Modularizing inference in large causal probabilistic networks

Modularizing Inference in Large CausalProbabilistic NetworksKristian G. Olesen,1,* Steen Andreassen,2,† Marko Suojanen2,‡

1Department of Computer Science and 2Department of Medical Informaticsand Image Analysis, Aalborg University, Frederik Bajers Vej 7,DK-9220 Aalborg Øst, Denmark

This article describes a number of implementation aspects of modular inference in large medicalexpert systems based on causal probabilistic networks. Examples are provided from the neuro-muscular diagnosting system the muscle and nerve inference network (MUNIN). The inferenceprocedure is outlined and the principal data structure underlying the inference procedure aredescribed. A condensed summary of selected technical details of the inference procedure incausal probabilistic networks (CPNs) is provided. This is required for understanding theimplemented modularization of the inference. The modularization of the inference implies a needfor transfer of information between modules, which is realized by establishing communicationchannels between modules. Modules are also used to perform inference by conditioning, amethod that reduces storage requirements to a manageable size and thereby prepares the way forMUNINs migration to common PCs. © 2003 Wiley Periodicals, Inc.

1. INTRODUCTION

The medical field is attractive to research projects in intelligent systems partlybecause of the challenges it offers in the form of the variety of tasks involved inmedical decision making and partly because of the potential impact of successfulapplications. The complexity of medical reasoning tends to be quite resourcedemanding, not in the least because of the uncertainty inherent in most problems.A promising approach to treat this uncertainty is causal probabilistic networks(CPNs), also known as Bayesian networks,1–3 because this formalism is able tointegrate the various aspects of medical reasoning under uncertainty in a coherentand theoretically well-founded framework. However, a drawback of the method-ology is that such systems often require computational resources that lie outside thelimits of what common modern computers offer. A frequently used approach tothis problem is to adopt the single disease assumption. In the context of CPNs, the

*Author to whom all correspondence should be addressed: e-mail: [email protected].†e-mail: [email protected].‡e-mail: [email protected].

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 18, 179–191 (2003)© 2003 Wiley Periodicals, Inc. Published online in Wiley InterScience(www.interscience.wiley.com). • DOI 10.1002/int.10082

single disease assumption forces all possible diseases to be states of one commonvariable and consequently all descendants of this variable have to be specified forall included diseases. This introduces a problem on the CPN level where thespecification of the system involves details of relations between diseases andsymptoms that are not related in the real world. One way to ease this problem issimilarity networks.4 We have chosen an alternative approach based on the fact thatthe single disease assumption often is wrong. Therefore, we allow several simul-taneous diseases to coexist. This eases the specification problem, but the drawbackis problems on the inference level, where it is frequently impossible to construct acomputational structure of a manageable size. Therefore, approximate methods areneeded. One such method for inference in large systems has been proposed bySuojanen et al.5 and in this article we focus on implementation details for thismethod. In the following section we introduce our application the muscle and nerveinference network (MUNIN) and outline the proposed method and give details onaspects of the implementation of the method including tricks that enable it forpractical use.

2. MUNIN

Our domain of application is diagnosis of muscle diseases and diseases of theperipheral nerve system. The system has been baptized as MUNIN. The currentprototype of the system is called “microhuman” to indicate that the anatomy of thesystem is limited to four nerves and three muscles on each side of the body. Despitethis limitation, the prototype describes all principal elements of a complete system,including general muscle and nerve diseases, local nerve diseases, both proximaland distal muscles, and pure motor and sensory nerves as well as mixed nerves.The system consists of about 1,100 variables describing 22 diseases and approx-imately 200 possible findings. In this study we are concerned primarily withinferential issues in large CPNs and we will not dig into the medical details ofMUNIN. Readers with special interest in these matters are referred to the existingliterature.6–9

2.1. The Structure of the MUNIN Network

Diseases in MUNIN are modelled by between one and five variables. Generalmuscle disorders are sufficiently described by a single variable, but for, e.g., localnerve lesions, more variables are needed in order to give an adequate descriptionof the disease pattern. This fact is not important in the current context, andtherefore we will map each disease as a single variable throughout this article. Theimpact of the diseases are modeled through pathophysiological contributions,which describe the effect of the diseases on specific muscles and nerves. Thesevariables are artificial in the sense that they are not directly observable. What is, atleast in principle, observable in the body is the total effect of the joint contributionsof the diseases in terms of pathophysiological changes. These pathophysiologicalchanges are modeled as variables that describe a feature such as the loss of axonsin a specific nerve. The pathophysiological features are adding up the contributions

180 OLESEN, ANDREASSEN, AND SUOJANEN

from each disease to that specific feature. Because it would be connected withunfortunate side effects to observe these changes directly, they are connected to thefourth and last principal layer of variables, the findings that describe the effect inmore convenient measurable quantities. This could be, e.g., a measurement of theconduction velocity of nerve impulses elicited by electrical stimulation of a nerve.The structure of the MUNIN is shown in Figure 1.

The extract of MUNIN shown in Figure 1 shows three diseases in the toplayer, two local nerve lesions, a median lesion at the left elbow and left wrist, anddiffuse neuropathy. Two nerves are included: the left sensory median nerve and theleft sensory ulnar nerve. The two local nerve lesions affect the left median nerveand diffuse neuropathy affects both the left sensory median nerve and the leftsensory ulnar nerve. The effect of each of the diseases on the nerves they affect aremodeled through the pathophysiological contributions in the second layer and thetotal effect on each nerve from all diseases are combined in the pathophysiologyin the third layer. The bottom layer represents the available findings on each of thenerves. All variables describing facets of a given nerve (or muscle) are collectivelyreferred to as an anatomical unit. Thus, in the example, two anatomical units arerepresented as shown in Figure 1.

2.2. Inference in MUNIN

The complex interdependency pattern of the diseases in MUNIN makes itintractable to maintain a representation of the joint probabilities for the completesystem. Even a modularized representation of this distribution has a size that ismeasured in terabytes and therefore is outside the scope of practical computing.Earlier versions of MUNIN tackled this problem by an approximate solution thatsplit the system into a number of partial networks, each describing only a subset ofthe diseases. These subnets were hand generated and although the approach yieldedviable results, it is not sustainable for a general system covering most relevant parts

Figure 1. An excerpt of MUNIN showing the overall structure of the system. Diseases on thetop level are linked through pathophysiological contributions and pathology to findings at thebottom level.

MODULARIZING INFERENCE 181

of the human anatomy. Therefore, we were forced to look for a systematic methodof approximate inference. The method we have proposed5 first considers singlediseases and then pairs of diseases and it continues by iterating over still larger setsof diseases. In each iteration we investigate the plausibility of each disease, and ifit meets a heuristic elimination criterion, we disregard it from further consideration.Thereby, the total number of diseases is reduced in each cycle and at the same timethe number of simultaneously considered diseases in each cycle is increased. Whenthe number of diseases left equals the number of diseases considered in the cycle,the final diagnosis is found. Algorithm 1 shows the method in more detail.

ALGORITHM 1 (iterative inference procedure). Given a CPN N and a case Cconsisting of a set of findings in N. Let D � {d1, . . . , dn} be the set of all diseasesin N and let n denote the number of diseases in D. For each subset S of D let NS

be the partial CPN only including the diseases in S.

Set i � 1while i � n

for each subset S of i elements in Dpropagate C in NS

update the probability for all d in Sif d in S meets the elimination criterionthen delete d from D

recalculate n and increase i by 1Report P(d) for all d in D

Observe that not all subsets of size i are necessarily investigated. If, e.g., D � {d1,d2, d3} and S1 � {d1, d2} is the first subset to be processed, this could result inthe elimination of d2. Then, we just need to investigate S2 � {d1, d3}, becauseS3 � {d2, d3} would never be generated. We are exploiting this fact in order toobtain efficient computations by sorting diseases in decreasing order with respectto posterior probability. The final diagnosis is typically consisting of the diseaseswith the highest intermediate probability and therefore these diseases are mostlikely to eliminate diseases with lower probability. Therefore, combinations ofdiseases with low probability should be considered at the end of each phase.

When two results exist for a particular disease, which, e.g., could be the casefor d1 after propagation of the nets for S1 � {d1, d2} and S2 � {d1, d3} in thesecond iteration, then the minimum is chosen as the current value. The rationalebehind this approach is elaborated by Soujanen et al.5

The heuristic elimination criterion we have used consists of two elements.

Elimination criterionIf the posterior probability of a hypothesis is below 1% and less than50% of the prior probability, it is excluded from further consideration.

The first part of the criterion ensures that the absolute probability of the hypothesisis low, and the second part ensures that the posterior probability for the hypothesishas decreased relative to the prior probability.


The construction of the partial networks NS is obtained through what we callsoft conditioning. Soft conditioning is an approximation scheme in which we deletea hypothesis from the CPN thereby ignoring some of the (potential) dependenciesbetween the diseases. Consider, e.g., the two diseases, median lesion at the leftwrist and diffuse neuropathy, in Figure 1. If a case includes a finding that is acommon ancestor of these diseases (e.g., in the left median sensory nerve), theybecome dependent. When diffuse neuropathy is removed from the network, thisdependency is neglected and consequently the computed posterior probability forthe median lesion at the left wrist results in an approximation of the true value.Remark that this approximation is merely an intermediate result that is refined inlater cycles. The partial networks NS used in Algorithm 1 are constructed by (1)removing all diseases not in S from the total network and (2) identifying thecomponent(s) containing diseases in S. These subnets are then compiled and storedseparately. The soft conditioning scheme discussed in more depth by Soujanen etal.5 also contains an empirical justification of the results.

2.3. Basic Data Structure for the Inference Scheme

The overall method for diagnosing multiple diseases includes numerouspropagations in a large number of nets. These nets are organized in lists as shownin Figure 2. The three levels of Figure 2 hold, respectively, single diseases, pairsof diseases, and triplets of diseases. These lists hold all necessary information tocompute the posterior marginals for the diseases in the first three phases (iterations)of the inference method.

In the first phase of the method, propagation is performed in all single diseasenets and the results constitute a first approximation of the posterior probabilities forthe diseases. After each propagation the elimination criterion is applied to thedisease to decide whether this disease should be considered in the following phase.In the second phase of the method, the list of pairs of diseases is traversed. For eachelement in the list it is determined whether any of the diseases have beeneliminated already. If one or both diseases have been eliminated, the proceduresimply continues with the next element in the list. When neither of the diseases hasbeen eliminated, propagation is performed in the double disease net, the posterior

Figure 2. The basic data structure for the inference method consists of lists of single diseases,pairs of diseases, and triplets of diseases. The figure shows these lists for four diseases.


probabilities for the diseases are updated, and the elimination criterion is appliedto decide whether any of the diseases is to be excluded from further consideration.Finally, the third phase runs through the triple disease list in a similar fashion.

3. INCREASING THE COMPUTATIONAL EFFICIENCY BYMODULARIZATION

A closer look at the method reveals that a number of computations arerepeated several times. This can be avoided by reuse of results that are alreadyavailable. To comprehend this, some details of the inference in CPNs are needed.When inference in CPNs is performed, a secondary structure, called a junction tree,is generated. The junction tree is constructed from a CPN by grouping nodestogether in cliques. The cliques are organized in a tree structure, in which the linksbetween the cliques are described by separators consisting of the intersectionbetween the nodes that appear in the two cliques they link together. The junctiontree construction is described in detail by, e.g., Jensen.3 As an example, considerthe single disease net of Figure 3(a). If, for simplicity, the entities of this networkare considered as simple nodes, the corresponding junction tree could look like theone shown in Figure 3(b). The ellipses in Figure 3(a) identifies the cliques of thenet.

The actual inference is carried out in the junction tree by local operations incliques or between neighboring cliques in the tree.3 Each clique and separator holdsa table of numbers with one entry for each element in the joint state space of allnodes in that clique or separator. Cliques can calibrate to each other through theseparators. When a clique C1 calibrates to a neighboring clique C2 it receivesinformation from C2 on shared variables through the separator S [Figure 3(b)].This is done in three steps. First, a marginal table on the nodes in S is calculatedfrom the table of C2. Next, the table of C1 is updated by dividing it with the currentvalue of the table of S and multiplying it with the new value of the table of Sprovided by C2. Finally, the old table of S is replaced by the new one provided byC2. A general full propagation in a junction tree consists of two calibrations overeach separator (one in each direction) in a restricted order.

Figure 3. (a) A single disease net for diffuse neuropathy affecting the left sensory mediannerve and the left sensory ulnar and (b) a corresponding junction tree when the entities in parta are considered as simple nodes.


3.1. Modularizing Inference

As we shall see, parts of the junction trees for the nets in MUNIN areidentical, and therefore the calculations performed in these parts are also identical.Thus, it is desirable to modularize the junction trees to avoid redundant compu-tations. A junction tree can be split into two modules by deleting a link betweentwo cliques. Such modularization is not supported by available software packagesand consequently the implementation of a modular inference method requires amethod for exchange of information between modules. This is done throughcommunication channels. Communication channels are tables of variables com-mon to the two modules that interact, and they are initialized by the marginal tablefor these variables. The actual transfer of information is performed by the follow-ing steps. First, the table of the clique of module 1 connected to the communicationchannel is divided by the table of the communication channel. Next, the table forthe communication channel of module 2 is copied to the table of the communica-tion channel in module 1, and, finally, the table of the clique of module 1 ismultiplied by the table of the communication channel. The result is equivalent tothe calibration operation of the propagation algorithm. This means that we canobtain exactly the same results by communicating between modules, as we obtainfrom a single complete junction tree. The requirement for this scheme to workcorrectly is that the communication channel between two modules must hold allvariables common to the underlying CPNs. To be able to obtain a joint table forthese nodes, both of the communicating modules must include a clique that has allvariables in a communication channel as members. This is almost always the casein MUNIN, where communication channels always consist of diseases. In a fewsituations, the requirement was not fulfilled and we had to force it to hold. This canbe done by adding a dummy variable with only one state as a child of all thevariables that should belong to such a clique. The junction tree construction thenensures that these variables appear in at least one clique of the junction tree. InMUNIN, this was unproblematic because the largest enforced clique contained fivevariables.

3.1.1. Single Disease Nets

In the network in Figure 3(a), diffuse neuropathy d-separates10 the anatomicalunit left sensory median nerve from the anatomical unit left sensory ulnar nerve.Loosely speaking, this means that the influence of one anatomical unit on the othergoes through diffuse neuropathy, but that influence is not of interest, because thefocus is on the posterior probability for diffuse neuropathy. Consequently, theinformation that influences diffuse neuropathy in each of the two anatomical unitscan be processed independently. The same pattern is found at several places in thenets, and to avoid repeated computations of the same contribution, we split the netsinto smaller modules and combine the partial results from these modules into thedesired results. The split of the single disease net in Figure 3(a) is illustrated inFigure 4.

Instead of performing propagation in the single disease net of Figure 3(a), wepropagate in the two nets of Figure 4 and combine the results into the posterior


belief of diffuse neuropathy. The resulting belief in diffuse neuropathy is collectedin a separate CPN that solely describes diffuse neuropathy, and partial results fromthe modules of Figure 4 are transferred to this CPN through communicationchannels as shown in Figure 5. This construction extends the standard inferencetechniques by having communication channels (separators) that are identical to theclique that is actually updated (diffuse neuropathy). Notice that all communicationchannels have to be created before the procedure is performed in order to ensureproper initialization of the tables of the communication channels.

3.1.2. Double Disease Nets

The inference pattern just described is not faster than standard inference;actually, it results in a slight overhead. The benefits of the approach are harvestedin the second phase of the method, where the results from the first phase are reused.For example, the double disease net of Figure 6 is similarly split into partial netsas shown in that figure. This split leads to two modules of which the rightmost isidentical to the rightmost module of Figure 4. Hence, the results from that modulecan be reused without further computation.

The junction trees for the two modules in Figure 6 consist of one cliqueholding all the nodes in the modules and the communication channel between themconsists of diffuse neuropathy. In the double disease nets the two diseases are

Figure 4. The single disease net from Figure 3(a) is split into two modules.

Figure 5. The system of communicating domains used to update diffuse neuropathy, consistingof three junction trees (one for diffuse neuropathy and one for each of the modules in Figure 4).


dependent because of findings in their common anatomical unit, e.g., the leftsensory median nerve in the example of Figure 6. Therefore, the result from themodule describing the influence of the left sensory ulnar nerve on diffuse neurop-athy has to be transferred to the double disease net before propagation in this netin order to get coherent posterior beliefs for both diseases.

3.1.3. Triple Disease Nets

A similar reuse of results can be obtained in the third phase of the methodwhere three diseases are considered simultaneously. There are five fundamentallydifferent patterns of three diseases as shown in Figure 7, in which child nodesdenote one or more anatomical units.

The first type is shown in Figure 7(a), in which all three diseases areindependent (no common nodes in the nets). Type two is where two diseases aredependent because of common findings but independent of the third disease[Figure 7(b)]. Types three and four is where two [Figure 7(c)] or three [Figure 7(d)]pairs of diseases have common findings and the fifth type is where all threediseases have findings in common [Figure 7(e)]. All diseases may have an exclu-sive effect on other anatomical units as well, and for type five [Figure 7(e)] any pairof diseases may also influence other anatomical units. For combinations of typeone and two the results are readily available. Type four does not appear in MUNINbut types three and five do. An example of type five is shown in Figure 8. This netbecomes relevant if none of the diseases were eliminated during the second phaseof the method. This is similar to the double disease case shown previously. In thespecific example the influence of the left sensory ulnar nerve on diffuse neuropathy

Figure 6. A double disease net is split into two modules.

Figure 7. The five fundamental patterns of three diseases. All diseases may have exclusiveeffect on other anatomical units as well, and for type five (part e in the figure) any pair of diseasesmay also influence other anatomical units.


can be reused by transfer to the triple disease net of Figure 8 before propagationis performed.

An example of a type three triple disease net is shown in Figure 9. In this type,there is no anatomical unit common to all three diseases, but two pairs of diseasesshare anatomical units. In this case, diffuse neuropathy again d-separates thevarious parts of the total net that can be split into three modules as shown in Figure9. These modules are identical to already existing modules from the first phases ofthe inference. Instead of actually constructing a triple disease net, we can reuse theexisting modules. Notice that although diffuse neuropathy d-separates the locallesions of the median nerve at the right and the left wrist, they are still dependentbecause the state of diffuse neuropathy is actually not known. Junction trees for thethree modules, including communication channels on diffuse neuropathy, areshown in Figure 10.

The following procedure solves the inference problem for this case. First,perform a propagation in the junction tree of Figure 10(a). Transfer the informationto the junction tree of Figure 10(b) and perform a propagation in this net. Then,transfer information on diffuse neuropathy from this junction tree to the junctiontree of Figure 10(c). Now, all information is glued together in this junction tree andthe posterior beliefs in diffuse neuropathy and the local lesion of the right mediannerve can be extracted. Next, transfer the updated belief in diffuse neuropathy backto the junction tree of Figure 10(b) and perform a new propagation in this junctiontree. Now, the junction trees of Figure 10(a, b) are completely updated and theposterior belief in the local lesion of the left median nerve is available. Readersfamiliar with the technical details of propagation3 will note that this descriptionincludes two propagations in the junction tree of Figure 10(b). This could beresolved by performing only the collect phase in the first propagation and post-poning the distribute phase to the second propagation. Unfortunately, these oper-ations are not directly accessible in commonly available software.

3.2. Propagation by Conditioning

Triple disease nets describing three diseases affecting the same anatomicalunit are very large and inference in such nets has to be done by conditioning.11 The

Figure 9. A triple disease net is split into three modules.

Figure 8. A triple disease net is split into two modules.


term conditioning in this context refers to the method for propagation originallyproposed by Pearl.12 Recall that approximation by the soft conditioning scheme(not to be confused by propagation by conditioning) is realized by disregardingdiseases from the complete net. The result of this is that the modules describingonly a subset of the diseases include the pathophysiological contribution ofdisregarded diseases. Consider, e.g., the triple disease net of Figure 8. Disregardingthe median lesion at the left elbow results in the double disease net of Figure 11.The double disease net of Figure 11 includes the pathophysiological contributionsof the median lesion at the left elbow. These pathophysiological contributionsappear as founding nodes in the upper left corner in Figure 11. The inclusion of thispart of the net is not necessary in order to process the double disease net, but itenables us to reuse this net for propagation by conditioning when we reach the thirdphase of the inference method where three diseases are considered simultaneously.Inference by conditioning is performed by a propagation for each possible state ofthe conditioned variable followed by a combination of the results (according toBayes theorem). For each state of the conditioned variable (the median lesion at theleft elbow) the corresponding distributions for its descendants (the pathophysio-logical contributions) are computed and these distributions are used as priors forthe propagation in the double disease net. The conditioning method for propagationis described in detail by Andreassen.13 With this method, all propagations in triple

Figure 10. Junction trees, with communication channels, for the three modules in Figure 9.

Figure 11. The triple disease net from Figure 8 where the median lesion at the left elbow isdisregarded as a result of soft conditioning.


disease nets with all diseases affecting the same anatomical unit can be performedby using already existing single and double disease modules. Thus, we do not haveto represent and store these triple disease nets.

4. DISCUSSION

We have outlined an approximative method for inference in large CPNs andhave described how this method can be realized through modularization of the netsand reuse of (partial) results.

The implemented method for modular inference actually is a bit more com-plicated than described. This is because of the introduction of the so-called foundin doctors lab (FIDL) factors that solved the problem with the choice of diseaseprevalences.14 FIDL can be seen as a summary of all the reasons that have led tothe referral of the patient to the electromyographic (EMG) examination. The FIDLfactors are introduced as a special anatomical unit with only one finding, which isalways known to have the value “true.” All diseases are linked to this variablethrough dedicated pathophysiological contributions. The construction enables adistinction between population prevalences, the prevalence of a disease in thegeneral population, and laboratory prevalences, the prevalence of a disease in thesubset of the population that is undergoing an EMG examination. The introductionof the FIDL factors greatly improved the diagnoses, relative to the version ofMUNIN without the FIDL variables.14 The complication introduced by the FIDLfactors is that it makes all diseases dependent. This dependence increases slightlythe complexity of the nets for already dependent diseases and implies that specialattention has to be devoted to combinations of diseases that are otherwise inde-pendent. This complication is resolved by variations of the described methods forinference.

Up to this point, our focus has been to design a practically feasible method.Future work will focus on performance aspects such as effective memory man-agement and speed up of the inference procedures. The current response time forthe most complicated cases are measured in minutes, and although this may beacceptable, we are confident that a considerably better response time can beobtained. Further optimization is needed before real-time experiments are relevant.

The main motivation for this work has been to establish a practically feasiblemethod for the MUNIN system. The proposed method is capable of diagnosing twosimultaneous diseases correctly but may have some false positive diseases for caseswith more than two diseases. This could be repaired by strengthening the elimi-nation criterion, but the current version seems to mirror the uncertainties of real lifequite well. The implementation details have indicated that inference in tripledisease nets can be performed in double disease modules, either by propagation byconditioning or by combining results from two double disease modules (and, inboth cases, by inclusion of results already available from single disease modules).This makes the representation and storage of triple disease nets superflous andconsequently a practical feasible method for inference in large CPNs is obtained.Consequently, MUNIN’s migration to a common PC is enabled.


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Modularizing inference in large causal probabilistic networks