AALBORG UNIVERSITY From Influence Diagrams to Junction Trees by Frank Jensen Finn V. Jensen Soren L. Dittmer R 94-2013 ISSN 0908-1216 March 1994 Institute for Electronic Systems Department of Mathematics and Computer Science Fredrik Bajers Vej 7E | DK 9220 Aalborg O | Denmark Tel. : +45 98 15 85 22 | TELEX 69 790 aub dk d From Influence Diagrams to Junction Trees ? Frank Jensen Finn V. Jensen Soren L. Dittmer Department of Mathematics and Computer Science Aalborg University Fredrik Bajers Vej 7E , DK-9220 Aalborg Ost , Denmark E-mail : -LSB- fj,fvj,dittmer]@iesd.auc.dk Abstract We present an approach to the solution of decision problems formulated as influence diagrams . This approach involves a special triangulation of the underlying graph , the construction of a junction tree with special properties , and a message passing algorithm operating on the junction tree for computation of probabilities and utilities . 1 Introduction Influence diagrams were introduced by Howard and Matheson -LRB- 1981 -RRB- as a formalism to model decision problems with uncertainty for a single decision maker . ? This work has been partially funded by the Danish research councils through the PIFT programme . 1 The original way to evaluate such problems involved unfolding the influence diagram into a decision tree and using the \ average-out and fold-back '' algorithm on that tree . Shachter -LRB- 1986 -RRB- describes a way to evaluate an influence diagram without tranforming it into a decision tree . The method operates directly on the influence diagram by means of the node-removal and arc-reversal operations . These operations successively transform the diagram , ending with a diagram with only one utility node that holds the utility of the optimal decision policy ; the policies for the individual decisions are computed during the operation of the algorithm -LRB- when decision nodes are removed -RRB- . Shenoy -LRB- 1992 -RRB- describes another approach to the evaluation of influence diagrams : the influence diagram is converted to a valuation network , and the nodes are removed from this network by fusing the valuations bearing on the node -LRB- variable -RRB- to be removed . Shenoys algorithm is slightly more efficient than Shachters algorithm in that it maintains a system of valuations , whereas Shachters algorithm maintains a system of conditional probability functions -LRB- in addition to the utility functions -RRB- , and some extra work -LRB- some division operations -RRB- is required to keep the probability potentials normalized . In -LRB- Shachter and Ndilikilikesha , 1993 -RRB- , the node-removal/arc-reversal algorithm is modified to avoid these extra divisions ; the result is an algorithm that is equivalent to Shenoys algorithm with respect to computational efficiency . Our work builds primarily on the work of Shenoy and Shachter and Peot -LRB- 1992 -RRB- , in addition to our previous work on propagation algorithms for the expert system shell Hugin -LRB- Andersen et al. , 1989 ; Jensen et al. , 1990 -RRB- . 2 Influence diagrams An influence diagram is a belief network augmented with decision variables and a utility function . The structure of a decision problem is determined by an acyclic directed graph G . The vertices of the graph represent either random 2 variables -LRB- also known as chance or probabilistic variables -RRB- or decision variables , and the edges represent probabilistic dependencies between variables . Decision variables represent actions that are under the full control of the decision maker ; hence , we do not allow decision variables to have parents in the graph . Let UR be the set of random variables , and let the set of decision variables be UD = fD1 ; : : : ; Dng , with the decisions to be made in the order of their index . Let the universe of all variables be denoted by U = UR -LSB- UD . We partition UR into a collection of disjoint sets I0 , : : : , In ; for < k < n , Ik is the set of variables that will be observed1 between decision Dk and Dk +1 ; I0 is the initial evidence variables , and In is the set of variables that will never be observed -LRB- or will be observed after the last decision -RRB- . This introduces a partial order OE on U : I0 OE D1 OE ? ? ? OE Dn OE In . We associate with each random variable A a conditional probability function OEA = P -LRB- AjPA -RRB- , where PA denotes the set of parents of A in G . The state space XV for V ? U is defined as the Cartesian product of the sets of possible outcomes/decision alternatives for the individual variables in V . A potential OEV for a set V of variables is a function from XV to the set of real numbers . The potential OEV can be extended to a potential OEW -LRB- V ? W -RRB- by simply ignoring the extra variables : OEW -LRB- w -RRB- = OEV -LRB- v -RRB- if v is the projection of w on V. Given two potentials , OE and . The product OE ? and the quotient OE = are defined in the natural way , except that 0 = 0 is defined to be -LRB- x = 0 for x < > is undefined -RRB- . The -LRB- a priori -RRB- joint probability function OEU is defined as OEU = Y A2UR OEA . 1By ` observed ' , we mean that the true state of the variable will be revealed . 3 For each instance of UD -LRB- i.e. , each element of XUD -RRB- , OEU defines a joint probability function on UR . The goal of the decision problem is to make a sequence of decisions that will maximize some objective function . Such a function is called a utility function . We shall assume that the utility function is a potential that may be written as a sum of simpler potentials : = m X k = 1 k We impose some restrictions on the decision problem , namely that a decision does not have an impact on a variable already observed . This means that we have the property P -LRB- Ik jI0 ; : : : ; Ik-1 ; D1 ; : : : ; Dn -RRB- = P -LRB- Ik jI0 ; : : : ; Ik-1 ; D1 ; : : : ; Dk -RRB- . -LRB- 1 -RRB- In words : we can calculate the joint distribution for Ik without knowledge of the states of Dk +1 , : : : , Dn . Graphical representation In Figure 1 , an example of an influence diagram is shown . Random variables are indicated by circles , and decision variables are indicated by rectangles . Moreover , we indicate each term of the utility function by a diamond , and the domain of the term is indicated by its parent set . The partial order OE is indicated by making Ik-1 the parent set of Dk , and we shall use the convention that the temporal order of the decisions are read from left to right . The independence restrictions imposed on the decision problem can be checked on the influence diagram by checking that there is no directed path from a decision node Dk to a decision node Di -LRB- i < k -RRB- . 3 Decision making Assume we have to choose an alternative for decision Dn -LRB- i.e. , the last decision -RRB- . We have already observed the random variables I0 , : : : , In-1 , 4 a b c d e f g h i j k ` D1 D2 D3 D4 Figure 1 : An influence diagram for a decision problem with four decisions . The set of variables is partitioned into the sets : I0 = fbg , I1 = fe ; fg , I2 = ; , I3 = fgg , and I4 = fa ; c ; d ; h ; i ; j ; k ; ` g . The utility function is a sum of four local utilities , three of which are associated with single variables -LRB- D1 , D3 , and ` -RRB- , and one associated with the pair -LRB- j ; k -RRB- . and we have chosen alternatives for decisions D1 , : : : , Dn-1 . The maximum expected utility principle2 says that we should choose the alternative that maximizes the expected utility . The maximum expected utility for decision Dn is given by aen = max Dn X In P -LRB- In jI0 ; : : : ; In-1 ; D1 ; : : : ; Dn -RRB- ? ; aen is obviously a function of previous observations and decisions . We calculate the maximum expected utility for decision Dk -LRB- k < n -RRB- in a 2There are good arguments for adhering to this principle . See , e.g. , -LRB- Pearl , 1988 -RRB- . 5 similar way : aek = max Dk X Ik P -LRB- Ik jI0 ; : : : ; Ik-1 ; D1 ; : : : ; Dk -RRB- ? aek +1 . -LRB- 2 -RRB- We note that aek is well-defined : because of -LRB- 1 -RRB- , it makes sense to talk about the probability P -LRB- Ik jI0 ; : : : ; Ik-1 ; D1 ; : : : ; Dk -RRB- . By expansion of -LRB- 2 -RRB- , we get aek = max Dk X Ik P -LRB- Ik jI0 ; : : : ; Ik-1 ; D1 ; : : : ; Dk -RRB- ? max Dk +1 X Ik +1 P -LRB- Ik +1 jI0 ; : : : ; Ik ; D1 ; : : : ; Dk +1 -RRB- ? aek +2 = max Dk X Ik max Dk +1 X Ik +1 P -LRB- Ik ; Ik +1 jI0 ; : : : ; Ik-1 ; D1 ; : : : ; Dk +1 -RRB- ? aek +2 . The last step follows from -LRB- 1 -RRB- and the chain rule of probability theory : P -LRB- AjB ; C -RRB- P -LRB- BjC -RRB- = P -LRB- A ; BjC -RRB- . By further expansion , we get aek = max Dk X Ik ? ? ? max Dn X In P -LRB- Ik ; : : : In jI0 ; : : : ; Ik-1 ; D1 ; : : : ; Dn -RRB- ? . From this formula , we see that in order to calculate the maximum expected utility for a decision , we have to perform a series of marginalizations -LRB- alternately sum - and max-marginalizations -RRB- , thereby eliminating the variables . When we eliminate a variable A from a function OE , expressible as a product of simpler functions , we partition the factors into two groups : the factors that involve A , and the factors that do not ; call -LRB- the product of -RRB- these factors OE+A and OE-A , respectively . The marginal P A OE is then equal to OE-A ? P A OE+A ; P A OE+A then becomes a new factor that replaces the product OE+A in the expression for OE . This also holds true for maxmarginalizations , provided OE-A does not assume negative values . The product OE may be represented by an undirected graph , where each maximal complete set -LRB- clique -RRB- of nodes -LRB- the nodes being variables -RRB- corresponds to a factor of OE with that set as its domain . Marginalizing 6 a variable A out of OE then corresponds to the following operation on the graph : the set of neighbors of A in the graph is completed , and A is removed . It is a well-known result that all variables can be eliminated in this manner without adding edges if and only if the graph is triangulated -LRB- Rose , 1970 -RRB- . Obviously , it is desirable to eliminate all variables without adding extra edges to the graph since this means that we do not create new factors with a larger domain than the original factors -LRB- the complexity of representing and manipulating a factor is exponential in the size of its domain -RRB- . However , in most cases , this is not possible : we have to add some edges , and the elimination order chosen will determine how many and hence also the size of the cliques . Unfortunately , it is NP-hard to find an optimal elimination order for all reasonable criteria of optimality . When we perform inference in a belief network -LRB- i.e. , calculation of the marginal probability of some variable given evidence on other variables -RRB- , the computation only involves sum-marginalizations . In this case , we can eliminate the variables in any order , since the order of two marginalizations of the same kind can be interchanged . However , the calculation of the maximum expected utility involves both max - and sum-marginalizations . In general , we can not interchange the order of a max - and a sum-marginalization . This fact imposes some restrictions on the elimination order . 4 Compilation of the influence diagram We first form the moral graph of G . This means adding -LRB- undirected -RRB- edges between vertices with a common child . We also complete the vertex sets corresponding to the domains of the utility potentials . Finally , we drop directions on all edges . Next , we triangulate the moral graph in such a way that it facilitates the computation of the maximum expected utility . This is equivalent to selecting a special elimination sequence for the moral graph : we always select a maximal vertex with respect to the partial order OE . 7 a b c d e f g h i j k ` D1 D2 D3 D4 Figure 2 : The moral graph for the decision problem in Figure 1 . Edges added by the moralization process are indicated by dashed lines . Finally , we organize the cliques of the triangulated graph in a strong junction tree : A tree of cliques is called a junction tree if for each pair -LRB- C1 ; C2 -RRB- of cliques , C1 \ C2 is contained in every clique on the path connecting C1 and C2 . For two adjacent cliques , C1 and C2 , the intersection C1 \ C2 is called a separator . A junction tree is said to be strong if it has at least one distinguished clique R , called a strong root , such that for each pair -LRB- C1 ; C2 -RRB- of adjacent cliques in the tree , with C1 closer to R than C2 , there exists an ordering of C2 that respects OE and with the vertices of C1 \ C2 preceding the vertices of C2 n C1 . This property ensures that the computation of the maximum expected utility can be done by local message passing in the junction tree -LRB- as we shall see later -RRB- . Construction of a strong junction tree Let be a numbering of U -LRB- i.e. , a bijection : U $ f1 ; : : : ; jUjg -RRB- such that for all u ; v 2 U , u OE v implies -LRB- u -RRB- < -LRB- v -RRB- . We assume that is 8 a b c d e f g h i j k ` D1 D2 D3 D4 Figure 3 : The triangulated graph of the moral graph in Figure 2 . Fill-in edges added during triangulation are indicated by dashed lines . the elimination ordering used to produce the triangulated graph of G : vertices with higher numbers are eliminated before vertices with lower numbers . Let C be a clique of the triangulated graph , and let v 2 C be a vertex such that there exists a vertex u = 2 C , -LRB- u -RRB- < -LRB- v -RRB- , u = ? v , and u ? w for each vertex w 2 C with -LRB- w -RRB- < -LRB- v -RRB- . If such a vertex v exists , we define the index for C as index -LRB- C -RRB- = -LRB- v -RRB- ; otherwise , we define index -LRB- C -RRB- = 1 . Intuitively , the index for a clique C indicates the step in the elimination process that cause C to \ disappear '' from the graph . It is easy to show that the index for a clique C is well-defined , and that no two cliques have the same index . Moreover , unless index -LRB- C -RRB- = 1 , the set f v 2C j -LRB- v -RRB- < index -LRB- C -RRB- g will be a proper subset of some other clique with a lower index than C. Let the set of cliques of the triangulated graph be C1 , : : : , Cm , ordered in increasing order according to their index . As a consequence of 9 the above construction , this ordering will have the running intersection property -LRB- Beeri et al. , 1983 -RRB- , meaning that for all k > 1 : Sk = Ck \ k-1 -LSB- i = 1 Ci ? Cj for some j < k . It is now easy to construct a strong junction tree : we start with C1 -LRB- the root -RRB- ; then we successively attach each clique Ck to some clique Cj that contains Sk . Consider the decision problem in Figure 1 . Figure 2 shows the moral graph for this problem : edges have been added between vertices with a common child -LRB- including utility vertices -RRB- , utility vertices have been removed , and directions on all edges have been dropped . Note that the time precedence edges leading into decision vertices are not part of the graph and are thus not shown . Figure 3 shows the strong triangulation of the graph in Figure 2 generated by the elimination sequence ` , j , k , i -LRB- fill-ins : D2 ? D4 and g ? D4 -RRB- , h -LRB- fill-in : f ? D3 -RRB- , a , c -LRB- fill-in : b ? e -RRB- , d -LRB- fill-ins : D1 ? e , D1 ? f , b ? f , and e ? f -RRB- , D4 , g -LRB- fill-in : e ? D2 -RRB- , D3 , D2 , f , e , D1 , and b . This graph has the following cliques : C16 = fD4 ; i ; ` g , C15 = fh ; k ; jg , C14 = fD3 ; h ; kg , C11 = fb ; c ; ag , C10 = fb ; e ; d ; cg , C8 = fD2 ; g ; D4 ; ig , C6 = ff ; D3 ; hg , C5 = fe ; D2 ; gg , and C1 = fb ; D1 ; e ; f ; dg . Using the above algorithm , we get the strong junction tree shown in Figure 4 for this collection of cliques . -LRB- There exists another strong junction tree for this collection , obtained by replacing the edge C5 ! C1 by the edge C5 ! C10 . This tree is computationally more efficient , but , unfortunately , it can not be constructed by the algorithm given in this paper . -RRB- 5 Using the strong junction tree for computations We perform computations in the junction tree as a special ` collect ' operation from the leaves of the junction tree to some strong root of the tree . 10 b ; D1 ; e ; f ; d C1 b ; e ; d ; c C10 b ; c ; a C11 f ; D3 ; h C6 e ; D2 ; g C5 D3 ; h ; k C14 D2 ; g ; D4 ; i C8 h ; k ; j C15 D4 ; i ; ` C16 Figure 4 : A strong junction tree for the cliques of the graph in Figure 3 . To each clique C in the junction tree , we associate a probability potential OEC and a utility potential C defined on XC . Let C be the set of cliques . We define the joint potentials OE and for the junction tree as OE = Y C2C OEC ; = X C2C C . We initialize the junction tree as follows : each variable A 2 UR is assigned to a clique that contains A -LSB- PA. . The probability potential for a clique is the product of the conditional probability functions for the variables assigned to it . For cliques with no variables assigned to them , the probability potentials are unit functions . In this way , the joint probability potential for the junction tree becomes equal to the joint probability function for the influence diagram . Similarly , each utility function k is assigned to some clique that can accommodate it . The utility potential for a clique is the sum of the utility functions assigned to it ; for cliques with no utility functions assigned to them , the utility potential is a null function . 11 We need a generalized marginalization operation that acts differently on random and decision variables . We denote the operation by ` P ' . For random variable A and decision variable D , we define X A OE = X A OE ; X D OE = max D OE . For a set V of variables , we define P V OE as a series of single-variable marginalizations , in the inverse order as determined by the relation OE ; although OE is only a partial order , P V OE is well-defined . Now , let T be a strong junction tree , and let C1 and C2 be adjacent cliques with separator S in T . We say that C1 absorbs from C2 if OEC1 and C1 change to OE C1 and C1 as follows : OE C1 = OEC1 ? OES ; C1 = C1 + S OES ; where OES = X C2nS OEC2 ; S = X C2nS OEC2 ? C2 . Note that this definition of absorption is different from the one given in -LRB- Jensen et al. , 1990 -RRB- . The present definition does not immediately specialize to a symmetric probability propagation algorithm with absence of decision variables . The complexity of an absorption operation is O?jXC1 j + jXSj + jXC2 j ? . Note in particular that the contribution from the division operation plays a much smaller r ^ ole than in -LRB- Shenoy , 1992 -RRB- . We will need the following lemma , which we shall state without proof . Lemma 1 Let D be a decision variable , and let V be a set of variables that includes all descendants of D in G , and let ' V be any nonnegative potential on XUn -LRB- V -LSB- fDg -RRB- . Then P V OEU ? ` V , considered as a function of D alone , is a non-negative constant . 12 Let T be a strong junction tree with at least two cliques ; let OET be the joint probability potential and T the joint utility potential on T. Choose a strong root R for T and some leaf L -LRB- 6 = R -RRB- ; let T n L denote the strong junction tree obtained by absorbing L into its neighbor N and removing L ; denote the separator between N and L by S. Theorem 1 After absorption of L into T , we have X LnS OET ? T = OETnL ? TnL . Proof : Let OEL = Y C2CnfLg OEC ; L = X C2CnfLg C . Since OEL does not assume negative values , we get X LnS OET ? T = OEL ? X LnS OEL ? ? L + L ? . We have to show that X LnS OEL ? ? L + L ? = OES ? ? S OES + L ? , where OES = X LnS OEL ; S = X LnS OEL ? L . We shall prove this by induction . Let X1 , : : : , X ` be some ordering of L n S that respects OE . Now , consider the equation : X Xk : : : X X ` OEL ? ? L + L ? = OE -LRB- k -RRB- ? ? -LRB- k -RRB- OE -LRB- k -RRB- + L ? , -LRB- 3 -RRB- where OE -LRB- k -RRB- = X Xk : : : X X ` OEL ; -LRB- k -RRB- = X Xk : : : X X ` OEL ? L. 13 -LRB- For k = 1 , -LRB- 3 -RRB- is equivalent to the desired result . -RRB- For k > ` , -LRB- 3 -RRB- is clearly true ; for 1 < = k < = ` , we have two cases : -LRB- 1 -RRB- Xk is a random variable . By induction , we get X Xk : : : X X ` OEL ? ? L + L ? = X Xk OE -LRB- k +1 -RRB- ? ? -LRB- k +1 -RRB- OE -LRB- k +1 -RRB- + L ? = X Xk -LRB- k +1 -RRB- + L ? X Xk OE -LRB- k +1 -RRB- = -LRB- k -RRB- + L ? OE -LRB- k -RRB- = OE -LRB- k -RRB- ? ? -LRB- k -RRB- OE -LRB- k -RRB- + L ? The correctness of the last step follows from the fact that OE -LRB- k -RRB- -LRB- x -RRB- = implies -LRB- k -RRB- -LRB- x -RRB- = -LRB- so that our division-by-zero convention applies -RRB- . -LRB- 2 -RRB- Xk is a decision variable . By induction , we get X Xk : : : X X ` OEL ? ? L + L ? = max Xk OE -LRB- k +1 -RRB- ? ? -LRB- k +1 -RRB- OE -LRB- k +1 -RRB- + L ? . Because of Lemma 1 , OE -LRB- k +1 -RRB- , considered as a function of Xk alone , is a non-negative constant , and we get ? max Xk OE -LRB- k +1 -RRB- ? ? ? maxXk -LRB- k +1 -RRB- maxXk OE -LRB- k +1 -RRB- + L ? = OE -LRB- k -RRB- ? ? -LRB- k -RRB- OE -LRB- k -RRB- + L ? . By successively absorbing leaves into a strong junction tree , we obtain probability and utility potentials on the intermediate strong junction trees that are equal to the marginals of the original potentials with respect to the universes of these intermediate trees . This is ensured by 14 the construction of the junction tree in which variables to be marginalized out early are located farther away from the root than variables to be marginalized out later . The optimal policy for a decision variable can be read off from the clique that is closest to a strong root and contains the decision variable -LRB- that clique may be a strong root itself -RRB- , since all variables that the decision variable may depend on will also be members of that clique . 6 Conclusion We have described an algorithm to transform a decision problem formulated as an influence diagram into a secondary structure , a strong junction tree , that is particularly well-suited for efficient computation of maximum expected utilities and optimal decision policies . The algorithm is a refinement of the work by Shenoy -LRB- 1992 -RRB- and Shachter and Peot -LRB- 1992 -RRB- ; in particular , the construction of the strong junction tree has been elaborated upon . This work forms the basis for an efficient computer implementation of Bayesian decision analysis in the expert system shell Hugin -LRB- Andersen et al. , 1989 -RRB- . We have not given an algorithm to construct the elimination sequence that generates the strong triangulation . However , the triangulation problem is simpler than for ordinary probability propagation , since the set of admissible elimination sequences is smaller ; at this stage , it appears that simple adaptations of the heuristic algorithms described by Kjaerulff -LRB- 1990 -RRB- work very well . Moreover , even given a triangulation , there might exist several strong junction trees for the set of cliques . Work regarding this problem is in progress . 15 References Andersen , S. K. , Olesen , K. G. , Jensen , F. V. , and Jensen , F. -LRB- 1989 -RRB- . Hugin | a shell for building Bayesian belief universes for expert systems . 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O

bayesian decision analysis
influence diagrams
local computations

strong junction trees and propagation algorithms