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Causal Inference and Direct Effects

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The format of this document is essentially &quot;two papers in one,&quot; with the first chapter focusing on mostly theoretical considerations (although illustrated with an example), while the second …

The format of this document is essentially &quot;two papers in one,&quot; with the first chapter focusing on mostly theoretical considerations (although illustrated with an example), while the second chapter provides a practical, real-world example presented in the form of a tutorial.

Methods of Causal Inference: We will first introduce the reader to the idea of formal causal inference using the well-known example of Simpson\’s Paradox. Secondly, we will provide a brief summary of the Neyman-Rubin model, which represents a traditional statistical approach in this context. Once this method is established as a reference point, we will introduce two methods within the Bayesian network paradigm, Pearl\’s Do-Operator, which is based on &quot;Graph Surgery&quot;, and a method based on &quot;Likelihood Matching&quot; algorithm (LM). LM allows fixing probability distributions and can be considered as a probabilistic extension of statistical matching.

Practical Applications of Direct Effects and Causal Inference: While our treatment of Neyman-Rubin is limited to the first chapter, the two Bayesian network-based methods will be further illustrated as practical applications in the second chapter. Special weight will be given to Likelihood Matching (LM), as it has not yet been documented in literature. We will explain the practical benefits of LM with a real-world business application and discuss observational and causal inference in the context of a marketing mix model. Using the marketing mix model as the principal example, we will go into greater detail regarding the analysis workflow, so the reader can use this example as a step-by-step guide to implementing such a model with BayesiaLab.

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• 1. Causal Inference and Direct EffectsPearl’s Graph Surgery and Jouffe’s Likelihood MatchingIllustrated with Simpson’s Paradox and a Marketing Mix ModelStefan Conrady, stefan.conrady@conradyscience.comDr. Lionel Jouffe, jouffe@bayesia.comSeptember 15, 2011Conrady Applied Science, LLC - Bayesia’s North American Partner for Sales and Consulting
• 2. Causal Inference and Direct EffectsTable of ContentsIntroduction Motivation & Objective 4 Overview 5 Notation 5I. Methods of Causal Inference Simpson’s Paradox Example 6 Neyman-Rubin Model of Causal Inference 7 Inference Based on Experimental Data 7 Inference from Observational Data 7 The Bayesian Network Representation 9 Pearl’s Do-Operator 10 Causal Networks 10 Intervention 11 Jouffe’s Likelihood Matching (LM) 13 Importance of Network Performance 15 Summary 16II. Practical Applications of Direct Effects and Causal Inference The Marketing Mix Model Example 17 CPG Example & Dataset 18 Model Development 18 Data Import 18 Supervised Learning 22 Network Performance 24 Model Analysis 25 Pearson’s Correlation 26 Mutual Information 27 Observational Inference 29www.conradyscience.com | www.bayesia.com ii
• 3. Causal Inference and Direct Effects Total Effects on Target 31 Causal Inference 33 Pearl’s Do-Operator 33 Direct Effects with Likelihood Matching (LM) 36 Causal Inference as an Afterthought 39 Causal Reasoning 40 Marketing Mix Optimization 40 Linear Marketing Mix Optimization 40 Non-Controllable Variables and Non-Confounders 40 Summary 44Appendix About the Authors 46 Stefan Conrady 46 Lionel Jouffe 46 References 47 Contact Information 48 Conrady Applied Science, LLC 48 Bayesia S.A.S. 48 Copyright 48www.conradyscience.com | www.bayesia.com iii
• 4. Causal Inference and Direct EffectsIntroductionMotivation & ObjectiveTo this day, randomized experiments remain the gold standard for generating models that permit causalinference. In many elds, such as drug trials, they are, in fact, the conditio sine qua non. Without rst hav-ing established and quanti ed the treatment effect (and any associated side effects), no new drug could pos-sibly win approval. This means that a drug must be proven in terms of its causal effect and hence the under-lying study must facilitate causal inference.However, in many other domains, such controlled experiments are not feasible, be it for ethical, economicalor practical reasons. For instance, it is obvious that the federal government could not create two differenttax regimes in order to evaluate their respective impact on economic growth. For lack of such experiments,economists have been traditionally be constrained to studying strictly observational data and, althoughmuch-desired, causal inference is much more dif cult to carry out on that basis. Causal inference from ob-servational studies typically requires an extensive range of assumptions, which may or may not be justi abledepending on one’s viewpoint. Being subject to such individual judgement, it should not surprise us thatthere is widespread disagreement among economic experts and government leaders regarding the effect ofeconomic policies.While economists and social scientists have been using observational data for over a century for policy de-velopment, the business world has only recently been discovering the emerging potential of “big data” and“competing on analytics.” As these terms are becoming buzzwords, and are rightfully expected to hold greatpromise, the strictly observational nature of most “big data” sources is often overlooked. The wide avail-ability of new, easy-to-use analytics tools may turn out to be counterproductive, as observational versuscausal inference are not explicitly differentiated. While the mantra of “correlation does not imply causa-tion” remains frequently quoted as a general warning, many business analysts would not know under whatspeci c conditions it can be acceptable to derive a causal interpretation from correlation in observationaldata. Consequently, causal assumptions are often made rather informally and implicitly and thus they typi-cally remain undocumented. The line between association and causation often becomes further blurred inthe eyes of the end users of such research. Given that the concept of causality remains ill-understood inmany practical applications, we seriously question today’s real-world business capabilities for deriving ra-tional policies from the newly-found “big data.”With these presumed shortcomings in business practice, it is our objective to provide a framework that fa-cilitates a much more disciplined approach regarding causal inference while remaining accessible to (non-statistician) business analysts and transparent to executive decision makers. We believe that Bayesian net-works are an appropriate paradigm for this purpose and that the BayesiaLab software package offers a ro-bust toolset for distinguishing observational and causal inference.www.conradyscience.com | www.bayesia.com 4
• 5. Causal Inference and Direct EffectsOverviewThe format of this document is essentially “two papers in one,” with the rst chapter focusing on mostlytheoretical considerations (although illustrated with an example), while the second chapter provides a prac-tical, real-world example presented in the form of a tutorial.I. Methods of Causal Inference We will rst introduce the reader to the idea of formal causal inference using the well-known example of Simpson’s Paradox. Secondly, we will provide a brief summary of the Neyman-Rubin model, which represents a traditional statistical approach in this context. Once this method is established as a refer- ence point, we will introduce two methods within the Bayesian network paradigm, Pearl’s Do- Operator, which is based on “Graph Surgery”, and a method based on Jouffe’s “Likelihood Match- ing” algorithm (LM). LM allows xing probability distributions and can be considered as a probabilis- tic extension of statistical matching.II. Practical Applications of Direct Effects and Causal Inference While our treatment of Neyman-Rubin is limited to the rst chapter, the two Bayesian network-based methods will be further illustrated as practical applications in the second chapter. Special weight will be given to Likelihood Matching (LM), as it has not yet been documented in literature. We will ex- plain the practical bene ts of LM with a real-world business application and discuss observational and causal inference in the context of a marketing mix model. Using the marketing mix model as the prin- cipal example, we will go into greater detail regarding the analysis work ow, so the reader can use this example as a step-by-step guide to implementing such a model with BayesiaLab.NotationTo clearly distinguish between natural language, software-speci c functions and example-speci c variablenames, the following notation is used:• Bayesian network and BayesiaLab-speci c functions, keywords, commands, etc., are capitalized and shown in bold type.• Names of attributes, variables, nodes and are italicized.www.conradyscience.com | www.bayesia.com 5
• 6. I. Methods of Causal InferenceI. Methods of Causal InferenceSimpson’s Paradox ExampleIn our recent white paper, Paradoxes and Fallacies, we have written about Simpson’s Paradox, which occa-sionally appears in the popular press as a rather enigmatic statistical anomaly. We use an admittedly con-trived example to illustrate this paradox:A hypothetical type of cancer equally effects men and women. A long-term, observational and non-experimental study nds that a speci c type of cancer therapy is associated with an increased remission rateamong all treated patients (see table). Based on the study, this particular treatment is thus recommended forbroader application. Remission Treatment Yes No Yes 50% 50% No 40% 60%However, when examining patient records by gender, the remission rate for male patients — upon treat-ment — decreases from 70% to 60% and for female patients the remission rate declines from 30% to 20%(see table). So, is this new therapy effective overall or not? Remission Gender Treatment Yes No Yes 60% 40% Male No 70% 30% Yes 20% 80% Female No 30% 70%The answer lies in the fact that — in this example — there was an unequal application of the treatment tomen and women. More speci cally, 75% of the male patients and only 25% of female patients received thetreatment. Although the reason for this imbalance is irrelevant for inference, one could imagine that sideeffects of this treatment are much more severe for females, who thus seek alternatives therapies. As a result,there is a greater share of men among the treated patients. Given that men also have a better recovery pros-pect with this type of cancer, the remission rate for the total treated population increases.So, what is the true overall effect of this treatment? There are actually several possible ways to compute theeffect, namely the statistical approach based on the Neyman-Rubin Model of Causal Inference, and the twoBayesian network-based approaches: Pearl’s Graph Surgery approach and the method based on Jouffe’s LMalgorithm, which are both implemented in BayesiaLab.www.conradyscience.com | www.bayesia.com 6
• 7. I. Methods of Causal InferenceNeyman-Rubin Model of Causal InferenceTo begin our discussion of causal inference, we will rst present matching as a statistical method for causalinference based on observational data. Our brief summary follows the framework that is widely known asthe Neyman-Rubin Model of Causal Inference (Rubin, 2006; Sekhon, 2007; Morgan and Winship, 2007;Rosenbaum, 2002).We closely follow Sekhon (2007) for this highly condensed summary:Inference Based on Experimental DataLet Yi1 denote the potential outcome for unit i if the unit receives treatment, and let Yi 0 denote the potentialoutcome for a unit in the control group. The treatment effect for unit i is de ned as: τ i = Yi1 − Yi 0Furthermore, let Ti be the a treatment indicator: 1 when unit i is in the treatment group and 0 when unit i isin the non-treatment control group.If assignment to treatment is randomized, causal inference is fairly simple because the two groups are drawnfrom the same population, and treatment assignment is independent of all baseline variables. As the samplesize grows, observed and unobserved confounders are balanced across treatment and control groups. Thatis, with random assignment, the distributions of both observed and unobserved variables in both groups areequal in expectation.Treatment assignment is independent of Y0 and Y1 — i.e., {Y i0 ,Yi1 ⊥ Ti } ,where “ ⊥ ” symbol represents independence.Hence, for j = 0, 1 E(Yij∣ i = 1) = E(Yij∣ i = 0) = E(Yi∣ i = j) T T TTherefore, the average treatment effect can be estimated by:τ = E(Yi1∣ i = 1) − E(Yi 0∣ i = 0) T T= E(Yi∣ i = 1) − E(Yi∣ i = 0) T Tτ can be estimated in an experimental setting because randomization can ensure that observations in treat-ment and control groups are exchangeable. Randomization ensures that assignment to treatment will not, inexpectation, be associated with the potential outcomes.Inference from Observational DataThe situation with observational data is much less straightforward as treatment and control groups are notnecessarily drawn from the same population. Hence, the average treatment effect τ cannot be estimated thesame way as was the case with experimental data.www.conradyscience.com | www.bayesia.com 7
• 8. I. Methods of Causal InferenceAs an alternative, we can pursue the average treatment effect for the treated, more formally expressed asτ∣ = 1) = E(Yi1∣ i = 1) − E(Yi 0∣ i = 1) (T T THowever, the challenge in this case is that Yi 0 is not observed for the treated, i.e. we simply cannot knowhow those, who were in fact treated, would have fared, had they not been treated.As a potential remedy for this quandary, one could assume that treatment selection depends on a set of ob-servable covariates X. Furthermore, we could assume that given X, treatment assignment is independent ofY.More formally, {Y ,Y ⊥ T∣X } , which is referred to as “unconfoundedness” 0 1A nal assumption is that there is a so-called “overlap:” 0 < P(T = 1 ∣X) < 1 .In particular case, where X ∈{male, female} , this implies that treatments must be observed for bothmales and females in order to obtain overlap. Together, unconfoundedness and overlap form the concept ofstrong ignorability, which are required for the estimation of the average treatment effect for the treated:τ∣ = 1) = E { E(Yi∣Xi ,Ti = 1) − E(Yi∣Xi ,Ti = 0) Ti = 1} (T ∣This means we condition on the observed covariates, Xi , and thus treatment and control groups are bal-anced.Conditioning on X can be a straightforward task and can typically be achieved by matching, which means nding exactly matching sets of covariates. In our case, matching is simple, as we only have one covariatewith two states, i.e. male and female. We can then compute the treatment effects for exactly matched sets oftreated and untreated units within each subset, i.e. within the male and female group.However, in most real-world applications, we have many more covariates and among those many may havecontinuous values rather than discrete states. Inevitably, this makes matching much more challenging and itactually may be impossible to perform exact matching.Given this challenge, propensity score matching (Rosenbaum and Rubin, 1983) and matching based on theMahalanobis distance (Cohran and Rubin, 1973) have emerged as commonly used methods. Both methodsperform matching based on covariate similarity, but it goes be beyond the scope of this paper to elaboratefurther on the details of these and related methods.www.conradyscience.com | www.bayesia.com 8
• 9. I. Methods of Causal InferenceThe Bayesian Network RepresentationTo illustrate Pearl’s Do-Operator based on Graph Surgery and subsequently the method based on Jouffe’sLikelihood Matching, we need to switch from the traditional statistical framework to the Bayesian networkparadigm. The starting point for both methods is a synthetically generated dataset with three variables,Gender, Treatment and Remission, with a total of 1,000 observations, which re ects the statistics describedin the tables provided in the description of Simpson’s Paradox.1 This dataset will serve as the basis for theBayesian network to be used for causal inference.For expositional simplicity, we will omit the steps required for importing the dataset into BayesiaLab andrefer the reader to the second chapter, which describe the import process in detail. Rather, we begin directlyin BayesiaLab’s Modeling Mode with the initially unconnected network consisting of three nodes, i.e. thevariables of interest:From theory we know that we can factorize a Joint Probability Distribution (JPD) into the product of condi-tional probability distributions (see Barber, 2011, for a detailed discussion). With the three nodes that wehave in our example, there are actually six different ways to do this:p(x1 , x2 , x3 ) = p(x1∣x2 , x3 )p(x2 , x3 ) = p(x1∣x2 , x3 )p(x2∣x3 )p(x3 )p(x1 , x3 , x2 ) = p(x1∣x3 , x2 )p(x3 , x2 ) = p(x1∣x3 , x2 )p(x3∣x2 )p(x2 )p(x2 , x1 , x3 ) = p(x2∣x1 , x3 )p(x1 , x3 ) = p(x2∣x1 , x3 )p(x1∣x3 )p(x3 )p(x2 , x3 , x1 ) = p(x2∣x3 , x1 )p(x3 , x1 ) = p(x2∣x3 , x1 )p(x3∣x1 )p(x1 )p(x3 , x1 , x2 ) = p(x3∣x1 , x2 )p(x1 , x2 ) = p(x3∣x1 , x2 )p(x1∣x2 )p(x2 )p(x3 , x2 , x1 ) = p(x3∣x2 , x1 )p(x2 , x1 ) = p(x3∣x2 , x1 )p(x2∣x1 )p(x1 )Given the semantics of Bayesian networks, this translates into six possible, equivalent Bayesian networks,that are all representing exactly the same JPD.When we perform one of BayesiaLab’s learning network algorithms on the sample dataset, we will indeedobtain one of the six possible networks shown above, as suggested by the theory. Without additional infor-mation on those variables, such as we might obtain from temporal indices, we will be unable to select onenetwork over the other and the network choice would have be entirely arbitrary.1 This dataset was created with BayesiaLab’s Generate Data function, based on the true Joint Probability Distribution(JPD).www.conradyscience.com | www.bayesia.com 9
• 10. I. Methods of Causal InferenceIn order to visualize that the arcs in these networks are invertible in their orientation, BayesiaLab can high-light the Essential Graph (Analysis>Graphic>Show the Edges). This will display the edges that can be ori-ented in either direction without modifying the represented JPD.For purposes of observational inference, any of these six equivalent networks would be suf cient. For in-stance, the probability of Remission=yes, given that we observe Treatment=yes, i.e. P(Remission=yes|Trea-tment=yes), can be computed with any of the six networks shown earlier.However, from the introduction of Simpson’s Paradox, we realize that a simple observation is not suf cientto establish the treatment effect. Observational inference may actually be misleading for interpretation pur-poses, which is at the very core of the paradox. So, our question remains, “what is the effect of treatment?”More speci cally, “what is the probability of remission, given that we do administer the treatment?” Thismeans that we want to see the effect of an intervention instead of merely observing that treatment has oc-curred.Graph Surgery and LM provide different ways to answers this question, which we will explain the followingtwo sections:Pearl’s Do-OperatorCausal NetworksTo introduce Pearl’s Do-Operator, we need to make a formal transition from a general Bayesian network toa causal network, because Bayesian networks describe a joint distribution over possible observed events butsay nothing about what will happen if an intervention occurs. A causal network is a Bayesian network withthe added property that the parents of each node are its direct causes. For example, Fire → Smoke is acausal network whereas Smoke → Fire is not, even though both networks are equally capable of represent-ing any joint distribution on the two variables.More formally, causal networks are de ned as a type of Bayesian network with special properties: uponsetting an intervention on a node in a causal network, the correct probability distribution is given by delet-ing the incoming arcs from the node’s parents, i.e. “cutting off” the direct causes of the node. Pearl hascharacterized this deletion of links rather graphically as “graph mutilation” or “graph surgery.”22 Interestingly, “intervenire”, the Latin origin of “intervention,” symbolizes this separation as it literally means “tocome in between.”www.conradyscience.com | www.bayesia.com 10
• 11. I. Methods of Causal InferenceWith this de nition, Pearl’s Graph Surgery approach requires us to provide a complete set of causal assump-tions regarding the network to compute the effect of an intervention. Given our background knowledge re-garding Simpson’s Paradox, we can make causal assumptions for all edges and thus declare, i.e. by at, ourBayesian network a causal network. As stated earlier, we assume that Gender has a causal effect on Remis-sion (rather than Remission on Gender), so we de ne the arc direction as Gender ➝ Remission. We alsoassume that Treatment has a causal effect (whether positive or negative) on Remission, which translates intothe (directed) arc Treatment ➝ Remission. Finally, we have learned that Gender in uences (causes) whetheror not one would undergo Treatment, so we have Gender ➝ Treatment. This eliminates ve of the six pos-sible Bayesian networks and leaves us with only one possible causal Bayesian network:Now that we have a causal Bayesian network we can make a distinction between observational inferenceand causal inference. This is because of the semantic difference of “given that we observe” versus “giventhat we do.” The former is strictly an observation, i.e. we focus on the patients who received treatment,whereas the latter is an active intervention. The answer to our question of the treatment effect then is infer-ring as to what would hypothetically happen, “given that we do”, i.e. given that we force the treatmentwithout permitting patients to self-select their treatment. In the semantics of Bayesian networks, this meansthat there must not be a direct relationship between Gender and Treatment. In other words, Treatmentmust not directly depend on Gender.InterventionIn our Bayesian network this can be done easily by “mutilating” the graph, i.e. deleting the arc connectingGender and Treatment. BayesiaLab offers a very simple function to achieve this, which is aptly named In-tervention (right-click on the node’s Monitor and then select Intervention).By intervening on the Treatment variable (and setting Treatment=yes), the causal Bayesian network ismodi ed (or “mutilated”) as follows:• The entering arcs of the node on which we want to perform intervention are “surgically” removed. With intervention, we cut the dependency between Treatment and Gender, i.e. administering the treatment will not affect Gender.• The original Chance Node (round) representing Treatment is transformed into a Decision Node (square). The associated Monitor will be highlighted in blue.www.conradyscience.com | www.bayesia.com 11
• 12. I. Methods of Causal InferenceNow we can observe what happens to Remission when we “do” Treatment, instead of just “observing”Treatment.Treatment=NoTreatment=YesAs we can see, the Gender probability distribution remains the same. However, Remission decreases from50% to 40%, given that we “do” Treatment. With this we have now obtained the treatment effect:τ = P(Remission = yes Treatment = yes) − P(Remission = yes Treatment = no) = −0.1 ∣ ∣www.conradyscience.com | www.bayesia.com 12
• 13. I. Methods of Causal InferenceTo answer our original question, we must conclude that this new treatment is detrimental to the patients’health.Jouffe’s Likelihood Matching (LM)We will now brie y introduce the Jouffe’s Likelihood Matching (LM) algorithm, which was originally im-plemented in the BayesiaLab software package for “ xing” probability distributions of an arbitrary set ofvariables, allowing then to easily de ne complex sets of soft evidence. The LM algorithm searches for a setof likelihood distributions, which, when applied on the Joint Probability Distribution (JPD) encoded by theBayesian network, allows obtaining the posterior probability distributions de ned (as constraints) by theuser.As we saw with Pearl’s Graph Surgery approach, the core idea of intervention is to set an evidence on thenode on which we wish to intervene, while all other ascending nodes remain unchanged. Using this verysame idea, we can then intervene on a node by xing the posterior probability distributions of its covariates.Casually speaking, this would be a kind “virtual mutilation.”Treatment=Nowww.conradyscience.com | www.bayesia.com 13
• 14. I. Methods of Causal InferenceTreatment=YesThese results are identical to what was obtained with Pearl’s Graph Surgery.τ = P(Remission = yes Treatment = yes) − P(Remission = yes Treatment = no) = −0.1 ∣ ∣However, two main differences exist between the methods:I. One important feature of the LM algorithm is that it returns the same result for all the instantiations of the Essential Graph, i.e. for any one of the six equivalent networks. For example, intervening on Treatment using the Bayesian network below, and using the LM algorithm, will lead to exactly the same posterior probability distribution for Remission, even though the arc directions are non-causal (and thus perceived counterintuitive). In comparison to the Do-Operator, the approach based on LM does not require any available causal knowledge to be formally translated into a causal structure in order to compute treatment effects. While it may be easy to specify all the causal directions in a simple model with only three nodes, such as in our example, it is obviously more of a challenge to do the same for a larger network, perhaps consisting of dozens or even hundreds of nodes. That is not to claim that we can avoid causal assump- tions altogether, however, we aim to defer the need for making such assumptions until a later point and then only make those assumptions that are directly related to the pair of variables for which we want to obtain the causal effect.www.conradyscience.com | www.bayesia.com 14
• 15. I. Methods of Causal InferenceII. The other difference between the Graph Surgery and LM is more subtle and may not always be obvi- ous. The Graph Surgery implies a modi cation of the representation of the JPD, whereas the approach based on LM always works on the original JPD. The mere graph mutilation can bring about a modi - cation of some marginal probability distributions, even without changing the marginal probability dis- tributions of the nodes on which we want to intervene. We need to brie y digress from our principal example in order to clarify this particular point: The two graphs below illustrate the mutilation impact on the marginal probability distribution of Customer Satisfaction.Importance of Network PerformanceWhile the statistical matching approach of the Neyman-Rubin Matching Model directly utilizes the originalobservations, the LM algorithm is based on the JPD encoded by the Bayesian network. This emphasizes therequirement that a Bayesian network to be used for this purpose must provide a good representation of thetrue JPD. While there is no hard-and-fast rule as to what constitutes a minimum t requirement, we canreview the overall network performance by selecting Analysis>Network Performance>Global:www.conradyscience.com | www.bayesia.com 15
• 16. I. Methods of Causal InferenceThe key metric here is the Contingency Table Fit (CTF). This measure can range between 0%, as if the JPDwere represented with a fully unconnected network (all the nodes are independent), and 100%, as if the JPDwere perfectly represented with a fully connected network. The network learned on the Simpson’s Paradoxdataset happens to be a complete (fully-connected) graph, thus the CTF is 100%.SummaryWe have provided a brief summary of the Neyman-Rubin model, which represents a traditional statisticalapproach for causal inference. Extending beyond the statistical framework, and now within the Bayesiannetwork paradigm, we illustrated Graph Surgery and Pearl’s Do-Operator, and a nally presented a methodbased on Jouffe’s Likelihood Matching algorithm (LM). Most importantly, working with the Bayesian net-work methods highlighted that formal causal assumptions are critical to correct causal inference.www.conradyscience.com | www.bayesia.com 16
• 17. II. Practical Applications of Direct Effects and Causal InferenceII. Practical Applications of Direct Effects and CausalInferenceThe Marketing Mix Model ExampleThe adage, “I know I waste half of my advertising dollars...I just wish I knew which half”, re ects acentury-old uncertainty about the effectiveness of marketing instruments.3 More formally, one could de-scribes this quandary as a domain with an unknown (or ill-understood) causal structure.While “big data”, especially in the eld of marketing, is expected to rapidly yield “actionable business in-sights,” we need to recognize that there are many steps to traverse to achieve this goal. Hence, we wouldlike to parse this overarching objective of “actionable insights” into distinct components, which will imme-diately highlight the central role of causal inference:• “Big data” most often refers to large amounts of observational data from a domain. Despite the ever- increasing amount of data, most measures collected do include noise and missing data points.• “Actionable insights” actually implies several things: Firstly, it requires an understanding of the domain, which can be used as a basis for reasoning about this domain. A key assumption in this context is that we must not only have a structure describing the observations we have gathered, but rather we must have a causal structure, so we can anticipate the consequences of actions we have not yet taken. If we have this ability to evaluate the results of our potential interventions in this domain, we can chose the rational course of action among all the possible alternatives. As an added complexity, most dynamics uncovered in a domain are probabilistic rather than deterministic in nature.Although it is typically a challenge, our chosen toolset, BayesiaLab, can implicitly handle missing values andcapture the probabilistic nature of the domain and hence we will not focus on that aspect. Rather, the cen-tral theme of this paper is the transition from observation to causation.As we have seen in the rst chapter with the introductory example of Simpson’s Paradox, Bayesian net-works provide two principal ways of moving from observational inference to causal inference, namelyGraph Surgery and Likelihood Matching. With the following example from the CPG industry, we will jux-tapose Graph Surgery and LM and then speci cally demonstrate how LM can be utilized for computingcausal effects, which can subsequently be used for performing marketing mix optimization.3 Various versions of this quote have been attributed to Henry Procter, Henry Ford, John Wanamaker and J.C. Penneywww.conradyscience.com | www.bayesia.com 17
• 18. II. Practical Applications of Direct Effects and Causal InferenceCPG Example & DatasetTo illustrate this approach we study daily ice cream sales of a European food distributor as a function ofenvironmental variables and marketing efforts.4Our sample data set includes the following variables:• Seasonally-adjusted daily sales in the local currency• Traditional advertising, such as print advertising (incl. coupons), TV, radio, in-store promotions, etc.• Online advertising, including banner ads, search engine marketing, online coupons• Competitive advertising (estimate of all competitive marketing efforts combined)• Temperature in °C• Number of open retail outlets• WeekdayModel DevelopmentWhile the focus of this example is to evaluate and to causally interpret a given marketing mix model, wewill spell out the steps one would take to generate such a model with BayesiaLab. This should enable cur-rent users of BayesiaLab to replicate the exercise in its entirety.Data ImportWe use BayesiaLab’s Data Import Wizard to load all 7 time series5 into memory from a comma-separated le (CSV). BayesiaLab automatically detects the column headers, which contain the variable names.4 For expository purposes, this dataset was synthetically generated based on actual market dynamics observed in anindustry and locale different from the example.5 Although the dataset has a temporal ordering, for expository simplicity we will treat each time interval as an inde-pendent observation.www.conradyscience.com | www.bayesia.com 18
• 19. II. Practical Applications of Direct Effects and Causal InferenceThe next step identi es the data types contained in the dataset. BayesiaLab will attempt to detect the type ofvariables in the dataset and assumes in this case all variables to be continuous, as indicated by the turquoisebackground color for all columns.Although Weekday appears continuous, i.e. 1 through 7, it must be treated as discrete so as to avoid bin-ning in the subsequent discretization function.6 Upon setting it to discrete, the Weekday variable will appearin red.6 In the original dataset the variable Weekday was coded into ordered numerical states, 1 through 7, representing Mon-day through Sunday. BayesiaLab could also have used text descriptions as state labels, in which case the variable wouldhave been automatically recognized as discrete.www.conradyscience.com | www.bayesia.com 19
• 20. II. Practical Applications of Direct Effects and Causal InferenceAs our dataset contains missing values, we need to specify the type of missing values imputation. We willchoose the Structural EM method, given that for the size of this dataset, the computational complexity ofthis algorithm will not be a burden.The following discretization step is very important for all models in BayesiaLab and thus we provide a bitmore detail here. Our objective of this model is to establish Sales as a function of the marketing instrumentsand other external factors. Thus we can take this objective into account for the discretization process. Morespeci cally, we will split the process into two parts. First, we will discretize the target variable, i.e. Sales, onits own. We highlight the Sales column in the data table and then choose Manual as the DiscretizationType. This provides us with probability density function of Sales.www.conradyscience.com | www.bayesia.com 20
• 21. II. Practical Applications of Direct Effects and Causal InferenceBy clicking Generate a Discretization, we are prompted to select the discretization type.We chose Type: K-Means and Intervals: 4.7 The chart will now display the results of this discretization.7 For a discussion of discretization algorithms and a guide for interval selection, please see the papers referenced in theappendix.www.conradyscience.com | www.bayesia.com 21
• 22. II. Practical Applications of Direct Effects and Causal InferenceNow that we have discretized the target variable by itself, we will discretize the remaining continuous vari-ables with the Decision Tree algorithm and use Sales as the target. This allows binning the continuous vari-ables in such a way that we gain a maximum amount of information from these variables with respect tothe target.Upon completion of the discretization, BayesiaLab will present all variables as nodes in an unconnectednetwork in the Graph Panel.Supervised LearningNow that we have an initial network, albeit unconnected, we can perform our rst Supervised Learning al-gorithm with the objective of characterizing the target node. However, we do need to rst specify the targetby right-clicking on Sales and selecting Set As Target Node (or pressing “T” while double-clicking on thenode).www.conradyscience.com | www.bayesia.com 22
• 23. II. Practical Applications of Direct Effects and Causal InferenceOnce this is set, the Sales node will appear in the graph as a bulls-eye, symbolizing a target.We now have an array of Supervised Learning algorithms available to apply here. Given the small numberof nodes, variables selection is not an issue and hence this should not in uence our choice. Furthermore, therelatively small number of observations does not create a challenge in terms of computational effort. Withthese considerations, and without going into further detail, we select the Augmented Naive Bayes algorithm.The “augmented” part in the name of this algorithm refers to the additional unsupervised search that is per-formed on the basis of the given naive structure.www.conradyscience.com | www.bayesia.com 23
• 24. II. Practical Applications of Direct Effects and Causal InferenceUpon learning, the newly generated network is now displayed in the Graph Panel.The prede ned naive structure is highlighted by the dotted arcs, while the additional (augmented) arcs fromthe unsupervised learning are shown in solid black.Network PerformanceWe could now spend some time to further re ne this model, such as balancing the degree of complexity ver-sus the overall model t. Furthermore, we could also specify this as a dynamic model.8 To maintain exposi-tional clarity, we will leave the model as is.However, we do wish to cover a few performance measures to assure the reader that the model presentedhere is a reasonable characterization of the underlying domain.With the relatively small number of observations, we chose not to set aside a hold-out sample (e.g. 20% ofobservations) during the data import process. As an alternative way of testing the out-of-sample networkperformance, we carry out Cross Validation by selecting (from within the Validation Mode) Tools>CrossValidation>Targeted:In terms of parameters for the Cross Validation, we select the same learning algorithm as before, i.e. Aug-mented Naive Bayes. Also, using a 10-fold validation is a typical choice in this context.8 Given the inherently dynamic nature of marketing effects, it would be very appropriate to model this as a temporalBayesian network. For instance, this would enable us to capture potential lags in the effects of marketing activities onthe target variable. The BayesiaLab framework can easily accommodate such a temporal speci cation.www.conradyscience.com | www.bayesia.com 24
• 25. II. Practical Applications of Direct Effects and Causal InferenceThe resulting Global Report provides a variety of metrics, including precision and R2.Sampling Method: K-FoldsLearning Algorithm: Augmented Naive BayesTarget: Sales <=20755 <=23387 <=25914 >259145Value 6.406 7.375 5.594 .594Gini Index 66% 41.75% 38.03% 69.52%Relative Gini Index 75.25% 62.92% 63.76% 80.63%Mean Lift 2.49 1.64 1.52 2.49Relative Lift Index 81.50% 78.29% 80.11% 84.09%Relative Gini Global Mean: 70.64%Relative Lift Global Mean: 81%Total Precision: 67.37%R: 0.76104342242R2: 0.57918709081Occurrences <=20755 <=23387 <=25914 >259145 Value 6.406 7.375 5.594 .594 (53) (142) (172) (59) <=207556.406 (56) 37 18 1 0 <=233877.375 (124) 15 86 22 1 <=259145.594 (213) 1 38 140 34 >259145.594 (33) 0 0 9 24Reliability <=20755 <=23387 <=25914 >259145 Value 6.406 7.375 5.594 .594 (53) (142) (172) (59) <=207556.406 (56) 66.07% 32.14% 1.79% 0% <=233877.375 (124) 12.10% 69.35% 17.74% 0.81% <=259145.594 (213) 0.47% 17.84% 65.73% 15.96% >259145.594 (33) 0% 0% 27.27% 72.73%Precision <=20755 <=23387 <=25914 >259145 Value 6.406 7.375 5.594 .594 (53) (142) (172) (59) <=207556.406 (56) 69.81% 12.68% 0.58% 0% <=233877.375 (124) 28.30% 60.56% 12.79% 1.69% <=259145.594 (213) 1.89% 26.76% 81.40% 57.63% >259145.594 (33) 0% 0% 5.23% 40.68%Even without further comparison, the reported values appear reasonable and suggest that we can proceedwith analyzing this network.Model AnalysisWe have accepted the network as plausible representation of this domain and will now interpret the struc-ture we obtained. To make it easier to understand the structure, we will rst apply one of BayesiaLab’sautomatic layout algorithms, which quite literally “disentangles” the network and thus provides a clearerpicture. Selecting View>Automatic Layout achieves this (or pressing the keyboard shortcut “P”).www.conradyscience.com | www.bayesia.com 25
• 26. II. Practical Applications of Direct Effects and Causal InferenceThe “Naive Bayes” versus the “Augmented” part of this network, shown in dotted arcs and solid arcs re-spectively, are now much more obvious in this layout.As that the naive structure was given by de nition, only the presence or absence of solid arcs provides in-formation about the existence of relationships between the predictors. Much more can be understood whenwe examine the magnitude and the sign of all relationships in the network.Pearson’s CorrelationAlthough correlation, as we will later emphasize, is not a central metric for network analysis in BayesiaLab,we will use it for a rst look, especially since all readers will be familiar with this measure. Selecting Analy-sis>Graphic>Pearson’s Correlation provides this information directly in the network graph.www.conradyscience.com | www.bayesia.com 26
• 27. II. Practical Applications of Direct Effects and Causal InferenceThe colors of the arcs indicate the sign of the relationship and the arc labels provide the correlation value.Many of the shown relationships seem intuitive, for instance that No. of Stores and both Trad. Adv. andOnline Adv. have a positive association with Sales. Equally plausible is the fact that Temperature is associ-ated with Sales (although one of the co-authors of this paper believes that one can eat ice cream rain orshine). The negative association between Competitive Adv. and Sales also seems expected. Less clear is thenegative correlation between Sales and Weekday, but the small value suggests either very weak link or per-haps a nonlinear relationship.Mutual InformationGiven that Pearson’s correlation is a strictly linear metric, its ability to characterize all these relationships isinherently limited. We will now turn to Mutual Information as a new measure, which can help overcomethis limitation.www.conradyscience.com | www.bayesia.com 27
• 28. II. Practical Applications of Direct Effects and Causal InferenceIn contrast to correlation, Mutual Information does not re ect the sign of the relationship, however, thismeasure captures the strength of relationships between variables, even if they are highly nonlinear.More speci cally, the Mutual Information I(X,Y) measures how much (on average) the observation of ran-dom variable Y tells us about the uncertainty of X, i.e. by how much the entropy of X is reduced if we haveinformation on Y. Mutual Information is a symmetric metric, which re ects the uncertainty reduction of Xby knowing Y as well as of Y by knowing X.In our example, knowing the value of Weekday on average reduces the uncertainty of the value of Sales by0.4802 bits, which means that it reduces its uncertainty by 26.3% (shown in red, in the opposite directionof the arc). Conversely, knowing Sales reduces the uncertainty of Weekday by 17.11% (shown in blue, inthe direction of the arc). It is interesting to see that, by looking at Mutual Information, Weekday and Salesnow have a very strong relationship whereas previously the correlation coef cient was near zero.www.conradyscience.com | www.bayesia.com 28
• 29. II. Practical Applications of Direct Effects and Causal InferenceObservational InferenceTo explore the nature of this relationship further, we can perform the Target Mean Analysis with Sales andWeekday (Analysis>Graphic>Target Mean Analysis).This prompts us to select the way we want to examine this relationship. In this context it seems appropriateto look at the delta mean of the target as a function of Weekday.The resulting plot con rms the previous hypothesis of nonlinearity.www.conradyscience.com | www.bayesia.com 29
• 30. II. Practical Applications of Direct Effects and Causal InferenceFor instance, we can interpret this as follows: given that Weekday=Friday, we observe that Sales reach theirhighest value. Furthermore we can infer, given that Weekday=Sunday, we observe that Sales have their low-est value, as many shops in Europe are closed on Sundays. We can further speculate that consumers perhapsbuy more ice cream on Fridays in preparation for leisure activities over the weekend.Returning to our interpretation of Mutual Information, it is now obvious why Weekday reduces the uncer-tainty of Sales by over 25%. There is quite apparently an intra-week seasonality. Another interpretation ofMutual Information is “importance” and we can use Analysis>Report>Target Analysis>Correlations withthe Target Node to obtain an overview of the importance of all nodes in the network with respect to thetarget, Sales.Node significance with respect to the information gain brought by the node to the knowledge of Sales Mutual Mutual Relative Degrees.of Degrees.of Node Mean.Value G:test p:value G:test.(Data) p:value.(Data) information information.(%) significance Freedom Freedom.(Data)Weekday 0.4802 26.30% 1 4.0047 283.5916 18 0.00% 283.5916 18 0.00%Competitive:Adv. 0.1293 7.08% 0.2692 514.9959 76.332 9 0.00% 76.332 9 0.00%Trad.:Adv. 0.0835 4.57% 0.1739 483.8701 49.307 9 0.00% 49.307 9 0.00%No.:of:Stores 0.081 4.44% 0.1686 3096.5023 47.8213 9 0.00% 47.8213 9 0.00%Online:Adv. 0.0764 4.18% 0.159 181.6759 45.0943 9 0.00% 45.0943 9 0.00%Temperature 0.0592 3.24% 0.1233 14.5441 34.9654 9 0.01% 34.9654 9 0.01%www.conradyscience.com | www.bayesia.com 30
• 31. II. Practical Applications of Direct Effects and Causal InferenceIt is important to stress that this is a form of observational inference and it does not imply a causal relation-ship with Sales. We assume that some of these variables “cause” Sales, but from this table we can only inferassociation, not causation.Total Effects on TargetThe same caveat also holds true for our next evaluation, Total Effects on Target (Analysis>Report>TargetAnalysis>Total Effects on Target):Total Effect is a linearized measure that shows the impact of a one-unit change in the mean (that is com-puted at the mean) of each node on the Target.Total Effects on Target Sales Standardized Degrees-of Degrees-of Node Total-Effect G:test p:value G:test-(Data) p:value-(Data) Total-Effect Freedom Freedom-(Data)Competitive)Adv. -0.3456 -32.0159 76.332 9 0.00% 76.332 9 0.00%Trad.)Adv. 0.2567 6.2351 49.307 9 0.00% 49.307 9 0.00%No.)of)Stores 0.1679 48.0703 47.8213 9 0.00% 47.8213 9 0.00%Online)Adv. 0.1482 22.4707 45.0943 9 0.00% 45.0943 9 0.00%Temperature 0.1291 323.6881 34.9654 9 0.01% 34.9654 9 0.01%Weekday -0.0501 -583.078 283.5916 18 0.00% 283.5916 18 0.00%This can be illustrated by performing the computation manually in the Monitor Panel. By default, the Moni-tors shows the marginal frequency distributions of the states of the nodes plus the mean value (expectedvalue) of those distributions:www.conradyscience.com | www.bayesia.com 31
• 32. II. Practical Applications of Direct Effects and Causal InferenceAs stated above, the Total Effect is computed on the basis of a one-unit change of each node. We can simu-late this by setting Competitive Adv. to a new mean value, i.e. changing its mean from 514.996 to 515.996.It must be noted that there is an in nite possibility of achieving a mean value of +1 in this distribution.BayesiaLab supports the analyst by choosing the particular distribution (of all possible distributions) that isclosest to the original distribution while achieving the targeted mean value of +1. We simply need to right-click on the Monitor for Competitive Adv. and select Distribution for Target Value/Mean.This prompts us to type in our desired value, i.e 515.996, to re ect the one-unit change.www.conradyscience.com | www.bayesia.com 32
• 33. II. Practical Applications of Direct Effects and Causal InferenceWe can now observe the impact on Sales as a result of changing Competitive Adv. by one unit. The resultingdelta of -32.104 is shown in parentheses. This con rms (within the possible numeric precision) the valuereported in the Total Effects table.However, the reader will notice that not only Sales was affected but also most of the other nodes, albeitwith very small changes. This means that, given that we observe a one-unit change of Competitive Adv., willalso observe a change in other nodes, which are too connected to the target and may thus contribute to achange in the target. This re ects the Bayesian network property of omnidirectional inference. As such, theone-unit change in Competitive Adv. is not an orthogonal impulse, which is very important to bear in mindfor interpretation purposes.Causal InferencePearl’s Do-OperatorTo move beyond the observational inference generated by the Total Effects function, we must now turn to acausal framework. Our rst option is to use Intervention with the Do-Operator, which requires us to con-vert our original network into a fully speci ed causal network.At it is immediately obvious that most of the original arc directions, which were found by the SupervisedLearning algorithm, cannot be interpreted causally, e.g. Sales does neither cause Temperature nor Weekday.www.conradyscience.com | www.bayesia.com 33
• 34. II. Practical Applications of Direct Effects and Causal InferenceHowever, using our domain knowledge, we can assume that Sales is the effect of all the other variables inthis model.So, we will need to encode these causal relationships manually, as shown in the following graph:While this causal representation is formally correct, it creates an immediate practical problem. As we do nothave any parametric representation of the relationship between Sales and the other 6 variables, the requiredCPT associated with Sales contains 28,672 cells. With only a few hundred observations, it is impossible toobtain a robust estimate all these parameters.BayesiaLab will actually highlight this problem as we build this network manually.www.conradyscience.com | www.bayesia.com 34
• 35. II. Practical Applications of Direct Effects and Causal InferenceFor now, however, we may want to ignore this constraint and proceed with this approach. We can useBayesiaLab’s Taboo Learning to search for additional probabilistic relationships after having xed themanually-encoded causal arc structure from above. Upon completion of this algorithm, and having appliedthe layout algorithm, we now have a more connected network:These newly established arcs, however, do not yet re ect our causal assumptions. We now need to gothrough them one by one to formalize the direction of causality. With some arcs, it is fairly obvious, such asWeekday ➝ No. of Stores (e.g. some stores are closed because it is Sunday).We can invert this arc from within BayesiaLab’s Validation Mode. We simply right-click the arc of inter-ested and select Invert Orientation within the Equivalence Class.99 For a discussion of equivalence class, see chapter 1.www.conradyscience.com | www.bayesia.com 35
• 36. II. Practical Applications of Direct Effects and Causal InferenceThe new structure, with the inverted arc highlighted in red, is shown below:However, a side effect of this arc inversion within the equivalence class was that the arc, Temperature ➝No. of Stores was automatically inverted in order to maintain the original JPD. We can resolve this by es-tablishing constraints that re ect our causal knowledge, e.g. a higher Temperature in summer causes ahigher No. of Stores to be open, meaning that only the arc Temperature ➝ No. of Stores is permissible butnot the inverse.While these constraints can be easily applied in BayesiaLab, we will omit these details and instead fast-forward to another issue, which, as it turns out will make all previous efforts futile: We have probabilisticrelationships in our domain for which, given our knowledge, we cannot resolve the causal direction. Forinstance, does Trad. Adv. cause Competitive Adv. or is it the other way around? Without nalizing thiscausal structure we are unable to proceed with Graph Surgery, which ultimately prevents us from carryingout causal inference.In conclusion, we have two major obstacles towards performing causal inference with Graph Surgery: rst,the intractable size of the CPT and, second, the incomplete causal structure.Direct Effects with Likelihood Matching (LM)As opposed to using the Do-Operator, we can move forward using LM, regardless of the arc directions, aslong as the network provides a good representation of the JPD of the underlying data.For this purpose, a new Direct Effect Analysis tool has recently been introduced in BayesiaLab 5.0.4. This issimilar to the Total Effects tool, however Direct Effects obtains, as the name implies, the “direct” impact ofa treatment variable on the target node by using the LM algorithm to x the confounders.The new approach with LM requires fewer prerequisites and may thus lead us to the desired causal infer-ence more quickly. We can return to the originally learned non-causal Bayesian network, which is computa-tionally entirely tractable.www.conradyscience.com | www.bayesia.com 36
• 37. II. Practical Applications of Direct Effects and Causal InferenceOn the basis of this non-causal network, we can perform Direct Effects (Analysis>Report>Target Analy-sis>Direct Effects on Target).The resulting table provides us with Standardized Direct Effect, Direct Effect, Contribution and Elasticity,with respect to Sales:Direct Effects on Target Sales Standardized Node Direct,Effect Contribution Elasticity Direct,EffectNo.\$of\$Stores 0.229 65.5416 32.72% 21.39%Trad.\$Adv. 0.1851 4.496 26.45% 19.41%Competitive\$Adv. ?0.13 ?12.041 18.57% ?9.67%Online\$Adv. 0.0982 14.8906 14.03% 9.03%Weekday 0.0305 354.755 4.36% 2.55%Temperature 0.027 67.7507 3.86% 2.09%The Direct Effect column represents the effect of a unit-change of each variable while holding all other vari-ables xed. One can think of each node (in turn and by itself) being considered a treatment, while all othernodes, except for the target, are being used as “likelihood-matched” sets of covariates. For instance, a one-unit change in No. of Stores is associated with +65.5 delta in Sales, everything else being equal.The Contribution column provides a breakdown of each variable’s individual contributions in percent(summing up to 100%). This means than an observed change in Sales should be attributed to the individualvariables as per the Contribution values.www.conradyscience.com | www.bayesia.com 37
• 38. II. Practical Applications of Direct Effects and Causal InferenceElasticity is shown in the rightmost column. The de nition of Elasticity is based on the mathematical notionof point elasticity. In general, the “x-elasticity of y”, also called the “elasticity of y with respect to x”, is: ∂ln y ∂y x %ΔyE y,x = = ⋅ = ∂ln x ∂x y %ΔxIn marketing, Elasticity is most often used in the context of price elasticity.It is important to point out that the Direct Effect is a linearized value and represents the derivative of theDirect Effects Function taken at the a-priori mean value of the respective variable. All the Direct EffectsFunctions can be shown with Analysis>Graphic>Target Mean Analysis>Direct Effect:www.conradyscience.com | www.bayesia.com 38
• 39. II. Practical Applications of Direct Effects and Causal InferenceTo make the graph easier to interpret, the values of all variables (except the target) are normalized. In thecase of Weekday, this means that the numerical values 1 through 7 (representing Monday through Sunday)are normalized to a 0 to 100 range.The nonlinear character of several of these variables is rather obvious and suggests that the linearized DirectEffect must be used with caution. For the near-linear variables Trad. Adv. and Competitive Adv. the DirectEffect may fully capture the nature of the relationship with Sales, whereas for the nonlinear Weekday itwould be misleading. This becomes particularly relevant in the context of optimization, which we will dis-cuss later.Causal Inference as an AfterthoughtDirect Effects per se carry no causal meaning. However, if we do provide causal assumptions, we can im-mediately interpret Direct Effects as causal effects. We can make the causal assumption after computing theDirect Effects, quite literally as an afterthought.www.conradyscience.com | www.bayesia.com 39