European Journal of Operational Research 128 (2001) 479±498 www.elsevier.com/locate/dsw

Theory and Methodology A Bayesian network approach to making inferences in causal maps

Sucheta Nadkarni, Prakash P. Shenoy * School of Business, University of Kansas, Summer®eld Hall, Lawrence, KS 66045-2003, USA Received 8 September 1998; accepted 19 May 1999

Abstract

The main goal of this paper is to describe a new graphical structure called ÔBayesian causal mapsÕ to represent and analyze domain knowledge of experts. A Bayesian causal map is a causal map, i.e., a network-based representation of an expertÕs cognition. It is also a Bayesian network, i.e., a graphical representation of an expertÕs knowledge based on probability theory. Bayesian causal maps enhance the capabilities of causal maps in many ways. We describe how the textual analysis procedure for constructing causal maps can be modi®ed to construct Bayesian causal maps, and we illustrate it using a causal map of a marketing expert in the context of a product development decision. Ó 2001 Elsevier Science B.V. All rights reserved.

Keywords: Causal maps; Cognitive maps; Bayesian networks; Bayesian causal maps

1. Introduction In the last few years, researchers have empha- sized that causal maps may be used as tools to Recently, there has been a growing interest in facilitate decision making and problem solving the use of causal maps to represent domain within the context of organizational intervention knowledge of decision-makers (Hu€, 1990; Eden (Eden, 1992; Laukkanen, 1996). Studies in arti®- et al., 1992; Laukkanen, 1996). Causal maps are cial intelligence have also emphasized the impor- cognitive maps that represent the causal knowl- tance of domain knowledge in constructing edge of subjects in a speci®c domain. Causal maps decision tools, especially in situations where do- (also called cognitive maps, cause maps, etc.) have main knowledge is crucial and availability of data been used extensively in the areas of policy anal- is scarce (Heckerman, 1996). Since causal maps ysis (Axelrod, 1976) and management sciences represent domain knowledge more descriptively (Klein and Cooper, 1982; Ross and Hall, 1980) to than other models such as regression or structural represent salient factors, knowledge, and condi- equations, they are more useful decision tools. An tions that in¯uence decision making. important aspect of a decision model is the ability to make inferences. Inference refers to drawing conclusions based on a premise. To make an in- ference is to come to believe a new fact based * Corresponding author. Tel.: +1-785-864-7551. on other information. Inference is important in

0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 368-9 480 S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498 decision analysis for two reasons. First, it allows causal (Shachter and Heckerman, 1987), and one to make predictions in case of interventions. sparse Bayesian networks allow us to make infer- For example, a marketing manager may want to ences eciently even when we have many variables know how a new marketing strategy will a€ect the in the network (Pearl, 1986; Lauritzen and Spei- company pro®ts. Second, it provides a prescriptive gelhalter, 1988; Shenoy and Shafer, 1990). Baye- framework for decision making which can be sian networks in which the dependence relations useful to individuals other than the expert in a are causal are also called causal belief networks, speci®c decision context. causal probabilistic networks, etc. We simply refer Current techniques used to analyze causal maps to such representations as Bayesian causal maps. provide a qualitative interpretation of the vari- Using the framework of Bayesian networks, ables representing a decision problem by focusing Bayesian causal maps can account for uncertainty on the structure of a causal map (Axelrod, 1976; associated with the variables in the map. Bayesian Bougon, 1992; Eden et al., 1979; Ross and Hall, causal maps can use the evidence propagation al- 1980). This qualitative approach is useful in de- gorithms of Bayesian networks to analyze the scribing a decision problem or in making infer- sensitivity of variables of interest (e.g. outcome ences in case of very simple maps with few variables) to additional ®ndings about other vari- variables. However, drawing inferences about ables (e.g. situational variables). variables in complex causal maps may be a non- An outline of the remainder of the paper is as trivial task because of two reasons. First, causal follows. In Section 2, we discuss the de®nition and maps do not model uncertainty associated with the analysis of causal maps. In Section 3, we discuss decision variables (Hu€, 1990; Laukkanen, 1996). Bayesian networks, their semantics, and the pro- All variables in the maps have the same level of cess of making inferences. In Section 4, we propose certainty. Identifying the level of uncertainty is a procedure for constructing a Bayesian causal important in making inferences because observa- map starting from a causal map. In Section 5, we tions of variables may be uncertain, information discuss a case study of product development de- may be incomplete, or the variables involved may cision in a technology organization. Finally, in be vague. Second, causal maps provide a static Section 6, we conclude with a summary and a representation of the decision variables. They do statement of future research. not depict how beliefs of decision-makers about some target variables change when decision-mak- ers learn additional information about relevant situational factors or decision options represented 2. Causal maps in the map. Such a dynamic approach is important in not only drawing inferences but also in learning Causal maps are directed graphs that represent about causal relations representing complex and the cause±e€ect relations embedded in managersÕ uncertain decisions (Heckerman, 1996). thinking (Fiol and Hu€, 1992). Eden et al. (1992) Recent advances in arti®cial intelligence such as de®nes a causal map as a ``directed graph char- Bayesian networks allow us to use causal maps to acterized by a hierarchical structure which is most make inferences for decision making. This study often in the form of a means/end graph.'' Causal uses the theory of Bayesian networks to suggest a maps express the judgment that certain events or quantitative way to make inferences in causal actions will lead to particular outcomes. The three maps. A Bayesian network is a speci®cation of a components of a causal map are node representing joint probability distribution of several variables in causal concept, link representing causal connection terms of conditional distributions for each vari- among causal concepts (can be positive or nega- able. In general, the relations represented in a tive), and strength representing causal value of a Bayesian network do not have to be causal rela- causal connection. Fig. 1 depicts a causal map of a tions. However, a Bayesian network representa- marketing expert in the context of a new product tion is sparse and ecient when the relations are decision. S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498 481

construct theory (Eden et al., 1979), and textual analysis approach (Axelrod, 1976). We use the textual analysis method for constructing Bayesian causal map. The textual analysis method is deemed appropriate because it does not require the re- searcher to identify causal concepts prior to or during the interview with subjects. The advantage of this method is that it enables the researcher to let the data drive the concepts rather than eliciting an individualÕs cognition from a prede®ned set of concepts (Carley and Palmquist, 1992). The unin- trusive nature of this method is especially impor- tant in representing an expertÕs description and understanding of an unstructured decision prob- Fig. 1. A causal map of a marketing expert for a New Product lem. Decision. Causal maps have been useful in practice. For example, Axelrod (1976) describes a causal map This causal map consists of four causal con- derived from text to represent a decision-makerÕs cepts: Market Dynamics, Product Life Cycle, beliefs concerning the relationships between fac- Market Leadership and Rate of Product Launch. tors in the public health system. Similarly, Swan The causal concepts are linked to each other (1995) describes textually derived causal maps of through causal connections represented by unidi- key managers to identify important factors a€ect- rectional arrows (either positive or negative). The ing the decision of implementing computer-aided concept at the tail of an arrow is the ÔcauseÕ of the production management technologies in manu- concept at the head of the arrow. For example, facturing ®rms. Such causal maps have also been high market dynamics leads to a short product life used to represent cause±e€ect beliefs of decision- cycle (negative relation). A short product life cycle makers in strategic alliance decisions (Fiol, 1990), causes the rate of product launch of a company to and marketing distribution decisions (Laukkanen, be high (negative relation). Finally, market lead- 1996). ership causes a company to launch products at a Eden and Jones (1980) point out that there is no high rate (i.e. a positive relation). general approach to analyze causal maps. In gen- Causal maps make the following three as- eral, causal maps can be analyzed qualitatively or sumptions about cognition in the context of deci- quantitatively. EdenÕs causal mapping technique, sion making (Hu€, 1990). most widely applied to problem construction, is an 1. Causal associations are a major way in which aid to de®ning the nature of policy problem. It decision problems can be described and under- involves con¯icting and qualitative views of the stood; problem. This has been widely applied in strategic 2. Causality is the primary form of post-hoc ex- management for organizations and is constantly planation of decision outcomes; and being developed (Eden, 1991; Eden, 1992; Eden 3. Choice among alternative decision actions in- et al., 1992; Eden and Ackermann, 1993). Causal volves causal relations. maps can also be analyzed quantitatively using The primary emphasis in causal maps is in di€erent analytical tools. For example, quantita- identifying dependence between variables in terms tive analysis could be used to explore the numeri- of explanation±consequence or means±end rela- cal implications of possible policy options based tionships. Di€erent techniques have been em- on constructed causal maps. Quantitative analysis ployed to construct causal maps including provides an objective basis for assessments of structured methods such as the self-Q method causal maps so that subjective biases in qualitative (Bougon, 1983), open method based on personal analysis could be reduced. 482 S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498

Quantitative tools used to analyze causal maps network starting from a causal map is described in largely di€er depending on the context of appli- Section 4. cation. Matrix algebra and network analytic Bayesian networks have their roots in attempts methods have been used to explore the structural to represent expert knowledge in domains where properties of causal maps (Axelrod, 1976; Bougon expert knowledge is uncertain, ambiguous, and/or et al., 1977; Eden et al., 1979). System dynamics incomplete. Bayesian networks are based on has been used to analyze causal maps from a dy- probability theory. A primer on Bayesian net- namic perspective (Forrester, 1961; Wolstenholme works is found in Speigelhalter et al. (1993). and Coyle, 1983). Chen et al. (1979) uses an un- A Bayesian network model is represented at structured variation of causal maps to construct two levels, qualitative and quantitative. At the decision trees. They use these decision trees for qualitative level, we have a directed acyclic graph decision analysis using probability assessments in which nodes represent variables and directed and utility functions. More recently, Wang (1996) arcs describe the conditional independence rela- describes a neural network model to compare the tions embedded in the model. Fig. 2 shows a dynamic properties of causal maps. In this study, Bayesian network consisting of four discrete vari- we use the probabilistic inference procedures of ables: Market Dynamics (D), Product Life Cycle Bayesian networks to make inferences about (C), Market Leadership (L), and Rate of Product variables in causal maps. Launch (R). At the quantitative level, the depen- dence relations are expressed in terms of condi- tional probability distributions for each variable in 3. Bayesian networks the network. Each variable X has a set of possible values called its state space that consists of mutu- In this section, we brie¯y describe the semantics ally exclusive and exhaustive values of the vari- of Bayesian networks and their use in making in- able. In Fig. 2, e.g., Market Dynamics has two ferences. A procedure for constructing a Bayesian states: ÔhighÕ and ÔlowÕ; Market Leadership has two

Fig. 2. A Bayesian network with conditional probability tables. S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498 483 states: ÔleaderÕ and ÔfollowerÕ; Product Life Cycle conditional independence assumptions. Thus the has two states: ÔshortÕ and ÔlongÕ; and Rate of lack of an arc from D to L signi®es that L is in- Product Launch has two states: ÔhighÕ and ÔlowÕ.If dependent of D; the lack of an arc from C to L there is an arc pointing from X to Y, we say X is a signi®es that L is independent of C; and the lack of parent of Y. For each variable, we need to specify a an arc from D to R signi®es that R is conditionally table of conditional probability distributions, one independent of D given C and L. for each con®guration of states of its parents. In general, there may be several sequences Fig. 2 shows these tables of conditional distribu- consistent with the arcs in a Bayesian network. In tions ± P D†; P CjD†; P L†, and P RjC; L†. such cases, the list of conditional independence assumptions associated with each sequence can be shown to be equivalent using the laws of condi- 3.1. Semantics tional independence (Dawid, 1979). Pearl (1988) and Lauritzen et al. (1990) describe other equiva- A fundamental assumption of a Bayesian net- lent graphical methods for identifying conditional work is that when we multiply the conditionals for independence assumptions embedded in a Baye- each variable, we get the joint probability distri- sian network graph. bution for all variables in the network. In Fig. 2, Unlike a causal map, the arcs in a Bayesian e.g., we are assuming that network do not necessarily imply causality. The (lack of) arcs represent conditional independence P D; C; L; R†ˆP D† P CjD† P L† assumptions. How are conditional independence P RjC; L†; and causality related? Conditional independence can be understood in terms of relevance. If Z is where denotes pointwise multiplication of ta- conditionally independent of X given Y, then this bles. The rule of total probability tells us that statement can be interpreted as follows. If the true state of Y is known, then in assigning probabilities P D; C; L; R†ˆP D† P CjD† P LjD; C† to states of Z, the states of X are irrelevant. In P RjD; C; L†: practice, the notion of direct causality is often used to make judgments of conditional independence. Comparing the two, we notice that we are making Consider a situation where X directly causes Y and the following assumptions: P LjD; C†ˆP L†, i.e., Y in turn directly causes Z, i.e., the causal e€ect of L is independent of D and C; and P RjD; C; L†ˆ X on Z is completely mediated by Y. Then it is P RjC; L†, i.e., R is conditionally independent of D clear that although X is relevant to Z, if we know given C and L. the true state of Y, further knowledge of X is ir- Notice that we can read these conditional in- relevant (for assigning probabilities) to Z, i.e., Z is dependence assumptions directly from the Baye- conditionally independent of X given Y. This sit- sian network graph as follows. Suppose we pick a uation is represented by the Bayesian network sequence of the variables such that for all directed X ® Y ® Z in which there is no arc from X to Z. arcs in the network, the variable at the tail of each As another example, consider the situation where arc precedes the variable at the head of the arc in X directly causes Y and X also directly causes Z. the sequence. Since the directed graph is acyclic, Although knowledge of Y is relevant to Z (if Y is there always exists such a sequence. In Fig. 2, e.g., true then it is more likely that X is true which in one such sequence is DCLR. Then, the condi- turn means that it is more likely that Z is true), tional independence assumptions can be stated as once we know the true state of X, then further follows. For each variable in the sequence, we are knowledge of Y is irrelevant to Z, i.e., Y is con- assuming that it is conditionally independent of its ditionally independent of Z given X. This situation predecessors in the sequence given its parents. The is represented by the Bayesian network Z ¬ X ® Y essential point here is that missing arcs (from a in which there is no arc from Y to Z or vice-versa. node to its successors in the sequence) signify Finally as a third example, consider the situation 484 S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498 where X and Y are two independent direct causes in the concept of local computation where we of Z, i.e., X and Y are unconditionally indepen- compute the marginal of the joint without actu- dent. But if we learn something about the true ally computing the joint distribution. A crucial state of Z, then X and Y are no longer irrelevant to feature of a Bayesian network is that it describes each other (if Z is believed to be true and X is false, a joint distribution as built out of local relation- then it is more likely that Y is true), i.e., Y is not ships within groups of variables ± such as a node conditionally independent of X given Z. This sit- and its parents. Instead of tackling the whole uation is represented by the Bayesian net collection of variables simultaneously, we can use X ® Z ¬ Y in which there is no arc from X to Y or the concept of factorization. Factorization in- vice-versa. volves breaking down the joint probability dis- tributions into subgroups called factors or belief universes in such a way that the naive computa- 3.2. Making inferences tions described above need only be performed within each belief universe. Since the state space Inference (also called probabilistic inference) in of such belief universes is much smaller than that a Bayesian network is based on the notion of evi- of the joint probability distribution, the calcula- dence propagation. Evidence propagation refers to tions become manageable. an ecient computation of marginal probabilities To ensure that we obtain the correct answers of variables of interest, conditional on arbitrary when considering all the variables together, we con®gurations of other variables, which constitute need to develop ways for the belief universes to the observed evidence (Speigelhalter et al., 1993). communicate with each other, so that (for exam- Probabilistic inference refers to the process of ple) the e€ect of conditioning on a variable in one computing the posterior marginal probability dis- belief universe can be felt by those in another. This tributions of a set of variables of interest after is called the procedure of evidence propagation, obtaining some observations of other variables in i.e., how information propagates between the dif- the model. ferent variables in the Bayesian network. Several Once a Bayesian network is constructed, it can propagation architectures have been developed be used to make inferences about the variables in which provide di€erent algorithms to pass mes- the model. The conditionals given in a Bayesian sages between the belief universes. Pearl (1986) network representation specify the prior joint dis- developed a message-passing scheme that updates tribution of the variables. If we observe (or learn the probability distributions for each node in re- about) the values of some variables, then such sponse to observations of one or more variables. observations can be represented by tables where Lauritzen and Speigelhalter (1988), Jensen et al. we assign 1 for the observed values and 0 for the (1990) and Shenoy and Shafer (1990) devised unobserved values. Then the product of all tables propagation algorithms that ®rst transforms the (conditionals and observations) gives the (unnor- Bayesian network into a tree where each node in malized) posterior joint distribution of the vari- the tree corresponds to a belief universe. The al- ables. Thus, the joint distribution of variables gorithm then exploits several mathematical prop- changes each time we learn new information about erties of this tree to perform probabilistic the variables. inference. In theory, the posterior marginal probability of These propagation algorithms can be used to a variable X, say P X †, can be computed from the make two types of inferences. Often we are inter- joint probability by summing out all other vari- ested in the values of some target variables. In this ables except X one by one. In practice, such a naive case, we make inference by computing the mar- approach is not computationally tractable when ginal of the posterior joint distribution for the we have a large number of variables because the variables of interest. Consider the situation de- joint distribution has an exponential number of scribed by the Bayesian network in Fig. 2. Suppose states and values. The key to ecient inference lies we are interested in the true state of Rate of S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498 485

Product Launch (R). Given the prior model (as 4. Constructing Bayesian causal maps per the probability tables shown in Fig. 2), the marginal distribution of R is 0.71 for high and In this section, we sketch a procedure for con- 0.29 for low. Now suppose we learn that Market structing Bayesian causal maps starting from a Dynamics is low. The posterior marginal distri- causal map based on concepts from Bayesian bution of R changes to 0.52 for high and 0.48 for networks and causal maps. The procedure for low. Suppose we further learn that the state of constructing Bayesian causal maps proceeds in Market Leadership is follower. Then the marginal two stages: qualitative and probabilistic. distribution of R changes to 0.34 for high and 0.66 for low. This type of inference is referred to as Ôsum propagationÕ. 4.1. Qualitative stage Sometimes we are more interested in the con- ®guration of all variables (``the big picture'') rather In this stage, the structure of the causal maps is than the values of individual variables. In this case, modi®ed for two reasons. First, to eliminate some we can make inferences by computing the mode of of the limitations of the modeling procedure used the posterior joint distribution, i.e., a con®gura- in the derivation of causal maps. Second, to make tion of variables that has the maximum probabil- the causal maps compatible with the Bayesian ity. Consider again the situation described by the network approach. In the qualitative stage, we Bayesian network in Fig. 2. Given the prior model address four major modeling issues. These issues (as per the probability tables shown in Fig. 2), the are discussed in the following paragraphs. mode of the prior joint distribution is (high market dynamics, leader, short product life cycle, high rate 4.1.1. Conditional independencies of product launch). Now suppose we learn that A network model can be either a dependence Market Dynamics is low. The mode of the poste- map (D-map) or an independence map (I-map) rior joint distribution changes to (low market dy- (Pearl, 1988). A D-map guarantees that vertices namics, leader, long product life cycle, high/low (concepts) found to be connected are indeed de- rate of product launch) (both con®gurations have pendent; however, it may display a pair of de- the same maximum probability). Suppose we fur- pendent concepts as a pair of separated vertices ther learn that the state of Market Leadership is (concepts). An I-map, on the other hand, guar- follower. Then the mode of the joint distribution antees that vertices (concepts) found to be sepa- changes to (low market dynamics, leader, long rated are indeed conditionally independent, given product life cycle, low rate of product launch). other variables. However, it may display a pair of This type of inference is referred to as Ômax prop- independent concepts as connected vertices (con- agationÕ. cepts). A model that is both a D-map and an The results of inference are more sensitive to I-map is called a perfect map. the qualitative structure of the Bayesian network A causal map is a directed graph that depicts than the numerical probabilities (Darwiche and causality between variables as perceived by indi- Goldszmidt, 1994). For decision making, the in- viduals. Since an arrow between two variables ference results are robust with respect to the nu- implies dependence, it is a D-map. The absence of merical probabilities (Henrion et al., 1994). an arrow between two variables does not imply a There are several commercial software tools lack of dependence. In other words, a causal map such as Hugin [www.hugin.com] and Netica does not guarantee that variables found to be [www.norsys.com] that automate the process of separated correspond to independent concepts, inference. These tools allow the user to enter the i.e., it is not an I-map. Bayesian network structure graphically, enter the Bayesian networks, on the other hand, are numerical details, and then do inference of either I-maps. Given a sequence of variables, an absence type. The results of the inference are then shown of arrow from a variable to its successors in the se- graphically using bar charts. quence implies conditional independence between 486 S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498 the variables. Conditional independence is an im- portant issue in making inferences since it speci®es the relevance of information on one variable in making inference on another. Thus if we are to regard a causal map as a Bayesian network, it is important to ®rst convert the causal map model from a D-map to an I-map. This can be done in consultation with the expert (whose causal map is at issue) by ensuring that all dependencies are de- picted in the causal map. An adjacency matrix can be used to achieve this. An adjacency matrix is the matrix of all possible direct cause±e€ect relations Fig. 3. Making a causal map a perfect map. among the variables where the i; j† entry indicates the existence of a direct causality relation from original causal map an I-map. These unmodeled variable i to variable j (Axelrod, 1976). The adja- dependencies a€ect the inferences made on vari- cency matrix is represented as an n by n matrix in ables ± Sales Uncertainty, Product Risk and Price which the columns and rows are labeled with the Level. In the original causal map, inference on names of the variables. The rows represent causes Sales Uncertainty is not conditional on any other and columns represent e€ects. For each pair of variables in the map. For example, Market dy- variables, the subject (whose causal map is at is- namics and Market Share Distribution are not sue) is asked to specify if there is a causal relation relevant in making inference on Sales Uncertainty. (scored initially as 1 for yes, and 0 for no) and if However, in the modi®ed causal map, inference on yes, to specify which is the cause and which is the Sales Uncertainty is conditional on Market Share e€ect. Subject should be instructed that if (s)he Distribution as well as Market Dynamics. This considers the two variables to have reciprocal in- example illustrates how unmodeled dependencies ¯uences, (s)he should indicate which one has the can change the inferences that can be made on the more dominant causal in¯uence (to eliminate the various variables in the causal maps. occurrence of direct reciprocal causality wherein a After incorporating all possible dependencies variable is both a direct cause and an e€ect of between variables, the revised causal map is a another variable). Once the subject has identi®ed perfect map ± a D-map as well as an I-map. 1 An the existence of a causal relation, (s)he can then be arrow between two variables in the map implies a asked to indicate nature of the relation (positive or causal relation between them; whereas missing negative). Thus when the subject is ®nished, each arrows imply conditional independence. cell in the adjacency matrix will have one of the three values: )1 for a negative causal relation, 0 for no causal relation, or +1 for a positive causal 4.1.2. Reasoning underlying cause±e€ect relations relation. Causal maps identify individualsÕ perceptions of Fig. 3 indicates a part of a Bayesian causal map cause±e€ect relationships between variables based of a marketing expert relating to a product de- on language rather than the reasoning processes velopment decision. The solid arrow (from Prod- (Carley and Palmquist, 1992). Studies in manage- uct Life Cycle to Rate of Product Launch) was rial cognition indicate that individuals reason by identi®ed in the original causal map. The dashed accumulating possibly signi®cant pieces of infor- arrows (from Market Dynamics to Product Life Cycle, from Market Dynamics to Sales Uncer- 1 tainty, from Market Share Distribution to Sales If we start with an arbitrary probability distribution, it is Uncertainty, from Sales Uncertainty to Product not always possible to ®nd a Bayesian network representation that is a perfect map (Pearl, 1988). Our starting point is an Risk, and from Competition Strategy to Price expertÕs causal model and we are assuming that we can always Level) were identi®ed in the process of making the ®nd a Bayesian network representation that is a perfect map. S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498 487 mation and organizing them in relation to each the arc from Product Risk to Sales Projection. other so as to be able to combine them into a Causal statement 2 involves abductive reasoning. conclusion and decision (Jaques and Clement, Since Product Risk is a latent variable, and Sales 1996). Individuals use such reasoning processes to Projection is one of the observable measures of put information together as a cause±e€ect series of Product Risk, the decision maker is making in- events leading to predicted future courses of ference about the latent variable Product Risk events. These reasoning processes are important in based on the observable variable Sales Projection. decision making and in making inferences about This does not imply that Sales Projection causes future decision outcomes. Product Risk. The reasoning in this causal state- Literature on logic suggests that individuals ment is in the direction opposite of causation. perceive cause±e€ect relationships based on two Causal statements involving abductive reasoning types of reasoning: deductive and abductive are misrepresented in a causal map by an arc from (Charniak and McDermott, 1985; Winston, 1984). e€ect to cause. Such misrepresentation can also A reasoning process is called deductive when we lead to redundant circular relations between vari- reason from causes to e€ects, i.e., in the direction ables in the causal map. For instance, both the of causation. For example, in the medical domain, arrows in Fig. 4 may be represented in a causal risk factors (e.g., smoking) are regarded as causes, map creating a loop. and the diseases (e.g., lung cancer) as e€ects. When A distinction between deductive and abductive a physician, confronted with a patient who has reasoning behind the causal linkages is essential been a smoker, reasons that the patient is at risk to establish accurate directions of linkages in for lung cancer, (s)he is reasoning deductively. causal maps. The emphasis in deriving causal A reasoning process is called abductive when maps should be on the causal theory underlying we reason from e€ects to causes, i.e., in the di- the causal statements rather than the language rection opposite to causation. For example, dis- used. eases (e.g., lung cancer) are regarded as causes of symptoms (e.g., positive X-ray). When a physician, 4.1.3. Distinguishing between direct and indirect after observing a patientÕs positive X-ray result, relationships concludes that the patient is probably su€ering The procedure for deriving causal maps does from lung cancer, (s)he is reasoning abductively. not provide for a distinction between ÔdirectÕ and The di€erence between deductive and abductive ÔindirectÕ relationships between concepts (Eden reasoning underlying causal statements and their et al., 1992; Laukkanen, 1996). This distinction is e€ect on representation of causal linkages are il- important to identify conditional independencies lustrated in Fig. 4. Causal statement 1 involves the in the causal maps. Fig. 5 depicts how a lack of use of logical deduction and the reasoning is in the distinction between direct and indirect relationship direction of causation. This is correctly re¯ected in a€ects conditional independence assumptions in a

Fig. 4. Distinguishing between deductive and abductive rea- soning. Fig. 5. Distinguishing between direct and indirect relations. 488 S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498 causal map. In Fig. 5, both Product Risk and Sales in previous paragraphs. However, if causal loops Projection directly in¯uence Product Decision. In exist despite these clari®cations, then they can be the modi®ed Bayesian causal map, there is no eliminated by aggregating the variables into a linkage between product risk and product decision single variable. In causal maps, causal assertions implying that product risk impacts product deci- of individuals are aggregated (or clustered) into sion strictly through Sales Projection. If we have broader concepts. All variables or nodes in cir- complete information on Sales Projection, any cular relations are of the same hierarchical status additional information on Product Risk would be and so if aggregated to a single node, the gen- irrelevant in making inferences about Product eral form of the causal map can still be hierar- Decision. chical. A clear distinction between direct and indirect In addition to coding mistakes, feedback loops cause±e€ect relations is important for three rea- may indicate dynamic relations between variables sons. First, it helps us understand the nature of overtime. In such cases, part of the linkages in the relations between variables. It tells us whether the loop pertains to a current time frame and some e€ect of a variable on another is completely linkages pertain to a future time frame. In such modeled by the e€ect of the ®rst on a third medi- cases, deaggregating the variables into two time ating variable (which in turn is a cause of the frames can often solve the problem of circularity. second). Second, if Product Risk (in Fig. 5) a€ects For example, Fig. 6 shows a reciprocal causal Product Decision only through Sales Projection relation between Market Leadership and Rate of Figures, then an arrow from product risk to Product Launch and reasoning underlying this product decision is redundant and increases the circular relation. Arrow t1 implies that companies complexity of the representation. Finally, distinc- having market leadership tend to introduce more tion between direct and indirect cause±e€ect rela- products that are new. Arrow t2 implies that in- tions allows incorporation of conditional troduction of new products a€ects the Market independencies in causal maps. As we have seen Leadership of the ®rm at some future point in earlier, conditional independencies are critical in time. The circular relation has resulted from ag- making inferences on the variables in large causal gregation of the variable Market Leadership maps. across two time frames: t1 and t2. After de-ag- gregating Market Leadership into two time 4.1.4. Eliminating circular relations frames, we get an acyclic relation between the Causal maps are directed graphs and are char- three variables. To make the causal map acyclic, acterized by a hierarchical (or acyclic) structure. we arbitrarily retain one of the two relations and However, circular relations or causal loops destroy exclude the other from the causal map. An acyclic the hierarchical form of a graph. Circular relations structure of the causal map is essential to the in the causal maps violate the acyclic graphical inference process and to make causal maps com- structure required in a Bayesian network. It is patible with Bayesian networks. Bayesian net- therefore essential to eliminate circular relations to works are unable to represent reciprocal causal make causal maps compatible with Bayesian net- relations. works. Causal loops can exist for two reasons (Eden et al., 1992; Hu€, 1990; Laukkanen, 1996). First, they may be coding mistakes that need to be corrected. Second, they may represent dynamic relations between variables across multiple time frames. Coding mistakes can be recti®ed by clarifying causal linkages between variables in terms of deductive versus abductive reasoning or direct versus indirect linkage; issues already discussed Fig. 6. Deaggregating variables over time. S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498 489

4.2. Probabilistic stage Treatment of uncertainty is necessary in causal maps because there are di€erent sources of un- In this stage, the numerical parameters of the certainty in a decision problem. The observations Bayesian causal map are assessed. Also, the causal may be uncertain, the information about variables Bayesian map is validated using validation proce- in the map may be incomplete, the relations in the dures. domain represented by the map may be of a non- deterministic type, or the variables involved may be vague. One common way of capturing uncer- 4.2.1. Assessment of numerical parameters tainty of the variables in a Bayesian network is to The parameters of the causal map consist of the measure a personÕs Ôdegree of beliefÕ for that vari- strength of the linkages between the di€erent able conditional on the states of its parents. This variables. However, in the context of inference, we uncertainty associated with the variables in a de- are interested in the uncertainty associated with cision model is sensitive to the context in which the every variable in the map and the interactive e€ects certainties have been established. The process of of multiple causal variables on e€ect variables. The measuring degrees of belief is commonly referred Bayesian network approach provides tools to ad- to as probability assessment or probability encoding dress both of these issues in de®ning the parame- procedure. ters of the map. Major issues in the derivation of Many di€erent probability encoding techniques the parameters of the causal map are discussed in are available (for a detailed review see (Spetzler the following sections. and Stael von Holstein, 1975)) wherein a subject One of the most challenging problems in responds to a set of questions either directly by generating a causal map is specifying the strength providing numbers or indirectly by choosing be- associated with the causal connections in the tween simple alternatives or bets. The choice of map. Traditional analytical methods to represent response mode (direct or indirect) as well as the the parameters of the causal maps are based on choice of a method within each mode depends on social network methods (Knoke and Kuklinski, the preferences of the subject. Spetzler and Stael 1982). The parameters of a causal map constitute von Holstein (1975) describe three direct response the strength of each pairwise linkage in the map encoding methods ± cumulative probability, frac- and are measured in terms of the frequency of tiles and verbal encoding ± to elicit probabilities. the linkage in the narrative. This measure has In the cumulative probability method, the subject been criticized by researchers for several reasons is asked to assign the cumulative probability as- (Hu€, 1990; Laukkanen, 1996). First, it does not sociated with a variable conditioned ton the states explicitly incorporate an individualÕs perception of its parent variables. The probability response of the strength associated with a linkage in the can be expressed either as an absolute number causal map. A number of factors such as the type (0.30), as a discrete scale (``three on a scale from of interview (structured versus unstructured), zero to ten''), or as a fraction using a discrete scale length of the narrative (either interview of ar- (``three in ten''). Verbal encoding uses verbal de- chival data), etc., can a€ect the frequency with scriptions to characterize events in the ®rst phase which concepts and linkages are mentioned in the of the encoding procedure. The descriptors used map. Second, the strength of the causal connec- are those to which the subject is accustomed to tion is based on pairwise comparison of each such as ``high'', ``medium'' or ``low''. The quanti- causal variable independently on the e€ect vari- tative interpretation of the descriptors is then en- able. It does not capture the interactive e€ects of coded in a second phase. The form chosen to causal variables on the e€ect variables. Finally, express the probability (absolute number, per- the uncertainty associated with each variable in centage, fraction or verbal) should be the one most the map is not represented in the frequency familiar to the subject. measure. All variables are assumed to have the When a variable has many parents, the num- same level of uncertainty. ber of probability assessments can be reduced by 490 S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498 assessing the nature of the relationship between pretations have contaminated the modi®cation the variable and its parents such as noisy-OR, process. noisy-AND, etc. (Henrion, 1989; Pradhan et al., Validation of Bayesian causal maps involve 1994). Once the parameters of the causal map are establishing that the modi®cations made to the identi®ed, propagation algorithms can be used to original causal map measure what they are in- make inferences about the variables in the causal tended to measure. This validation has to be es- maps. tablished at the qualitative level as well as the quantitative level. Typical practice suggests that 4.2.2. Validation researchers check the interpretation of variables in We propose validation procedures for both a causal map with the subjects involved. This can qualitative and quantitative stages of constructing be extended to the modi®cation procedure in the Bayesian causal maps. In the qualitative stage, we qualitative phase wherein modi®cations can be extend the reliability and validity techniques made to the original causal maps after consulta- suggested by causal mapping studies (Axelrod, tion with the subjects involved. 1976; Carley, 1997; Eden et al., 1992; Hu€, 1990) At the quantitative level, sensitivity analysis to the modi®cation procedures proposed in our approach from operations research can be used to study. A typical method used to establish reli- validate the map (Churchman et al., 1957; Howard ability of the cognitive mapping procedure is to and Matheson, 1981). Sensitivity analysis consists achieve a consensus among multiple raters for all of examining the posterior marginals of decision the stages of the cognitive mapping procedure. variables under di€erent scenarios and then cor- Accordingly, the modi®cations to the original roborating these marginals with the domain ex- causal map (described in Section 4.1) can be pert. The sensitivity analysis procedure is made through a consensus between multiple rat- illustrated in the next section in the context of a ers. For example, modi®cations agreed to by two case study. or more raters can be retained, whereas those for Bayesian causal maps combine the strengths of which agreement cannot be reached can be causal maps and Bayesian networks and reduce thrown out. Such inter-rater reliability also re- the limitations of either. A comparison of the three duces the possibility that researchersÕ own inter- network models is shown in Table 1.

Table 1 A comparison of causal maps, Bayesian networks and Bayesian causal maps Features Causal map Bayesian network Bayesian causal map Graphical structure: 1. Type of network model Dependence map Independence Map Perfect map 2. Type of graph Directed graph Directed acyclic graph Directed acyclic graph 3. De®nition of relations Based on causality Based on conditional Based on causality and independence conditional independence

Model parameters: 4 .Uncertainty of variables Not represented Represented Represented 5. Background noise or Not represented Represented Represented unmodeled factors 6. Interactive e€ects of multiple Not represented Represented through Represented through causal variables on e€ect variables conditional probabilities conditional probabilities 7. Validation Qualitative Quantitative using Qualitative and quantitative sensitivity analysis Inference procedure: 8. Inference procedure Qualitative based on the Quantitative based on Qualitative and quantitative graphical structure evidence propagation S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498 491

5. A case study: Product development decision cated its con®dence in solving the problems, but there is a slight chance that devising a workable This section describes a case study of a con- solution may take a much longer than predicted struction of a Bayesian causal map. The main goal resulting in a signi®cant delay in introducing this of the case study is to provide a test bed for modi®ed product. The best case pro®t scenario for the construction of a Bayesian causal map using the the third option is much higher than the second. technique described in the previous section. The Subject. The subject was a marketing professor case study is also useful for illustration purposes. whose prior work experience and current line of We describe how starting from a causal map, we research was relevant to the decision context. construct the qualitative structure of a Bayesian Data collection. The subject was given the de- causal map. We show how additional information cision case a week before the interview. An inten- can be collected from a subject to address the sive interview (two to three hours) was conducted modeling issues discussed in Section 4.1 as well as with the subject to elicit his/her analysis of the case to derive the numerical parameters of the Bayesian and decision recommendations. The subject was causal map. In addition, we describe how Bayesian asked to identify factors that he/she would con- network software can be used to draw probabi- sider as the marketing executive responsible for the listic inferences in a Bayesian causal map. new product decision. The subject was encouraged Decision context. We used a modi®ed version of to identify factors not identi®ed in the decision the New Product Decision case (Clemen, 1996) scenario that, in his/her opinion, might be relevant from decision analysis literature as a decision to the decision. The narrative yielded by this in- context. We chose this case because it was su- terview was used to construct the original causal ciently unstructured to allow an expert to develop map shown in Fig. 7 below using a textual analysis his/her own framework in diagnosis, analyses, and method (for details see Carley and Palmquist, recommendations of decision options. Also, the 1992; Eden et al., 1979; Hu€, 1990; Laukkanen, decision alternatives, outcomes, and situational 1996). Follow-up interviews were conducted with factors all involved uncertainty. the subject to modify the structure of the causal The decision scenario presented in the case is as map to construct the qualitative structure of the follows. A marketing executive of a computer- Bayesian causal map. The next few paragraphs manufacturing ®rm is faced with a decision of in- discuss this procedure in detail. troducing a new computer model. There are three Causal map. The original causal map shown in di€erent options. The ®rst option is to continue Fig. 7 describes the subjectÕs causal perceptions of with the existing product line that is essentially a the decision problem in the new product decision risk-free proposition with stable expected pro®ts. case. There are 18 variables in the map represent- The second option, which is riskier than the ®rst, ing industry factors, ®rm factors, product charac- involves introducing a modi®ed version of the ex- teristics, and decision options. The three major isting model. This option involves signi®cant ini- industry factors are Market Dynamics, Product tial investment and very little information is Life Cycle, and Market Share Distribution. Firm available on sales predictions. However, a working factors represent the prior market experience of prototype of the modi®ed product has already the ®rm and long term strategies of the ®rm and been developed and pricing decisions have been include Competitive Strategy, Brand Loyalty, made. In other words, it is ready to be produced Market Leadership, Cross-Functional Teams, and marketed. Moreover, this product, if success- R&D, Rate of Product Launch and Prior Presence fully marketed, has the potential of yielding pro®ts (in the market). The product characteristics are much higher than the ®rst product. The third op- represented by Sales Uncertainty, Production De- tion, which is riskier than the second, consists of lay, Initial Investment, Product Risk, and Sales introducing a product that has not yet been fully Projection. The three decisions the manager has to developed. The prototype developed has some make in this speci®c decision context are Willing- problems, and the engineering division has indi- ness to Take Risk, Product Pricing, and Product 492 S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498

Fig. 7. The original causal map of a marketing expert for the product development decision.

Decision. The manager has to evaluate each of the sion. Competitive Strategy has a direct a€ect on three products options in terms of go/no go deci- Product Pricing, which a€ects Sales Projection. sion. Sales Projection, in turn, a€ects Product Decision. Constructing a Bayesian causal map. The causal In other words, an indirect relationship between map in Fig. 7 was used to construct a Bayesian Competitive Strategy, Sales Projection and Prod- causal map. The procedure proceeded in two uct Decision was depicted as a direct relationship stages: qualitative and probabilistic. in the original causal map. Second, the original In the qualitative stage, we conducted two fol- causal map shows a direct causal relation between low-up interviews to address the four modeling Product Risk and Product Decision. However, in issues discussed in Section 4.1. In the ®rst follow- the modi®ed causal map, Product Risk a€ects up interview, the subject was provided with the Sales Projection, which in turn a€ects Product original causal map and was requested to provide Decision. The indirect relation between Product clari®cations of the cause±e€ect relations in the Risk and Product Decision was shown as a direct map based on the following four instructions. relation in the original map. Finally, Market 1. Direct causality between variables. The sub- Leadership and Product Decision were directly ject was instructed that an arrow between two related in the original map. In the modi®ed causal variables in the map should represent a direct map, Market Leadership directly a€ects Willing- cause±e€ect relationship only. This resulted in ness to Take Risk, which in turn directly a€ects three major changes in the linkages in the original Product Decision. The linkages in the modi®ed map. First, in the original causal map, Competitive causal map depict only direct relations between Strategy leads to Sales Projection as well as variables. Product Decision. However, in the modi®ed causal 2. Conditional independence. The subject was map, Competitive Strategy does not have a direct instructed that an absence of an arrow between e€ect on either Sales Projection or Product Deci- two variables in the map should represent a lack of S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498 493 dependence between the two variables. This re- follow-up interview) to check if the modi®ed map sulted in ®ve additional links in the modi®ed map. correctly re¯ected the modi®cations made to the First is the link between Market Dynamics and original map. The ®nal modi®cations were made Product Life Cycle that did not exist in the original to the map through close consultations with the causal map. Second is the link between Market subject. The resultant structure of the Bayesian Dynamics and Sales Uncertainty. These variables causal map is shown in Fig. 8. were shown as conditionally independent in the In the probabilistic stage, the parameters of the original map. Similarly, the modi®ed map shows model were assessed. The parameters of a Baye- links between Market Share Distribution and Sales sian causal map consist of marginal probabilities Uncertainty, Market Share Distribution and Sales and conditional probabilities. To assess the mar- Uncertainty, and Market Share Distribution and ginal probabilities, the expert was asked to provide Prior Presence that did not exist in the original the following information. causal map. 5. To rate the marginal and conditional prob- 3. Deductive reasoning. The subject was in- abilities on a discrete scale (0 to 10); and structed that an arrow should be directed from 6. To identify the type of interactive e€ects of cause to e€ect only (deductive reasoning). The multiple causal variables on e€ect variables. For original map shows a link from Sales Projection to example, whether each causal variable a€ects the Product Risk. However, as shown in Fig. 4, the e€ect variable independently (noisy-OR model), or direction of the arrow is based on abductive rea- whether each causal variable a€ect the e€ect vari- soning from the observable measure (Sales Pro- ables through interactions of two or more vari- jection) to the latent cause (Product Risk). The ables (noisy-AND), or some combination of the direction of this arrow changed in the modi®ed two (Henrion, 1989; Pradhan et al., 1994). causal map from Product Risk to Sales Projection. Making inferences. We used Netica [www.nor- 4. Similar time frames. The subject was in- sys.com] to make probabilistic inferences using structed that all variables should pertain to a both sum propagation and max propagation. speci®c time frame t1. We de®ned t1 as the period Sum propagation. The sum propagation com- until the ®nal decision of marketing the new putes the marginal probabilities of all the model product was reached (to eliminate circular rela- variables and updates the marginals with all ad- tions due to relations pertaining to di€erent peri- ditional evidence received about other variables. ods). This resulted in the elimination of a In our case study, we can evaluate each product reciprocal relationship between Market Leader- decision option under di€erent scenarios. The ship and Rate of Product Launch shown in Fig. 6. scenarios were de®ned in consultation with three At the end of the ®rst interview, the subject was marketing experts (including the subject), and they asked to provide a brief rationale or explanation represent situations in which there are unambigu- for every link added or deleted in the original ous prescriptions for product decisions in the causal map. This explanation provided a validity marketing literature. We illustrate how predictions check for modi®cations made to the map. Based can be made about our subjectÕs perceptions of on the suggestions made and the explanation product option 1 (risk free proposition) in terms of provided by the subject, two raters (one researcher ÔgoÕ or Ôno goÕ under di€erent information condi- and another marketing expert) modi®ed the orig- tions. The same procedure can be used to evaluate inal causal map through consensus. Modi®cations product options 2 and 3. Product option 1 is to agreed to by both the raters were made to the continue with the current product line and is a original causal map, and those not agreed to were risk-free proposition with stable expected pro®ts. excluded. This option can be represented by specifying the A second follow-up interview was conducted to states of two variables in the Bayesian causal map: validate the modi®cation process. Accordingly, the Initial Investment ˆ low, and Production De- subject was provided with a modi®ed causal map lay ˆ no delay. Since the product is similar to the (based on the information collected in the ®rst one already being marketed, the initial cost of 494 S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498

Fig. 8. The ®nal bayesian causal map of the product development decision. development and commercialization is minimal. variables directly a€ecting product decision when Similarly, the chance of delay in production due to all other variables (except Initial Investment and development problems is also very low. Table 2 Production Delay) in the map are unknown. The shows the prior marginal probabilities of the three marginal probabilities of these variables are as

Table 2 Prior and posterior marginal probabilities under two di€erent scenarios Variables Prior marginals Posterior marginals in Posterior marginals in scenario 1a scenario 2b 1. Rate of Product Launch: High 0.70 0.15 0.80 Low 0.30 0.85 0.20 2. Willingness to Take Risk: High 0.66 0.13 0.83 Low 0.34 0.87 0.17 3. Sales Projection: High sale high risk 0.49 0.32 0.41 Low sale low risk 0.51 0.68 0.59 4. Product Option 1c: Go 0.51 0.63 0.44 No go 0.49 0.37 0.56 a Scenario 1: Market Share Distribution ˆ low share, Competitive Strategy ˆ cost-eciency, Market Leadership ˆ follower and R&D ˆ low. b Scenario 2: Market Share Distribution ˆ high share, Competitive Strategy ˆ cost-eciency, Market Leadership ˆ leader and R&D ˆ high. c Product Option 1: Production Delay ˆ no delay, and Initial Investment ˆ low. S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498 495 follows. Rate of Product Launch: high ˆ 0.70, As shown in Table 2, the posterior marginal of low ˆ 0.30; Willingness to Take Risk: high ˆ 0.66 Rate of Product Launch ˆ high increases from and low ˆ 0.34; Sales Projection: high sale high 0.70 to 0.80, that of Willingness to Take risk ˆ 0.49, low sale low risk ˆ 0.51; Product De- Risk ˆ high increases from 0.66 to 0.83, and that cision: go ˆ 0.51, no go ˆ 0.49. Since the industry of Sales Projection ˆ low sale low risk increases speci®ed in the decision case is the computer in- from 0.51 to 0.59. The posterior marginal proba- dustry in which market dynamics are high, the bility of Product Decision ˆ no go increases from priors show that any company in the industry 0.50 to 0.56. This implies that in scenario two, our would be willing to take risk and have a high rate marketing expert is more likely to reject Product of product launch. However, if we have additional Option 1. information about speci®c ®rm characteristics, A relevant question is which product option then our inferences about Product Decision may will our marketing expert choose in the second change. scenario. To ®nd an answer to this question, we We specify two di€erent scenarios and show evaluated both product options two and three how our inferences about product decision option following the procedure described above. The re- one change depending on the ®rm factors. In the sults show that the highest probability of ÔgoÕ in ®rst scenario, we consider a struggling ®rm with a scenario two is for Product Option 3. Product low market share that follows a price-based Option 3 can be described in the map by specifying strategy. Accordingly we specify the states of the Initial Investment ˆ high and Delay in Produc- following four ®rm variables in the map: Market tion ˆ high. This increases the probability of Sales Share Distribution ˆ low, Competitive Strategy ˆ Projection ˆ high sale high risk to 0.56, whereas cost-eciency, Market Leadership ˆ follower, and the probabilities of Willingness to Take Risk and R&D ˆ low. Based on this information, we prop- Rate of Product Launch remain the same. The agate the information to compute the posterior posterior marginal probability of Product Option marginals of variables of interest. A comparison of 3 ˆ go is 0.62. This implies that in scenario two, prior and posterior marginals of Rate of Product our subject is more likely to go for a high sale high Launch, Willingness to Take Risk, Sales Projec- risk proposition such as Product Option 3 since he/ tion and Product Decision is shown in Table 2. she has an organizational infrastructure which al- When additional information is received about lows him/her to take risk in order to go for a new ®rm factors, the posterior probability of Rate of product with potentially very high sales. Product Launch ˆ low increases from 0.30 to 0.85, Max propagation. The max propagation com- that of Willingness to Take Risk ˆ low increases putes the most likely con®guration of the states of from 0.37 to 0.87 and that of Sales Projec- model variables under di€erent scenarios. This is tion ˆ low sale low risk increases from 0.51 to 0.68. useful in getting a macro picture of the situation. These posterior marginals change our inference We illustrate max propagation for Product Option about the state of Product Decision. The posterior 1 (risk free proposition with stable pro®ts) under probability of ÔgoÕ for product option 1 is 0.63 in the same two scenarios described in the section on comparison to a prior of 0.51. Under the condi- sum propagation. Table 3 shows the most likely tions described in scenario 1, our marketing expert con®guration of the variables when all variables is likely to select product option 1 since the ®rm (except Initial Investment ˆ low and Production factors make him/her risk averse. Delay ˆ no delay) are unknown. However, when In the second scenario, we consider a market- we specify Scenario 1, the most likely con®gura- leading ®rm with a high market share and that tion of the variables changes. The most likely state follows a technology-based strategy. Accordingly, of Rate of Product Launch changes from high to the states of the four ®rm variables in the map are low, that of prior presence in the market changes speci®ed as follows: Market Share Distribu- from yes to no, that of Sales Uncertainty changes tion ˆ high share, Competitive Strategy ˆ innova- from high to low. The most likely states of all the tion, Market Leadership ˆ leader and R&D ˆ yes. three decision variables change: Willingness to 496 S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498

Table 3 Most likely con®guration of variables under two di€erent scenarios Variables Prior model Scenario 1 Scenario 2 Industry variables: 1. Market Dynamics High High High 2. Product Life Cycle Short Short Short 3. Market Share Distribution High share Low share High share Firm variables: 4. Competitive Strategy Technology-based Price based Technology-based 5. Brand Loyalty High High High 6. Market Leadership Leader Follower Leader 7. Cross Functional Teams Yes No Yes 8. R&D High Low High 9. Rate of Product Launch High Low High 10. Prior Presence in the Market Yes No Yes

Product variables: 11. Sales Uncertainty High Low High 12. Production Delay No delay No delay No delay 13. Initial Investment Low Low Low 14. Product Risk Low Low Low 15. Sales Projection Low sale low risk Low sale low risk Low sale low risk Decision variables: 16. Willingness to Take Risk High Low High 17. Product Pricing High Low High 18. Product Decision No go Go No go

Take Risk changes from high to low, Product maps using the Bayesian network approach. Ac- Pricing from high to low and Product Decision cordingly, we describe a new graphical structure Option one changes from no go to go. This change called Bayesian causal maps. Bayesian causal in the states of the ®rm variables and product maps combine the strengths of causal maps and characteristics help us explain why the state of Bayesian networks and reduce the limitations of Product Option 1 changes from no go to go. When both. Using concepts from the literature on causal Scenario 2 is speci®ed, the most likely con®gura- modeling and logic, Bayesian causal maps clarify tion of the variables changes and the Product the cause±e€ect relations depicted in the causal Option 1 changes to no go. The change in the maps. They depict dependence between variables con®guration is shown in Table 3. The pattern of based on causal mapping approach (D-map) as change in the con®guration of the map helps us well as a lack of dependence between variables understand how and why the overall decision based on the Bayesian network approach (I-map). changes when ®rm characteristics change. A Bayesian causal map is therefore a perfect map. Making inferences using sum and max propa- Bayesian causal maps consider the reasoning (de- gation is useful in predicting decision outcomes as ductive versus abductive) underlying the cause± well as in evaluating and analyzing decision factors e€ect relations perceived by individuals. This that may have a€ected speci®c decision outcomes. strengthens the validity of the direction of causal relations represented in the map. Bayesian causal maps provide a framework for representing the 6. Summary and conclusions uncertainty of variables in the map as well as the e€ect of variables not modeled in the map. Using The main goal of this paper is to propose a evidence propagation algorithms, Bayesian causal formal procedure for making inferences in causal maps allow us to make inferences about the vari- S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498 497 ables in the map. Finally, using a case study, we Bougon, M.K., Weick, K.E., Binkhorst, D., 1977. Cognition in have illustrated how Bayesian causal maps can be organizations: An analysis of the Utrecht Jazz Orchestra. constructed starting from a causal map, and how it Administrative Science Quarterly 22, 606±639. Carley, K., 1997. Extracting team mental models through can be used to make inferences. textual analysis. Journal of Organizational Behaviour 18, There are some interesting implications of our 533±558. study. First, constructing Bayesian networks has Carley, K., Palmquist, M., 1992. Extracting, representing and been a big bottleneck in knowledge engineering analyzing mental models. Social Forces 70 (3), 601±636. (Henrion, 1989; Pradhan et al., 1994; Laskey and Charniak, E., McDermott, D., 1985. Introduction to Arti®cial Intelligence. Addison-Wesley, Reading, MA. Mahoney, 1997). If our procedure is tested and Chen, K., Mathes, J.C., Jarboe, K., Wolfe, J., 1979. Value re®ned, it could be used as a basis for constructing oriented social decision analysis: Enhancing mutual under- Bayesian networks based on an expertÕs knowl- standing to resolve public policy issues. IEEE Transactions edge. Second, most of the emphasis in causal maps on Systems, Man and Cybernetics 9, 567±580. has been in its use for knowledge representation. Churchman, C.W., Acko€, R.L., Arno€, E.L., 1957. Introduc- tion to Operations Research. Wiley, New York. This study enables decision-makers to use causal Clemen, R.T., 1996. Making Hard Decisions ± An Introduction maps for decision making. In¯uence diagrams to Decision Analysis. Duxbury Press, Belmont, CA. proposed by Howard and Matheson (1981) use Darwiche, A., Goldszmidt, M., 1994. On the relation between Bayesian network models of uncertainty in addi- kappa calculus and probabilistic reasoning. In: Mantaras, tion to decision nodes, utility functions, and in- R.L., Poole, D. (Eds.), Uncertainty in Arti®cial Intelligence: Proceedings of the 10th Conference. Morgan Kaufmann, formation constraints. Thus, Bayesian causal San Francisco, CA, pp. 145±153. maps can be use for normative decision making Dawid, A.P., 1979. Conditional independence in statistical using the framework of in¯uence diagrams. theory (with discussion). Journal of the Royal Statistical Society Series B 41 (1), 1±31. Eden, C., 1991. Working on problems using cognitive mapping. In: Littlechild, S.C., Shutler, M. (Eds.), Operations Re- Acknowledgements search Management. Prentice-Hall, London, pp. 236±259. Eden, C., 1992. On the nature of cognitive maps. Journal of We are grateful to Rohini Ahluwalia, V.K. Management Studies 29, 261±266. Narayanan, Anurag Pant, and Surendra Singh for Eden, C., Jones, S., Sims, D., 1979. Thinking in Organizations. Macmillan, London. their comments and discussions. The research re- Eden, C., Jones, S., 1980. Publish or perish?: A case study. ported in this paper has been supported by two Journal of Operational Research Society 31, 131±139. grants from the Kansas University Business Eden, C.F., Ackermann, F., Cropper, S., 1992. The analysis of School PhD Summer Research Fund, and by a cause maps. Journal of Management Studies 29 (3), 309± grant from the Kansas University General Re- 323. Eden, C., Ackermann, F., 1993. Evaluating strategy: Its role search Fund to the second author. The manuscript within the context of strategic control. Journal of Opera- has bene®ted from the comments of an anony- tional Research Society 44, 853±865. mous reviewer. Fiol, M., 1990. Explaining strategic alliance in the chemical industry. In: Hu€, A.S. (Ed.), Mapping Strategic Thought. Wiley, Chichester, UK, pp. 227±249. Fiol, M., Hu€, A.S., 1992. Maps for managers: Where are we? References Where do we go from here? Journal of Management Studies 29, 269±285. Axelrod, R., 1976. Structure of Decision: The Cognitive Maps Forrester, J.W., 1961. Industrial Dynamics. MIT Press, Cam- of Political Elites. Princeton University press, Princeton, bridge, MA. NJ. Heckerman, D., 1996. Bayesian networks for data mining. Data Bougon, M., 1983. Uncovering cognitive maps: The Ôself-QÕ Mining and Knowledge Discovery 1, 79±119. technique. In: Morgan, G. (Ed.), Beyond Method: A Study Henrion, M., 1989. Some practical issues in constructing belief of Organizational Research Strategies. Sage, Beverley Hills, networks. In: Kanal, L.N., Levitt, T.S., Lemmer, J.F. CA. (Eds.), Uncertainty in Arti®cial Intelligence, vol. 3. North- Bougon, M., 1992. Congregate cognitive maps: A uni®ed Holland, Amsterdam, pp. 161±173. dynamic theory of organization and strategy. Journal of Henrion, M., Provan, G., Favero, B.D., Sanders, G., 1994. An Management Studies 29, 369±389. experimental comparison of numerical and qualitative 498 S. Nadkarni, P.P. Shenoy / European Journal of Operational Research 128 (2001) 479±498

probabilistic reasoning. In: Mantaras, R.L., Poole, D. Pearl, J., 1988. Probabilistic Reasoning in Intelligent Systems: (Eds.), Uncertainty in Arti®cial Intelligence: Proceedings Networks of Plausible Inference. Morgan Kaufmann, San of the 10th Conference. Morgan Kaufmann, San Francisco, Mateo, CA. CA, pp. 319±326. Pradhan, M., Provan, G., Middleton B., Henrion, M., 1994. Howard, R., Matheson, J., 1981. In¯uence diagrams. In: Knowledge engineering for large belief networks. In: Lopez Howard, R., Matheson, J. (Eds.), Readings on the Princi- de Mantaras, R., Poole, D. (Eds.), Uncertainty in Arti®cial ples and Applications of Decision Analysis, vol. 2. Strategic Intelligence: Proceedings of the 10th Conference. Morgan Decisions Group, Menlo Park, CA, pp. 721±762. Kaufmann, San Francisco, CA, pp. 484±490. Hu€, A.S., 1990. Mapping Strategic Thought, Wiley, Chiches- Ross, L.L., Hall, R.I., 1980. In¯uence diagrams and organiza- ter, UK. tional power. Administrative Science Quarterly 25, 57±71. Jaques, E., Clement, S.C., 1996. Executive leadership ± A Shachter, R.D., Heckerman, D.E., 1987. A backwards view for practical guide to managing complexity. Cason, Hall and assessment. AI Magazine 8 (3), 55±61. Company. Shenoy, P.P., Shafer, G., 1990. Axioms for probability and Jensen, F.V., Lauritzen, S.L., Olesen, K.G., 1990. Bayesian belief-function propagation. In: Shachter, R.D., Levitt, updating in causal probabilistic networks by local compu- T.S., Lemmer, J.F., Kanal, L.N. (Eds.), Uncertainty in tation. Computational Statistics Quarterly 4, 269±282. Arti®cial Intelligence, vol. 4. North-Holland, Amsterdam, Klein, J.H., Cooper, D.F., 1982. Cognitive maps of decision pp. 169±198. Reprinted in: Shafer, G., Pearl, J. (Eds.), workers in complex game. Journal of Operational Research Readings in Uncertain Reasoning. Morgan Kaufmann, San Society 33, 63±71. Mateo, CA, 1990, pp. 575±610. Knoke, K., Kuklinski, J.H., 1982. Network Analysis. Sage, Speigelhalter, D.J., Dawid, A.P., Lauritzen, S.L., Cowell, R.G., Beverley Hills, CA. 1993. Bayesian analysis in expert systems. Statistical Science Laskey, K.B., Mahoney, S.M., 1997. Network fragments: 8 (3), 219±283. Representing knowledge or constructing probabilistic mod- Spetzler, C.S., Stael von Holstein, C.S., 1975. Probability els. In: Geiger, D., Shenoy, P.P. (Eds.), Uncertainty in encoding in decision analysis. Management Science 22 (3), Arti®cial Intelligence: Proceedings of the 13th Conference. 340±358. Morgan Kaufmann, San Francisco, CA, pp. 334±341. Swan, J.A., 1995. Exploring knowledge and cognitions in Laukkanen, M., 1996. Comparative cause mapping of organi- decisions about technological innovation: Mapping mana- zational cognition. In: Meindl, J.R., Stubbart C., Porac J.F. gerial cognitions. Human Relations 48 (11), 1241±1270. (Eds.), Cognition Within and Between Organizations. Sage, Wang, S., 1996. A dynamic perspective of di€erences between Beverley Hills, CA, pp. 1±45. cognitive maps. Journal of the Operational Research Lauritzen, S.L., Speigelhalter, D.J., 1988. Local computations Society 47, 538±549. with probabilities on graphical structures and their appli- Wolstenholme, E.F., Coyle, R.G., 1983. The development of cation to expert systems (with discussion). Journal of Royal system dynamics as a methodology for systems description Statistical Society Series B 50 (2), 157±224. and qualitative analysis. Journal of Operational Research Lauritzen, S.L., Dawid, A.P., Larsen, B.N., Leimer, H.-G., Society 34, 569±681. 1990. Independence properties of directed Markov ®elds. Winston, P.H., 1984. Arti®cial Intelligence. Addison-Wesley, Networks 20 (5), 491±505. Reading, MA. Pearl, J., 1986. Fusion, propagation, and structuring in belief networks. Arti®cial Intelligence 29, 241±288.