h i N }. We usually use H to denote the hypergraph (N, H). efinition 7 We say that a hyperedge h i in a hypergraph H is a twig if there exists another hyperedge h j in H, distinct from h i, such that ( h H {h i } h) h i = h i h j. We call any such h j a branch for the twig h i. he notions of twig and branch capture the scenario that the intersection of a hyperedge, namely, the twig, with the rest of the hypergraph is contained by a single hyperedge, namely, the branch. (iii)

efinition 8 hypergraph H is a hypertree (hafer 1991) or acyclic if its hyperedges can be ordered, say h 1, h 2,..., h n, such that h i is a twig in {h 1, h 2,..., h i }, for i = 2,..., n. We call any such ordering a tree (hypertree) construction ordering for H. efinition 9 Given a tree construction ordering h 1, h 2,..., h n, we can choose, for i from 2 to n, an integer j(i) such that 1 j(i) i 1 and h j(i) is a branch for h i in {h 1, h 2,..., h i }. We call a function j(i) that satisfies this condition a branching function for H and the tree construction ordering h 1, h 2,..., h n. Note that for a given hypertree, there might exist multiple tree construction orderings; for a given tree construction ordering, there might exist multiple choices of branching functions (hafer 1991). Given a tree construction ordering h 1, h 2,..., h n for a hypertree H and a branching function j(i) for this ordering, we can construct a set denoted H whose elements are the intersections of the twigs and their respective branches, i.e., H = {h j(2) h 2, h j(3) h 3,..., h j(n) h n }. Note that H is the same for any tree construction ordering of a given hypertree (hafer 1991) and we call H the separator set of H. Note also that for any hypertree H, there is a unique corresponding triangulated undirected graph denoted G H which has the same nodes as H and whose edges are constructed by connecting every two nodes that belongs to the same hyperedge of H. On the other hand, for any triangulated graph G, its cliques, being considered as hyperedges, constitute a unique hypertree denoted H G (Maier 1983). In other words, there is a one-to-one correspondence between triangulated undirected graphs and hypertrees. ince a hypertree corresponds to a triangulated undirected graph and a triangulated undirected graph may produce multiple junction trees, it is not surprising that given a hypertree, there exists a set of junction trees each of which corresponds to a particular tree construction ordering and a branching function (hafer 1991). his property implies that hypertrees are more versatile than junction trees. It is also trivial to see that given a junction tree, there always exists a unique corresponding hypertree representation denoted H whose hyperedges are the cliques in the junction tree. herefore, we will treat a junction tree as if it is a hypertree H whenever appropriate. urthermore, the separator set H of the hypertree H is exactly the same as the separator set of the junction tree, and both separator sets correspond exactly to the minimal complete separators of the triangulated graph G H. herefore, a junction tree in essence is a hypertree (with a particular tree constructing ordering and branching function) and the problem of constructing junction trees can be considered as a more general problem of constructing hypertrees. xample 3 onsider the hypertree H in igure 3 (i). It can be easily verified that the tree construction ordering, h 1 =, h 2 =, h 3 =, h 4 = H, h 5 =, h 6 = G, together with the branching function j(2) = 1, j(3) = 2, j(4) = 2, j(5) = 3, j(6) = 3, defines the junction tree in igure 3 (ii). he same ordering with a different branching function, namely, j(2) = 1, j(3) = H (i) G H (ii) G igure 3: (i) he hypertree H. (ii), (iii), (iv) hree possible junction tree representations of H in (i), respectively. 2, j(4) = 3, j(5) = 3, j(6) = 3, defines a different junction tree in igure 3 (iii). onsider a different tree construction ordering, i.e., h 1 =, h 2 =, h 3 =, h 4 =, h 5 = G, h 6 = H, together with the branching function j(2) = 1, j(3) = 2, j(4) = 3, j(5) = 3, j(6) = 5, it defines the junction tree in igure 3 (iv). It can also be easily verified that the separator sets of those junction trees in igure 3 (ii), (iii), and (iv) are not only the same, but also identical to the separator set of the hypertree H in igure 3 (i). 3. Ns and Rs It has been noticed that there exists a strong relationship between Ns and Rs. here are two previously obtained results that contribute to the new proposed algorithm. (1) he relationship between full Is and MVs. (2) onstructing an acyclic database scheme. et R be a finite set of symbols, called attributes. We define a database scheme R = {R 1, R 2,..., R n } to be a set of subsets of R, where R = R 1... R n. ach R i in R is a relation scheme. relation defined over a scheme R i is denoted r[r i ]. database scheme R can be treated as if it is a hypergraph each of whose hyperedges is one of the relation schemes in R. On the other hand, for a hypergraph, we can treat each of its hyperedge as a relation scheme and all its hyperedges constitute a database scheme. database scheme R is acyclic if its corresponding hypergraph is acyclic (Maier 1983). Multivalued ependence (MV) is an important class of data dependence that has been intensively studied in Rs. efinition 10 relation r[r] is said to satisfy the (full) multivalued dependency (MV) (Maier 1983), denoted, Y Z, if r[r] = r[y ] r[y Z], where denotes the natural join operator in relational algebra, R = Y Z, r[y ] and r[y Z] are projections (Maier 1983) of r[r] onto schemes Y and Y Z, respectively. Y is called the key of this MV. efinition 11 Given a set M of MVs, the left hand sides of the MVs in M are the keys of M. MV Y Z is said to split a set W of attributes if W and W Z. efinition 12 set M of (full) MVs defined over a set R of attributes is conflict free (Maier 1983) if (i) the keys of M are not split by any MV in M, and (ii) M satisfies H (ii) G H (iv) G

the intersection property (Maier 1983), that is, if ZY W and ZW Y are in M, then Z Y W M, where Y ZW = R. Recently, a thorough study (Wong, utz, & Wu 2000) has revealed that MVs and full Is, even though in different domains, correspond exactly to each other such that their implication problems coincide. his coincidence implies that the notions of keys and split can be carried over to full Is so that we can define conflict free for full Is in the exact same fashion as we did for MVs. efinition 13 (Wong, utz, & Wu 2000) set of full Is defined over a set V of variables is conflict free if (i) the keys of are not split by any I in, and (ii) satisfies the intersection property, that is, if ZY W and ZW Y are in, then Z Y W, where Y ZW = V. he significance of conflict free MVs is that they can be used to construct an acyclic database scheme (a hypertree) (Maier 1983). n efficient algorithm (ien 1982), referred to as ien-lgorithm-mvs in this paper, has been developed to construct a unique hypertree H from a set of conflict free MVs. 4. he New Relational atabase Method In this section, we will present a new method for constructing the MP-J of a N motivated by the I information encoded in the MP-J. We first briefly review the existing graphical algorithm in (Olesen & Madsen 2002), referred to as Olesen-Madsen-MP-J, using an example. he observations revealed by the examination motivate the development of the new method. he Olesen-Madsen-MP-J algorithm takes a G as input, moralizes and triangulates it. he algorithm then removes any redundant fill-in edges so that a minimal triangulation is obtained and a normal junction tree is constructed. fter that, it checks each separator of the resulting junction tree to decide whether it needs to aggregate the incidental cliques connected by the separator. onsider the sia N in xample 1. fter constructing the junction tree, the algorithm picks a junction tree separator and tests whether induces a complete subgraph of M (i.e., test whether M () is a complete subgraph of M ). If M () is not a complete subgraph, then the incidental cliques of, say, and in the junction tree, will be aggregated to obtain a bigger node which is the union of and. or instance, when testing the separator in igure 1 (iv), it can be verified that the subgraph induced by nodes and is not a complete subgraph of M because and are not connected in the moralized graph M as shown in igure 1 (ii). ince is connecting cliques and, they have to be aggregated to obtain a bigger node, i.e,, to replace the cliques, and the separator, which results in igure 2 (iii). he new structure in igure 2 (iii) is still a junction tree though. It is important to note that after the cliques and have been aggregated, the separator has been eliminated. his process repeats until all the separators in have been examined. It can be easily verified that all the other separators in the separator set induce complete subgraphs so that there are no more aggregations. herefore, the final resulting MP-J after all necessary aggregations, shown in igure 2 (iii), keeps only those separators originally in which induce complete subgraphs of M. It is well known that for any junction tree, each of its separators induces a full I satisfied by the JP defined by the N (Pearl 1988). More specifically, consider a junction tree consisting of cliques i, i = 1,..., k, with its separator set. If we delete a separator and its incidental edges from, will be separated into two disconnected parts. Without loss of generality, assuming one part contains 1,..., m, the other part contains the rest, the I induced by is m i=1 i k j=m+1 j. lso note that each induced I corresponds to a decomposition ( m i=1 k i,, j=m+1 j ) of the triangulated graph from which the junction tree is constructed, with being a minimal complete separator. xample 4 onsider the MP-J shown in ig. 2 (iii) consisting of cliques 1 =, 2 =, 3 =, 4 =, and 5 =, with its separator set = {,,, }. he I induced by the separator is 1 3 5 2 4, or. imilarly, the Is induced by the separators, and, are and,, and, respectively. Recall the ien-lgorithm-mvs algorithm which constructs an acyclic database scheme from a conflict free set of MVs, because of the correspondence between full Is and MVs, this algorithm has been carried over to the domain of Ns. In (Wong & utz 2001), an algorithm referred to as ien-lgorithm-is in this paper, was designed to construct a unique hypertree H from a set of conflict free full Is 2. Recall the fact that a junction tree has a corresponding hypertree and a junction tree can be derived from a hypertree, this turns the problem of constructing an MP-J into the problem of constructing a hypertree from which this MP- J can be derived. urthermore, the problem of constructing a hypertree, according to the algorithm ien-lgorithm-is, can be turned into the problem of obtaining a conflict free set of full Is. We therefore propose an alternative algorithm for constructing MP-J. We put our focus on how to obtain a set of conflict free full Is that can be used to construct the hypertree that corresponds to an MP-J. In xample 4, we demonstrated how to obtain a set of full Is from a junction tree. It was shown that the full Is identified as in xample 4 are conflict free and they are responsible for constructing the hypertree that corresponds to the junction tree from which this set of full Is is identified. In the context of MP-Js, if we can obtain a set of conflict free full Is from an MP-J without actually constructing it in the first place, then we can use 2 ee reference (Wong & utz 2001) for detailed discussions on ien-lgorithm-is and examples.

this set as input to the algorithm ien-lgorithm-is to construct a hypertree that corresponds to the MP-J we want to construct. he following theorem guarantees that such a conflict free set of Is is obtainable. onsider a G and its MP-J, let denote the set of conflict free full Is induced by separators in. heorem 1 Y Z if and only if (, Y, Z) is a decomposition of M with Y being the minimal complete separator. Proof 1 ( ) If Y Z, then Y is a minimal complete separator of the triangulated graph G (from which is constructed) and (, Y, Z) is a decomposition of G. If (, Y, Z) is a decomposition of G, then it implies that for every path in G from i to j Z, this path must pass through at least one vertex in Y. However, since every path from i to j in the moralized graph M is also a path in G, hence every path in M from to Z passes through Y, therefore (, Y, Z) is a decomposition of M with Y being the minimal complete separator. ( ) In (Wong, Wu, & utz 2002), it was proved that the decomposition (, Y, Z) of M with Y being a minimal complete separator is also a decomposition in every triangulated graph of M. herefore, Y is a junction tree separator in any junction tree that may be produced from M, including the MP-J. ince the algorithm Olesen-Madsen- MP-J only eliminates junction tree separators which do not induce complete subgraphs in M, the decomposition (, Y, Z) will be kept in the final resulting MP-J. herefore, Y Z. heorem 1 implies that the set of conflict free full Is encoded in an MP-J can be obtained in advance from the moralized graph of the original given G without constructing the MP-J. More precisely, the set can be obtained by examining every decomposition (, Y, Z) of the moralized graph such that Y is a minimal complete separator. fter the set is obtained, we can then use it as input to the algorithm ien-lgorithm-is to construct a hypertree and afterwards the MP-J. his analysis naturally results in the following algorithm for constructing an MP-J. lgorithm Relational-MP-J Input: a ayesian network with its G ; Output: an MP-J of. tep 1. Moralize the G to obtain M. tep 2. Identify each decomposition (, Y, Z) of M where Y is a minimal complete separator of M, and put the I Y Z in a set. tep 3. Invoke the ien-lgorithm-is with as input to obtain a hypertree H. tep 4. Return a junction tree constructed from G H. hronologically, prior to the algorithm Olesen-Madsen- MP-J, there existed another graphical algorithm (emer 1993), referred to as Graphical-MP, which can find all the mp-subgraphs of an arbitrary undirected graph (without going through the auxiliary junction tree as Olesen-Madsen- MP-J did). he Graphical-MP algorithm was designed as a pure graphical procedure which has applications in some graph problems and has the time complexity of O(ne), where n is the number of vertices and e is the number of edges in the undirected graph. he Olesen-Madsen-MP-J algorithm constructs the MP-J in the context of Ns via a junction tree obtained by minimal triangulation. his algorithm is also a graphical algorithm and specifically designed for Ns. he worst case complexity of this algorithm is dominated by triangulating the moralized graph which has the time complexity O(ne) while the time complexity of all other steps are no worse than this (Olesen & Madsen 2002). In other words, the algorithm Olesen-Madsen-MP-J does not exhibit any improvement over the algorithm Graphical-MP, as far as the complexity is concerned. Nevertheless, it provides a different graphical perspective for constructing an MP-J in the domain of Ns, namely, making full use of the intermediate junction tree obtained instead of directly working on the underlying moralized graph. onsider the new algorithm Relational-MP-J, the complexity of moralizing the input G to obtain M in step 1 is O(n 2 ) (Olesen & Madsen 2002). he complexity of identifying minimal complete separator decompositions in M to form in step 2 is O(ne) (emer 1993). onstructing the hypertree H from in step 3 takes O( n). inally, constructing the MP-J from H in step 4 takes O(n 2 ) to finish (Olesen & Madsen 2002). gain, the time complexity of the whole algorithm is dominated by step 2, i.e., O(ne). In other words, it has the same worst case time complexity with the Graphical-MP and Olesen-Madsen- MP-J algorithms. However, we want to emphasize that the new proposed algorithm makes the following two contributions: (1) It provides a semantic explanation for the graphical construction of the MP-J in (emer 1993; Olesen & Madsen 2002) based on the information encoded in an MP-J. hat is, constructing an MP-J is equivalent to obtaining a set of conflict full Is from the moralized graph of a given G. (2) ince the notion of conflict free Is is inherited and adapted from the notion of conflict free MVs in Rs, the new proposed algorithm makes full use of the existing algorithm ien-lgorithm-is, which was inherited from the database algorithm ien-lgorithm-mvs, to construct an MP-J. In other words, the core construction step of the algorithm Relational-MP-J is an application of the result on constructing a hypertree in database theory being applied in the N domain. his revelation farther confirms the intriguing relationship between Ns and Rs. 5. iscussion he bottleneck of the algorithm Olesen-Madsen-MP-J lies in the triangulation step whose complexity dominates the whole algorithm. ven though the new proposed algorithm does not need triangulation step, it needs to identify all the decompositions with minimal complete separators. It is perhaps worth pursuing how to shift the bottleneck of triangulation to decomposition identification which may result in a more efficient (perhaps approximate) algorithm.

lthough the new proposed algorithm is within the context of Ns, the idea of identifying and obtaining a conflict free set of full Is from the moralized graph of a G can also be applied to an arbitrary undirected graph, not necessarily a moralized graph of a G. herefore, the proposed method for constructing an MP-J is not confined to the domain of Ns. he relationship betwen conflict free MVs (Is) and hypertrees suggests that any graphical problem of constructing a hypertree can be equivalently considered as the problem of how to obtain a set of conflict free set of MVs (Is). his provides a new (algebraic) perspective for solving the graphical problem of constructing hypertrees (junction trees) in future research. or instance, the proposed algorithm may be possibly applied to obtain a hierarchical Markov network representation of a N (Wong, utz, & Wu 2001). I ransactions on Knowledge and ata ngineering 13(3):395 415. Wong,.; utz,.; and Wu,. 2000. On the implication problem for probabilistic conditional independency. I ransactions on ystem, Man, ybernetics, Part : ystems and Humans 30(6):785 805. Wong,.; utz,.; and Wu,. 2001. On undirected representations of bayesian networks. In M IG/IR Workshop on Mathematical/ormal Models in Information Retrieval (M/IR), 52 59. Wong,.; Wu,.; and utz,. 2002. riangulation of bayesian networks: a relational database perspective. In he hird International onference on Rough ets and urrent rends in omputing (R 2002), NI 2475, 389 397. cknowledgment he authors wish to thank the referees for their helpful and constructive criticism. References Huang,., and arwiche,. 1996. Inference in belief networks: procedural guide. International Journal of pproximate Reasoning 15(3):225 263. auritzen,., and piegelhalter,. 1988. ocal computation with probabilities on graphical structures and their application to expert systems. Journal of the Royal tatistical ociety 50:157 244. emer, H. 1993. Optimal decomposition by clique separators. iscrete Mathematics 113:99 123. ien, Y. 1982. On the equivalence of database models. Journal of the M 29(2):336 362. Madsen,.., and Jensen,. V. 1998. azy propagation in junction trees. In ooper, G.., and Moral,., eds., Proceedings of the 14th onference on Uncertainty in rtificial Intelligence (UI-98), 362 369. an rancisco: Morgan Kaufmann. Maier,. 1983. he heory of Relational atabases. Rockville, Maryland: omputer cience Press. Olesen, K., and Madsen,. 2002. Maximal prime subgraph decomposition of bayesian networks. I ransactions on ystems, Man and ybernetics, Part : ybernetics 32(1):21 31. Pearl, J. 1988. Probabilistic Reasoning in Intelligent ystems: Networks of Plausible Inference. an rancisco, alifornia: Morgan Kaufmann Publishers. hafer, G. 1991. n axiomatic study of computation in hypertrees. chool of usiness Working Papers 232, University of Kansas. Wen, W. 1991. rom relational databases to belief networks. In eventh onference on Uncertainty in rtificial Intelligence, 406 413. Morgan Kaufmann Publishers. Wong,., and utz,. 2001. onstructing the dependency structure of a multi-agent probabilistic network.