J. Pearl, Causal diagrams for empirical research, Biometrika 82 (4) (1995) 669--710.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....it follows from the UNC supposition (i) that U (s #) U (s) which, substituting, implies U (s) U (s) U (s # )# which is false since both U (s # ) and # are positive. APPENDIX B The causal DAG 1 in Figure A1 represents a non parametric structural equation model (NPSEM) as de#ned in Pearl [32]. Arrows on the graph represent direct causal relations. I [33] show that a NPSEM has a nearly equivalent interpretation as a member of a class of counterfactual causal models proposed in reference [3] 36] With the exception of U , all variables on the DAG are assumed to have been recorded for ....

....common cause of survival at time 1 and 2. Because there are no arrows from any unmeasured variable U directly into treatments Z 0 or Z 1 , DL s sequential ignorability assumption holds [34] Further, the absence of an arrow directly from Z 0 to Y 2 is equivalent to DL s UNC hypothesis. Pearl s [32] graphical d separation procedures can be used to determine all the conditional and unconditional independencies implied by the underlying causal structure that obtain in the observed data. In particular, Z 0 is not d separated from Y 2 conditional upon the observed variables Y 1 , Z 1 , X 1 , ....

Pearl J. Causal diagrams for empirical research. Biometrika 1995; 82(4):669 -- 688.

....methods for handling missing data where absence is independent of state are simpler than those where absence and state are dependent. In this tutorial, we concentrate on the simpler situation only. Readers interested in the more complicated case should see Rubin (1978) Robins (1986) and Pearl (1995). Continuing with our example using unrestricted multinomial distributions, suppose we observe a single incomplete case. Let Y ae X and Z ae X denote the observed and unobserved variables in the case, respectively. Under the assumption of parameter independence, we can compute the posterior ....

....by whichwe can learn causal relationships. In this section, we examine these semantics, and provide a basic discussion on how causal relationships can be learned. We note that these methods are new and controversial. For critical discussions on both sides of the issue, see Spirtes et al. 1993) Pearl (1995), and Humphreys and Freedman (1995) For purposes of illustration, suppose we are marketing analysts who want to know whether or not we should increase, decrease, or leave alone the exposure of a particular 40 advertisement in order to maximize our profit from the sales of a product. Let ....

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Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, 82:669-- 710.

....methods for handling missing data where absence is independent of state are simpler than those where absence and state are dependent. In this tutorial, we concentrate on the simpler situation only. Readers interested in the more complicated case should see Rubin (1978) Robins (1986) and Pearl (1995). 19 Continuing with our example using unrestricted multinomial distributions, suppose we observe a single incomplete case. Let Y C X and Z C X denote the observed and unob served variables in the case, respectively. Under the assumption of parameter independence, we can compute the posterior ....

....by which we can learn causal relationships. In this section, we examine these semantics, and provide a basic discussion on how causal relationships can be learned. We note that these methods are new and controversial. For critical discussions on both sides of the issue, see Spirtes at el. 1993) Pearl (1995), and Humphreys and Freedman (1995) For purposes of illustration, suppose we are marketing analysts who want to know whether or not we should increase, decrease, or leave alone the exposure of a particular 40 advertisement in order to maximize our profit from the sales of a product. Let ....

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Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, 82:669- 710.

....needs empirical demonstration, which is easier said than done. What brings you to Chicago Econometrics. There is a lot of that going around. Overheard by Arnold Zellner Nonlinear models: Figure 1 revisited Graphical models can be set up with nonlinear versions of equation (1) as in Pearl (1995, 2000) The specification would be something like Y i,x = f(x,# i ) where f is some fairly general (unknown) function. The same questions about interventions and counterfactual hypotheticals would then have to be considered. Instead of rehashing such isues, I will indicate how to formulate ....

....distributions, rather than specific numerical values. There will be some interesting new questions about identifiability. And the plausibility of causal interpretations can be assessed separately, as will be shown later. I will organize most of the discussion around two examples used by Pearl (1995); also see Pearl (2000, pp.66 68 and 83 85) But first, consider Figure 1. In the nonlinear case, the exogenous variables have to be assumed independent and identically distributed in order to make sense out of the mathematics; otherwise, there are substantial extra complications, or we have to ....

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Pearl, J. (1995). Causal Diagrams for Empirical Research. Biometrika 82: 669--710 (with discussion) .

.... and between probability of causation [Greenland and Robins, 1988] and attributable fractions [Schlesselman, 1982] have taken heavy tolls before they were finally resolved [Greenland et al. 1999b; Pearl 2000, Chapter 6; Greenland 1999; Tian and Pearl 2000] 7 Commenting on my set(x) notation [Pearl, 1995a, b] a leading statistician wrote: Is this a concept in some new theory of probability or expectation If so, please provide it. Otherwise, metaphysics may remain the leading explanation. Another statistician, commenting on the do(x) notation used in Causality [Pearl, 2000] insisted: the ....

....= yjdo(x) into do free expressions derivable from P (z; x; y) since only do free expressions are estimable from non experimental data. When such a transformation is feasible, we say that the causal quantity is identifiable. A calculus 11 for performing such transformations was developed in [Pearl, 1995a] This calculus permits the investigator to inspect the causal diagram and 1. Decide whether the assumptions embodied in the model are sufficient to obtain consistent estimates of the target quantity; 2. Derive (if the answer to item 1 is affirmative) a closed form expression for the target ....

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J. Pearl. Causal diagrams for empirical research. Biometrika, 82(4):669--710, December 1995.

....observables, is a set of terms. P 0 is a function [C 0 [0; 1] such that 8 2 C 0 , P ff2 P 0 (ff) 1: i.e. P 0 is a probability measure over the alternatives controlled by nature. 8 This mapping also lets us see the relationship between the causation that is inherent in Bayesian networks (Pearl 1995) and that of the logical formalisms. See Poole (1993) for a discussion on the relationship, including the Bayesian network solution to the Yale shooting problem and stochastic variants. See also Section 3.1.2. 17 F called the facts, is an acyclic logic program such that no atomic choice (in an ....

....decision is make. If we were using this as a representation of another agent, we could have a probability distribution over these actions. The parents of these decision nodes are the information available when the action was performed. ffl The second tradition is to treat actions as interventions (Pearl 1995). An action changes the value of a variable externally (effectively cutting the parents from the variable, and giving it a new value) The actions in the Bayesian network built from fragments like that of Figure 3 are more like hypotheticals. For example, the variable carrying(key ; S ) for ....

Pearl, J. (1995). Causal diagrams for empirical research, Biometrika 82(4): 669--710.

....to B in every DAG G(O,S,L) represented by PAG p contains a member of S. Theorem 4: If p is a partial ancestral graph, and every semi directed path from A to B contains some member of C in p, then every directed path from A to B in every DAG G(O,S,L) represented by PAG p contains a member of SC. Pearl (1995) showed how in some cases to use the Causal Calculus (equivalent to Theorem 7.1 of Spirtes, Glymour, and Scheines 1993) to calculate the effects of interventions from a DAG of completely known structure and a marginal observed distribution, even if the DAG contains latent variables. PAGs ....

Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, 82, 669-709.

....Yishay Mansour Tel Aviv University mansour math.tau.ac. il February 1998 1 Introduction In the literature on graphical models, there has been increased attention paid to the problems of learning hidden structure (see Heckerman [H96] for a survey) and causal mechanisms from sample data [H96, P95, S93]. In most settings we should expect the former to be difficult, and the latter potentially impossible without experimental intervention. In this work, we examine some restricted settings in which the ideal can be obtained: efficient algorithms that perfectly reconstruct the hidden causal structure ....

Judea Pearl. Causal Diagrams for Empirical Research. Biometrika 82:669-710, 1995.

....have been formally described as a representation of conditional independence, as we noted in Section 2, people often construct them using notions of cause and effect. Recently, several researchers have begun to explore a formal causal semantics for Bayesian networks (e.g. Pearl and Verma, 1991, Pearl, 1995, Spirtes et al. 1993, Druzdzel and Simon, 1993, and Heckerman and Shachter, 1994) They argue that the representation of causal knowledge is important not only for assessment, but for prediction as well. In particular, they argue that causal knowledge unlike statistical knowledge allows one ....

....seen and by assessing an equivalent sample size. Most remarkable, we show that Dirichlet assumption (Assumption 4) is not needed to obtain the BDe metric. 6 Note that, in some circumstances, we can identify causes and effects from network structure even when there are hidden common causes. See Pearl (1995) for a discussion. Learning Bayesian Networks, MSR TR 94 09 21 5.1 Informative Priors In this section, we show how the added assumption of likelihood equivalence simplifies the construction of informative priors. Before we do so, we need to define the concept of a complete network structure. A ....

Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, to appear. Learning Bayesian Networks, MSR-TR-94-09 49

....of attack. Two classes of technique have arisen: Bayesian causal discovery, which focuses on learning complete causal models for small data sets (Balke and Pearl, 1994, Cooper and Herskovits, 1992, Heckerman, 1995, Heckerman, 1997, Heckerman et al. 1994, Heckerman et al. 1997, Pearl, 1994, Pearl, 1995, Spirtes et al. 1993) and an offshoot of the Bayesian learning method called constraint based causal discovery, which use the data to limit sometimes severely the possible causal models (Cooper, 1997, Spirtes et al. 1993, Pearl and Verma, 1991) While techniques in the first class are ....

J. Pearl. Causal diagrams for empirical research. Biometrika, 82(1995): 669-709.

....validates the intuition that conditioning on all potentials confounders produces an unbiased estimate. The set of potential confounders does not contain any descendant of Y, because a potential confounder must occur prior to Y; hence 2 Note this criterion is similar to Pearl s back door criterion (Pearl, 1995), except that the back door criterion was proposed as a means of estimating the total effect of X on Y. 13 no descendant of Y is conditioned on. Suppose, contrary to the hypothesis that there is a path U d connecting X and Y given the set of potential confounders. U either contains an edge into ....

Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, 82, 669-709.

....a deeper understanding of the underlying structure of relationships between variables. A quick glance at the Bayesian network for the contraceptive study reveals the structure of these relationships. On the other hand, the formal definitions of spurious association, potential and genuine causality [6,7] may tax the casual reader or require a leap of faith as to their meaning. If the graph is not causally sufficient, as is the case with the CMC study, then Tetrad produces no estimates of conditional probabilities. Therefore, while one can see that the wife s educat ion is a genuine cause of ....

Pearl, J. Causal Diagrams for Empirical Research. (1995) Biometrika, 82:4, 669-P

....statistical literature, concerned with the exploitation of this language to clarify and extend causal concepts. Among these we This is Research Report R 99 2021, Department of Mathematical Sciences, Aalborg University. 1 mention in particular books by Spirtes et al. 1993) Shafer (1996) and Pearl (2000) as well as the collection of papers in Glymour and Cooper (1999) Very briefly, but fundamentally, the important distinction to be made is the distinction between two types of conditional probability. We refer to these as conditioning by intervention and conditioning by observation and suggest ....

....set containing all the variables involved is shown in the second graph of Fig. 1. It is immediate that S separates a from b in this moral graph, implying a b j S. 2 An alternative formulation of the global, directed Markov property was given by Pearl (1986a) with a formal treatment in Verma and Pearl (1990). 12 Recall that a trail in D is a sequence of vertices that forms a path in the undirected version D of D, i.e. when the directions of arrows are ignored. A trail from a to b in a directed, acyclic graph D is said to be blocked by S if it contains a vertex fl 2 such that either fl 2 S and ....

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Pearl, J. (1995a). Causal diagrams for empirical research. Biometrika, 82, 669--710.

....of a direct link between two features is an assertion of their conditional independence given the other features appearing in the network. Links in the network can be interpreted as causal relations between features though this is not always straightforward, as exemplified by the discussion in [15] which can yield particularly informative inferences. For realistic problems, graphical models involve large numbers of parameters and do not fit well into the framework of classical statistical inference. Nearest neighbor methods. At its simplest, the k nearest neighbor procedure assigns a ....

Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, 82, 669--710.

....i.e. the weakest sufficient condition of the observation. ffl Definability It is often necessary to determine whether a given theory yields a definition of a proposition in terms of a set of other propositions. For example, such computation is essential in both Simon s [12] and Pearl s [10] approaches to causation. This is also what is needed in order to compute successor state axioms [11] from causal theories. We believe definability is best handled using our proposed two conditions. While a proposition may or may not be definable in terms of a set of base propositions, our ....

J. Pearl. Causal diagrams for empirical research. Biometrika, 82(4):669--709, 1995.

....concept of causality is different of what is known as Granger causality . The latter concept implies an ordering in time such that a cause must be prior to the effect. The causal graph approach is a way to formalize the notion of instantaneous causality discussed, e.g. in Lutkepohl (1993) and Pearl (1995). The causal ordering given in (21) implies that E( it kt j jt ) 0 for i = k; n and i j k. Accordingly, Swanson and Granger (1996) suggest to test the partial correlation between it and kt conditional on jt in order to recover the causal ordering empirically. To motivate ....

Pearl, J. (1995), Causal Diagrams for Empirical Research, forthcoming in: Biometrika.

....concept of causality is different of what is known as Granger causality . The latter concept implies an ordering in time such that a cause must be prior to the effect. The causal graph approach is a way to formalize the notion of instantaneous causality discussed, e.g. in Lutkepohl (1993) and Pearl (1995). TABLE 1: Tests for the cointegration rank H 0 max. eigenvalue crit. val. trace crit. val. r 3 1.473 3.962 1.473 3.962 r 2 5.509 14.04 6.982 15.20 r 1 33.31 20.78 40.29 29.51 r = 0 50.06 27.17 90.35 47.18 Note: max. eigenvalue and trace indicate Johansen s LR statistics for the ....

Pearl, J. (1995), Causal Diagrams for Empirical Research, forthcoming in: Biometrika.

....do not appear in this definition. This is analogous to defining the first order predicate calculus without 6 In terms of (Poole 1997) all of the alternatives are controlled by nature. 7 This mapping also lets us see the relationship between the causation that is inherent in Bayesian networks (Pearl 1995) and that of the logical formalisms. See Poole (1993) for a discussion on the relationship, including the Bayesian network solution to the Yale shooting problem and stochastic variants. 12 any need to define situations. Situations will provide a standard interpretation for some of the terms. ....

Pearl, J. (1995). Causal diagrams for empirical research, Biometrika 82(4): 669--710.

....Similarly other concepts of sufficiency and ancillarity in the presence of nuisance parameters can be introduced and studied using CI [5, 6] Other applications CI has been also been fruitfully applied to clarify and study a wide range of other statistical problems. These include: causal inference [3, 8, 22]; selected or missing data [2, 3, 10] and model building [3, 7] The general usefulness of CI as a way of expressing assumptions about data generating and inferential processes, and of extracting their consequences, is well illustrated in [12] which deals with statistical problems of forensic ....

....of guilt in the light of the evidence. The above graphical reasoning techniques, and extensions, have had particularly important applications to PROBABILISTIC EXPERT SYSTEMS [20] G1 N A C X3 Y1 B G2 R Figure 3: Moralized ancestral subgraph G 0 GRAPHICAL MODELS [11, 16] and CAUSAL MODELLING [22, 14]. ....

Pearl, J. (1995). Causal diagrams for empirical research (with Discussion). Biometrika 82, 669--710. Extends graphical manipulation methods to address questions of causal inference from non-experimental data.

....of the variables. For example, these methods are not appropriate for a medical database where data about drug response is missing in those patients who became too sick to take the drug. Methods for addressing dependencies in omissions have been explored by (e.g. Rubin (1978) Robins (1986) and Pearl (1995). 8.1 Fill In Methods First, let us consider the simple situation where we observe a single incomplete case C in domain U . Let Y denote the variables not observed in the case. Under the assumption of parameter independence, we can compute the posterior distribution of Theta ij as follows: ....

Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, to appear.

....measure that is the focus of the present analyses or other more complex ones. It is useful to think about these issues by trying to construct a causal model for health carre outcomes so that we can consider causal connections among variables (e.g. see Spirtes, Glymour, and Scheines, 1993; and Pearl, 1996) for illustrations and discussion) as represented in directed acyclic graphs (DAG) In a DAG, nodes (circles) correspond to variables or groups of variables, and directed edges (arrows) correspond to known or hypothesized causal links (association with a causal explanation) the absence of an ....

Pearl, J. (1996). Causal diagrams for empirical research. Biometrika, 82, (in press).

....nor possible in most applications of data mining. Fortunately, recent research in statistics and Bayesian learning communities provide some avenues of attack. Two classes of technique have arisen: Bayesian causal discovery, which focuses on learning complete causal models for small data sets [BP94, CH92, H95, H97, HGC94, HMC97, P94, P95, SGS93], and an offshoot of the Bayesian learning method called constraint based causal discovery, which use the data to limit sometimes severely the possible causal models [C97, SGS93, PV91] While techniques in the first class are still not practical on very large data sets, a limited version ....

J. Pearl. Causal diagrams for empirical research. Biometrika, 82(1995): 669-709.

....affect whether we get lung cancer. Work by artificial intelligence researchers, statisticians, and philosophers have emphasized the importance of identifying causal relationships for purposes of modeling the effects of actions. For example, Simon (1977) Robins (1986) Spirtes et al. 1993) and Pearl (1993, 1995) have developed graphical models of cause and effect, and have demonstrated how these models are important for reasoning about the effects of actions. In addition, Robins (1986) Rubin (1978) Pearl and Verma (1991) and Spirtes et al. 1993) have developed approaches that embrace causality for ....

....are important for reasoning about the effects of actions. In addition, Robins (1986) Rubin (1978) Pearl and Verma (1991) and Spirtes et al. 1993) have developed approaches that embrace causality for learning the effects of actions from data. One useful framework for causal reasoning is that of Pearl (1993, 1995) herein Pearl. Using his framework, we construct a causal graph G. The nodes in G correspond to a set of variables U that we wish to model. Each variable has a set of mutually exclusive and collectively exhaustive values or instances. The arcs in G represent (informal) assertions of cause in ....

Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, in press.

....some structure S, but we are uncertain about the identity of S. We write M = m s when the true distribution 2 Sometimes, an additional causal interpretation is given to the arcs in S. Namely, an arc from X i to X i reflects the assertion that X i is a direct cause of X j (Spirtes et al. 1993; Pearl, 1995). factors according to S. 3 Second, we parameterize the local probability distributions with a finite number of parameters. Explicitly conditioning on the model and its parameters, we rewrite Equation 3 as p(xj s ; m s ) n Y i=1 p(x i jpa i ; i ; m s ) where i are the parameters ....

Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, 82:669--710.

....Families: Bun95a] Whi90] Issues of Conjugacy: Dawid s response to [LS88] Continuous Belief Function Densities: WD94b] WD94a] others: AFS94] KSC84] GH94a] GH95a] Ken86] 1.10 Philosophical Issues 1. 10.1 Causality and Control refs: Pea88a] Pea94b] Pea94a] Pea95b] Pea95a] SGS93] DS93] HS94] CS92] BP94] BP95] GP95] GP96] HB94b] Pea96] Sto93] 1.10.2 Modeling Various critiques and or criticisms on the problems and issues involved in designing or using a graphical model: Spe90] 1.10.3 Other Counterfactuals [Bal95] 1.11 Important ....

Judea Pearl. Causal diagrams for empirical research. Biometrika, 82(4):669--709, December 1995.

....representation of choice among researchers interested in uncertainty in AI. One often cited merit of Bayesian networks is that they have formal probabilistic semantics and yet can serve as a natural mirror of knowledge structures in the human mind [ Spirtes et al. 1993; Heckerman et al. 1995; Pearl, 1995 ] A Bayesian network consists of two components. The first is a directed acyclic graph in which each vertex corresponds to a random variable. This graph represents a set of conditional independence properties of the represented distribution: each variable is probabilistically independent of ....

....together with the fact that data is becoming increasingly available and cheaper to acquire has led to a growing interest in using data to learn both the structure and probabilities of a Bayesian network. Several groups have worked on learning structure from scratch [ Spirtes et al. 1993; Pearl, 1995; Friedman et al. 1997 ] or with weak constraints such as variable ordering [ Cooper and Herskovits, 1992, for example ] while others have worked on learning structure by refining an initial model [ Heckerman et al. 1994 ] Learning probabilities, which is non trivial when the network ....

Pearl, J. (1995). Causal diagrams for empirical research.

....nor possible in most applications of data mining. Fortunately, recent research in statistics and Bayesian learning communities provide some avenues of attack. Two classes of technique have arisen: Bayesian causal discovery, which focuses on learning complete causal models for small data sets [8, 12, 14, 15, 16, 17, 21, 22, 23, 25, 26, 27], and an offshoot of the Bayesian learning method called constraint based causal discovery, which use the data to limit sometimes severely the possible causal models [11, 26, 24] While techniques in the first class are still not practical on very large data sets, a limited version of the ....

J. Pearl. Causal diagrams for empirical research. Biometrika, 82(1995): 669-709.

....common causes, then we can interpret learned networks as causal networks. In our example, under these assumptions, we can infer that x 1 and x 2 are causes for x 3 . Note that, in some circumstances, we can identify causes and effects from network structure even when there are hidden common causes [Pearl, 1995]. For causal networks, we must modify the definition of B h s . For such networks, we say that hypothesis B h s is true iff the parameters Theta U satisfy the conditional independence assertions of B s , and each nonroot node x is the immediate causal effect of its parents. Consequently, as ....

Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, to appear.

....the model are unobservable. For simplicity, we will restrict the discussion in this section to the case where the availability or otherwise of an observation is independent of the actual states of the variables. Methods for addressing dependencies on omissions have been studied in, for example, [69], 75] and [78] We will come back to this in the next section) The two variable case illustrates the general situation quite simply. We have the variables X 1 and X 2 with states x 1 , x 1 and x 2 , x 2 respectively. X 2 is observed to be in state x 2 , whilst the state of X 1 is unknown. ....

Pearl J. 1995 Causal diagrams for empirical research. Biometrika, 82, 669-710.

....quantity depends upon the causal relations between C and D. If C is a cause of D, then forcing the value c on C will in general have an effect on the value of D, while if C is an effect of D, then forcing a value of c on C will not have an effect on the value of D. See Spirtes et al. 1993) and Pearl (1995) for details. In this particular case, it is possible to make both qualitative and quantitative predictions about the effects on the value of D of interventions that set the value of C from the PAG and the measured conditional distribution of D on C = c. This is because every DAG in O Equiv(Cond 2 ....

Pearl, J. (1995). "Causal Diagrams for empirical research." Biometrika, 82, 669-710.

....likelihoods. Thus, if S h 1 and S h 2 are independence equivalent, then S h 1 = S h 2 . This property, which we call hypothesis equivalence, implies likelihood equivalence. Nonetheless, some researchers give Bayesian network structure a causal interpretation (e.g. Spirtes et al. 1993; Pearl, 1995) In this case, we can modify the definition of S h to include the assertion that if X i X j in S, then X i is a direct cause of X j . Consequently, hypothesis equivalence does not hold. Nonetheless, the weaker assumption of likelihood equivalence is sometimes reasonable. For a detailed ....

Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, to appear.

....considerations to construct the Bayesian network structure shown in Figure 1. The network is used for troubleshooting printing problems within the Windows tm operating system. The connection between causation and conditional independence is discussed in detail in (e.g. Spirtes et al. 1993, Pearl, 1995, Heckerman and Shachter, 1995] Statistical techniques for learning Bayesian network structure from data or a combination of data and expert knowledge are also available [Cooper and Herskovits, 1992, Spiegelhalter et al. 1993, Buntine, 1994, Madigan and Raftery, 1994, Heckerman et al. 1995b] ....

....Philosophers call such an assumption a counterfactual [Lewis, 1973, Holland, 1986] a statement that can not be verified by observation. Although this assumption may seem unusual, this and other counterfactual assumptions can be made rigorous in the context of a causal model [Rubin, 1978, Pearl, 1995, Heckerman and Shachter, 1995] We note that amechanistic causal independence has several model restrictions. Namely, each intermediate node e i must have the same number of states as e. Also, let e 0 denote the state of e when all causes are in their distinguished state. Then, by definition of ....

Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, to appear.

....methods for handling missing data where absence is independent of state are simpler than those where absence and state are dependent. In this tutorial, we concentrate on the simpler situation only. Readers interested in the more complicated case should see Rubin (1978) Robins (1986) and Pearl (1995). Continuing with our example using unrestricted multinomial distributions, suppose we observe a single incomplete case. Let Y ae X and Z ae X denote the observed and unobserved variables in the case, respectively. Under the assumption of parameter independence, we can compute the posterior ....

....by which we can learn causal relationships. In this section, we examine these semantics, and provide a basic discussion on how causal relationships can be learned. We note that these methods are new and controversial. For critical discussions on both sides of the issue, see Spirtes et al. 1993) Pearl (1995), and Humphreys and Freedman (1995) For purposes of illustration, suppose we are marketing analysts who want to know whether or not we should increase, decrease, or leave alone the exposure of a particular advertisement in order to maximize our profit from the sales of a product. Let variables ....

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Pearl, J. (1995). Causal diagrams for empirical research. Biometrika, 82:669-- 710.

.... modifiable sets of functions [Galles and Pearl, 1997, Galles and Pearl, 1998, Halpern, 1998, Pearl, 2000] The structural models semantics, as we shall see in Section 2, leads to effective procedures for computing probabilities of counterfactual expressions from a given causal theory [Balke and Pearl, 1994, 1995]. Additionally, this semantics can be characterized by a complete set of axioms [Galles and Pearl, 1998, Halpern, 1998] which we will use as inference rules in our analysis. The central aim of this paper is to estimate probabilities of causation from frequency data, as obtained in experimental ....

....example, how the results presented in this paper can be applied to resolve issues of attribution in legal settings. Section 6 concludes the paper. 3 2 Structural Model Semantics This section presents a brief summary of the structural equation semantics of counterfactuals as defined in Balke and Pearl (1995), Galles and Pearl (1997, 1998) and Halpern (1998) Related approaches have been proposed in Simon and Rescher (1966) see footnote 4) and Robins (1986) For detailed exposition of the structural account and its applications see [Pearl, 2000] Structural models are generalizations of the ....

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J. Pearl. Causal diagrams for empirical research. Biometrika, 82(4):669--710, December 1995.

....the nature of nonlinear interactions: both direct and indirect effects may be small and, still, the total effect can be large. 5 Appendix Causal Models and Counterfactuals This appendix presents a brief summary of the structural equation semantics of counterfactuals as defined in Balke and Pearl (1995), Galles and Pearl (1997, 1998) and Halpern (1998) Related approaches have been proposed in Simon and Rescher (1966) see footnote 11) and Robins (1986) For detailed exposition of the structural account and its applications see [Pearl, 2000] Causal models are generalizations of the structural ....

.... z) f(x; z 0 ) 10 Structural modifications date back to Marschak (1950) and Simon (1953) An explicit translation of interventions into wiping out equations from the model was first proposed by Strotz and Wold (1960) and later used in Fisher (1970) Sobel (1990) Spirtes et al. 1993) and Pearl (1995). A similar notion of sub model is introduced in Fine (1985) though not specifically for representing actions and counterfactuals. 27 Definition 11 (Effect of action) Let M be a causal model, X be a set of variables in V , and x be a particular realization of X. The effect of action do(X = x) ....

J. Pearl. Causal diagrams for empirical research. Biometrika, 82(4):669--710, December 1995.

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J. Pearl. Causal diagrams for empirical research. Biometrika, 82(4):669--710, December 1995.

....6 concludes with remarks on the role of counterfactual calculus vis a vis structural equations and graphs. 2 Causal Models 2.1 Definitions A causal model is a mathematical object that provides an interpretation (and effective computation) of every causal query about the domain. Following [Pearl, 1995a] we adopt here a construct named modifiable structural equations, that generalizes most causal models used in engineering, biology, and economics. Definition 1 (causal model) A causal model is a triple M = U; V; F where (i) U is a set of variables, called exogenous, that are determined ....

.... of some local actions (e.g. creating a world where kangaroos have no tails) are invited to replace the word action with the word modification (see [Leamer, 1985] The advantages of using hypothetical external interventions to convey the notion of local change are emphasized in [Pearl, 1995a, p. 706] 4 Definition 5 (counterfactual) Let Y be a variable in V , and let X a subset of V . The counterfactual sentence The value that Y would have obtained, had X been x is interpreted as denoting the potential response Y x (u) 3 Two special cases are worth noting. First, if Y = V i ....

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J. Pearl. Causal diagrams for empirical research (with discussion). Biometrika, 82(4):669--710, 1995.

....proving statements about causal irrelevance. 2 2 Causal Theories A causal theory is a fully specified model of the causal relationships that govern a given domain, namely, a mathematical object that provides an interpretation (and computation) of every causal query about the domain. Following [Pearl, 1995a] we will adopt here a definition that generalizes most causal models used in engineering and economics. Definition 1 (Causal Theory) A causal theory is a 4 tuple T = V; U; P (u) ff i g where (i) V = fX 1 ; X n g is a set of endogenous variables determined within the system, ii) ....

....statistical literature [Rubin, 1974] to stand for the counterfactual sentence The value that Y would take in person u, had X been x, where X stands for a type of treatment that a person can receive. There is a strong connection between the the sentence above and our interpretation of Y x (u) Pearl, 1995a] Definition 2 interprets the abstract, counterfactual sentence above in terms of the processes responsible for Y taking on the value Y x (u) as X changes to x. It treats u not merely as an index of an individual but, rather, as the set of attributes u that characterize the individual, the ....

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J. Pearl. Causal diagrams for empirical research (with discussion). Biometrika, 82(4):669--709, 1995.

....reasons why the effect has been (and still is) considered paradoxical and why its resolution has been so late in coming. The report is extracted from a forthcoming book Causality [Pearl, 2000] and assumes some familiarity with causal diagrams and the do( Delta) orset( Delta) notation (e.g. [Pearl, 1995]) 0.1 A Tale of a Non Paradox Simpson s paradox [Simpson, 1951# Blyth, 1972] first encountered by Pearson in 1899 [Aldrich, 1995] refers to the phenomenon whereby an event C increases the probability of E in a given population p and, at the same time, decreases the probability of E in every ....

....for some misguided yet persistent illusion, what is so shocking about inequalities reversing direction 2 Cartwright (1983, p. 37) states, though, that the third factor F should be held fixed if and only if F is causally relevanttoE# the correct (back door) criterion is somewhat more involved [Pearl, 1995]. 3 Pearson et al. 1899) and Yule (1903) reported a weaker version of the paradoxinwhich (2) 3) are satisfied with equality. The reversal was discovered later by Cohen and Nagel (1934, p. 449) 3 Pearson understood that the shock originates with distorted causal interpretations, which he ....

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J. Pearl. Causal diagrams for empirical research. Biometrika, 82(4):669--710, December 1995.

....be improved without strengthening the assumptions. 2 Probabilities of Causation: Definitions In this section, wepresent the definitions for the three aspects of causation as defined in [Pearl, 1999] Weuse the language of counterfactuals in its structural model semantics, as given in BalkeandPearl (1995), Galles and Pearl (1997, 1998) and Halpern (1998) Weuse Y x = y to denote the counterfactual sentence Variable Y would have the value y,hadX been x. One property that the counterfactual relationships satisfy is the consistency condition [Robins, 1987] X = x) Y x = Y ) 1) stating that ....

....process, these quantities mayeven be identified. 1 The causal effects P (yx)andP (y x 0 ) can be estimated reliably from controlled experimental studies, and from certain observational (i.e. nonexperimental) studies which permit the control of confounding through adjustmentof covariates [Pearl, 1995]. 3 Bounds and Conditions of Identification In this section we will assume that experimental data will be summarized in the form of the causal effects P (y x ) and P (y x 0 ) and nonexperimental data will be summarized in the form of the joint probability function: PXY = fP (x# y)#P(x 0 ....

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J. Pearl. Causal diagrams for empirical research. Biometrika, 82(4):669--710, December 1995.

.... 4) and personal decision making (Section 5) 2 Probabilities of Causation: Definitions In this section, we present the definitions for the three aspects of causation as defined in [Pearl, 1999] We use the language of counterfactuals in its structural model semantics, as given in Balke and Pearl (1995), Galles and Pearl (1997, 1998) and Halpern (1998) We use Y x = y to denote the counterfactual sentence Variable Y would have the value y, had X been x. The structural model interpretation of this sentence reads: Deleting the equation for X from the model and setting the value of X to a ....

.... 001 = P (x; y 0 ) 7) p 110 p 010 = P (x 0 ; y) 1 The causal effects P (yx) and P (y x 0 ) can be estimated reliably from controlled experimental studies, and from certain observational (i.e. nonexperimental) studies which permit the control of confounding through adjustment of covariates [Pearl, 1995]. and the causal effects, P (y x ) and P (y x 0 ) impose the constraints: P (y x ) p 111 p 110 p 101 p 100 P (y x 0 ) p 111 p 110 p 011 p 010 (8) The quantities we wish to bound are: PNS = p 101 p 100 (9) PN = p 101 =P (x; y) 10) PS = p 100 =P (x 0 ; y 0 ) 11) ....

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J. Pearl. Causal diagrams for empirical research. Biometrika, 82(4):669--710, December 1995.

.... distinct models that cannot be distinguished on the basis of past empirical observation should lead to indistinguishable inference regarding future observation (which may be obtained under new experimental conditions) This is none other but the requirement of identifiability (see e.g. [Pearl, 1995]) It requires, for example, that if our data are nonexperimental, then two models that are indistinguishable on the basis of those data entail the same value of the average causal effect (ACE) a quantity that is discernible in experimental studies. It likewise requires that, if our data come ....

J. Pearl. Causal diagrams for empirical research. Biometrika, 82(4):669--710, December 1995.

....is assumed. Under the stability assumption, probabilistic causal irrelevance is equivalent to path interception in cyclic graphs. The deterministic definition allows for all of the axioms of path interception in cyclic graphs, with the exception of transitivity. Introduction In (Geiger, Verma, Pearl 1990), a set of axioms was developed for a class of relations called graphoids. These axioms characterize informational relevance among observed events based on the semantics of conditional independence in probability calculus. This paper develops a parallel set of axioms for causal relevance, that ....

....Once we hold Z fixed (at z) changing X will not affect the probability of Y . If we remove the hat from the definition above, we get the standard definition of conditional independence in probability calculus, denoted I(X; Z; Y ) which are governed by the graphoid axioms (Geiger, Verma, Pearl 1990) below. 1.1 (Symmetry) I(X; Z; Y ) I(Y; Z; X) 1.2 (Decomposition) I(X; Z; Y W ) I(X; Z; Y ) 1.3 (Weak union) I(X; Z; Y W ) I(X; ZW;Y ) 1.4 (Contraction) I(X; Z; Y W ) I(X; ZW;Y ) 1.5 (Intersection) I(X; ZY; W ) I(X; ZW;Y ) I(X; Z; Y W ) Intersection requires a strictly positive ....

Pearl, J. 1995a. Causal diagrams for empirical research.

....sample of P (y; djz) will be observed, but since our task is one of identification, not estimation, we make the large sample assumption and consider P (y; djz) as given. Causal analysis of treatment effects formalizes changes in the joint distribution which are induced by external interventions [Pearl, 1994]. In particular, the analysis concerns the effect of local interventions, relative to which P (yjd; u) is a stable quantity, i.e. the probability that an individual with characteristics U = u given treatment D = d will respond with Y = y remains the same, regardless of how the treatment was ....

....the corresponding variable in a hypothetical counterfactual world, and a value annotated with a hat ( indicates that the corresponding variable has been set to the value by an external local action. Details of the semantics and evaluation of counterfactual probabilities may be found in [Balke and Pearl, 1994] If we are interested in estimating the average change in Y due to treatment, we can similarly define the average causal effect, ACE(D Y ) Holland, 1988] as ACE(D Y ) P (y 1 j d 1 ) Gamma P (y 1 j d 0 ) 4) The task of causal inference is then to estimate or bound the ....

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Judea Pearl. Causal diagrams for empirical research. Technical Report R-218L, Revision I, UCLA Cognitive Systems Laboratory, May 1994. To appear in Biometrika.

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J. Pearl, Causal diagrams for empirical research, Biometrika 82 (4) (1995) 669--710.

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J. Pearl. Causal diagrams for empirical research. Biometrika, 82(4):669--709, 1995.

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Judea Pearl. Causal Diagrams for Empirical Research. Biometrika 82:669-710, 1995.

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J. Pearl, Causal diagrams for empirical research, Technical Report, R-218-B, Computer Science Department, University of California, Los Angeles, 1995.

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P Pearl, J. "Causal diagrams for empirical research." Biometrika 82:669-688 hilips, A.W. 1950. "Mechanical models in economic dynamics." Economica, 17:282305.

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