Experimental Indistinguishability of Causal Structures Experimental Indistinguishability of Causal Structures Author(s): Frederick Eberhardt Source: Philosophy of Science, Vol. 80, No. 5 (December 2013), pp. 684-696 Published by: The University of Chicago Press on behalf of the Philosophy of Science Association Stable URL: http://www.jstor.org/stable/10.1086/673865 . Accessed: 27/02/2014 10:44 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . The University of Chicago Press and Philosophy of Science Association are collaborating with JSTOR to digitize, preserve and extend access to Philosophy of Science. http://www.jstor.org This content downloaded from 131.215.71.79 on Thu, 27 Feb 2014 10:44:28 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/action/showPublisher?publisherCode=ucpress http://www.jstor.org/action/showPublisher?publisherCode=psa http://www.jstor.org/stable/10.1086/673865?origin=JSTOR-pdf http://www.jstor.org/page/info/about/policies/terms.jsp http://www.jstor.org/page/info/about/policies/terms.jsp Experimental Indistinguishability of Causal Structures Frederick Eberhardt* Using a variety of different results from the literature, I show how causal discovery with experiments is limited unless substantive assumptions about the underlying causal struc- ture are made. These results undermine the view that experiments, such as randomized controlled trials, can independently provide a gold standard for causal discovery. More- over, I present a concrete example in which causal underdetermination persists despite ex- haustive experimentation and argue that such cases undermine the appeal of an interven- tionist account of causation as its dependence on other assumptions is not spelled out. 1. Introduction. Causal search algorithms based on the causal Bayes net representation ðPearl 2000; Spirtes, Glymour, and Scheines 2000Þ have pri- marily focused on the identification of causal structure using passive obser- vational data. The algorithms build on assumptions that connect the causal structure represented by a directed ðacyclicÞ graph among a set of vertices with the probability distribution of the data generated by the causal structure. Two of the most common such bridge principles are the causal Markov as- sumption and the causal faithfulness assumption. The causal Markov as- sumption states that each causal variable is probabilistically independent of its ðgraphicalÞ nondescendents given its ðgraphicalÞ parents. Causal Markov enables the inference from a probabilistic dependence between two variables to a causal connection and from a causal separation to a statistical indepen- dence. The precise nature of such causal separation and connection relations is fully characterized by the notion of d-separation ðGeiger, Verma, and Pearl 1990; Spirtes et al. 2000, 3.7.1Þ. The causal faithfulness assumption can be seen as the converse to the Markov assumption. It states that all and only the independence relations true in the probability distribution over the set of variables are a consequence of the Markov condition. Thus, faithfulness *To contact the author, please write to: California Institute of Technology; e-mail: fde@ caltech.edu. Philosophy of Science, 80 (December 2013) pp. 684–696. 0031-8248/2013/8005-0013$10.00 Copyright 2013 by the Philosophy of Science Association. All rights reserved. 684 This content downloaded from 131.215.71.79 on Thu, 27 Feb 2014 10:44:28 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp permits the inference from probabilistic independence to causal separation and from causal connection to probabilistic dependence. Together causal Markov and faithfulness provide the basis for causal search algorithms based on passive observational data. For the simplest case they are combined with the assumptions that the causal structure is acyclic and that the measured variables are causally sufficient, that is, that there are no unmeasured com- mon causes of the measured variables. For example, given three variables x, y, and z, if we find that the only ðconditional or unconditionalÞ independence relation that holds among the three variables is that x is independent of z given y, then causal Markov and faithfulness allow us to infer that the true causal structure is one of those presented in figure 1. Causal Markov and faithfulness do not determine which of the three causal structures is true, but this underdetermination is well understood for causal structures in general. It is characterized by the so-called Markov equivalence classes of causal structures. These equivalence classes consist of sets of causal structures ðgraphsÞ that have the same independence and dependence relations among the variables. The three structures in figure 1 are one such equivalence class. To identify the true causal structure uniquely there are two options: one can make stronger assumptions about the under- lying causal model or one can run experiments. Here I will first focus on the latter to then show that one cannot really do without the former. I will take an experiment to consist of an intervention on a subset of the variables under consideration. While there are a variety of different types of interventions, I will focus here on experiments involving so-called surgical interventions ðPearl 2000Þ. In a surgical intervention the intervention com- pletely determines the probability distribution of the intervened variable and thereby makes it independent of its normal causes. Such an intervention is achieved ðat least in principleÞ by a randomized controlled trial: whether or not a particular treatment is administered is determined entirely by the randomizing device and not by any other factors. In a causal Bayes net a surgical intervention breaks the arrows into the intervened variable, while leaving the remaining causal structure intact. It is possible to perform an experiment that surgically intervenes on several variables simultaneously and independently. In that case, of course, all information about the causal relation among intervened variables is lost. For the three Markov equivalent structures in figure 1, a single-intervention experiment intervening only on y would distinguish the three causal struc- tures: it would make x independent of y if the first structure is true, but not Figure 1. EXPERIMENTAL INDISTINGUISHABILITY OF CAUSAL STRUCTURES 685 This content downloaded from 131.215.71.79 on Thu, 27 Feb 2014 10:44:28 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp for the second and third. And it would make y independent of z if the second structure is true, but not for the first and the third. Together these two con- siderations show that such an experiment on y would resolve the under- determination of this Markov equivalence class completely. Ever since Ronald A. Fisher’s work in the 1930s, experiments have come to be seen as the gold standard for causal discovery ðFisher 1935Þ. This view suggests that if one can perform experiments, then causal discovery is ðthe- oreticallyÞ trivial. The recent rise of the interventionist account of causation in philosophy appears to endorse this view, since it holds that just what it is to stand in a causal relation, is the possibility of performing the appropriate kind of experiment ðWoodward 2003Þ. 2. Underdetermination despite Experiments. First the hopeful news: Eber- hardt, Glymour, and Scheines ð2005Þ showed that one can generalize the strategy used to identify the true causal structure in figure 1 to arbitrary causal structures over N variables: assuming that causal Markov, faithful- ness, and causal sufficiency hold, and that the causal structure is acyclic, one can uniquely identify the true causal structure among a set of variables given a set of single-intervention experiments. Generally such a procedure will re- quire several experiments intervening on different variables, but a sequence of experiments that guarantees success can be specified. Similar results can be obtained without experiments but by instead strength- ening the assumptions one makes about the underlying causal structure. Shimizu et al. ð2006Þ show that if causal sufficiency holds, the causal re- lations are linear, and the error distributions on the variables are non- Gaussian, then the causal structures can also be uniquely identified. A set of causal variables is related linearly when the value of each variable is de- termined by a linear function of the values of its parents plus an error term. Each error variable has a disturbance distribution, and as long as these dis- tributions are not Gaussian ðand not degenerateÞ, then the same identifi- ability of causal structure is guaranteed as would be obtained by not making the assumptions about the causal relations, but instead running a set of single- intervention experiments. In either case, whether by strengthening assumptions or using experi- ments, the results rely on the assumption of causal sufficiency—that there are no unmeasured common causes. In many discovery contexts it is im- plausible that such an assumption is appropriate. Moreover, part of the rationale for randomized controlled trials in the first place was that a ran- domization makes the intervened variable independent of its normal causes, whether those causes were measured or not. Thus, if there is an unmeasured common cause u—a confounder—of x and z, then randomizing x would break the ðspuriousÞ correlation observed between x and z that is due to 686 FREDERICK EBERHARDT This content downloaded from 131.215.71.79 on Thu, 27 Feb 2014 10:44:28 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp the confounder u. However, without the assumption of causal sufficiency, underdetermination returns despite the possibility of experiments. In figure 2, x, y, and z are observed ðand can be subject to interventionÞ, while u and v are unobserved. If only causal Markov, faithfulness, and acyclicity are assumed, the two causal structures in figure 2 cannot be dis- tinguished by any set of experiments that intervene on only one variable in each experiment ðor by a passive observationÞ. Since u and v are not ob- served, no variable is ðconditionallyÞ independent of any other variable under passive observation. The same is true when x is subject to an intervention, even though the surgical intervention would break the influence of u on x: x is not independent of z conditional on y, since conditioning on y induces a dependence via v ðconditioning on a common effect makes the parents de- pendentÞ. In an experiment intervening on y only, x and y are independent, but x and z remain dependent for both causal structures ðbecause of u in structure 2 and because of u and the direct effect x → z in structure 1Þ. In an experiment intervening on z, the edge x → z that distinguishes the two causal structures is broken, so both structures inevitably have the same in- dependence and dependence relations. The problem is that no set of single- intervention experiments is sufficient to isolate the x → z edge in structure 1, and so the underdetermination remains. This underdetermination can, of course, be resolved: if one could inter- vene on x and y simultaneously, then x will be independent of z if the sec- ond structure is true, but dependent if the first is true. So, assuming only causal Markov, faithfulness, and acyclicity, the two causal structures are ex- perimentally indistinguishable for single-intervention experiments but dis- tinguishable for double-intervention experiments. How does this generalize to arbitrary causal structures? The resolution of the underdetermination of the causal structures in figure 2 depended on an experiment that intervened on all but one variable simultaneously. This is true in general: assuming causal Markov, faithfulness, and acyclicity, but not Figure 2. EXPERIMENTAL INDISTINGUISHABILITY OF CAUSAL STRUCTURES 687 This content downloaded from 131.215.71.79 on Thu, 27 Feb 2014 10:44:28 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp causal sufficiency, there exist at least two causal structures over N vari- ables that are indistinguishable on the basis of the independence and depen- dence structure for all experiments that intervene on at most N22 variables, where N is the number of observed variables. That is, at least one experi- ment intervening on all but one variable is necessary to uniquely identify the true causal structure. In fact, the situation is worse, because a whole set of experiments, each intervening on at least N2 i variables, for each integer i in 0 < i < n, is in the worst case necessary to ensure that the underdeter- mination is resolved ðsee app. A for a proofÞ. So, even when multiple si- multaneous interventions are possible, a large number of experiments each intervening on a large number of variables simultaneously are necessary to resolve the underdetermination. Again, one need not pursue this route. One could instead strengthen the search space assumptions. Part of why single-intervention experiments were not sufficient to resolve the underdetermination of the causal structures in figure 2 is that independence tests are a general but crude tool of analysis. If one could separate the causal effect of the x → y → z pathway from the direct causal effect of x → z in the structures in figure 2, then the two causal structures could be distinguished. For linear causal relations this is possible, since so-called trek rules specify that the correlation between two variables in a linear model is given by the sum-product of the correlations along the ðactiveÞ treks that connect the variables. Suppose that the linear coefficient of the x → y edge is a, of the y → z edge is b and of the x → z edge is c in figure 2. If the second structure is true, then in an experiment that intervenes on x, we have corðx, zÞ 5 ab, while if the first structure is true, then corðx, zÞ 5 ab 1 c in the same experiment. We can measure the cor- relations and compare the result to the predictions: in an experiment that intervenes on y, we can determine b by measuring corðy, zÞ. In an experi- ment intervening on x, we can determine a by measuring corðx, yÞ, and we can measure corðx, zÞ. If corðx, zÞ 5 corðx, yÞcorðy, zÞ 5 ab, then the second structure is true, while if the first structure is true, then corðx, zÞ ≠ corðx, yÞcorðy, zÞ, and we can determine c 5 corðx, zÞ 2 corðx, yÞcorðy, zÞ. Thus, on the basis of single-intervention experiments alone we are able to resolve the underdertermination. But we had to assume linearity. Eberhardt, Hoyer, and Scheines ð2010Þ show that this approach gener- alizes: if the causal model is linear ðwith any nondegenerate distribution on the error termsÞ, but causal sufficiency does not hold, then there is a set of single-intervention experiments that can be used to uniquely identify the true causal structure among a set of variables. This result holds even when the assumptions of acyclicity and faithfulness are dropped. It shows just how powerful the assumption of linearity is. Linearity is sufficient to achieve identifiability even for single-intervention experiments, but it is known not 688 FREDERICK EBERHARDT This content downloaded from 131.215.71.79 on Thu, 27 Feb 2014 10:44:28 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp to be necessary. Hyttinen, Eberhardt, and Hoyer ð2011Þ have shown that similar results can be achieved for particular types of discrete models—so- called noisy-or models. It is currently not known what type of parametric assumption is necessary to avoid single-intervention experimental indistin- guishability. However, there is a weaker result: appendix B contains two discrete ðbut faithfulÞ parameterizations, one for each of the causal structures in figure 2 ðadapted from Hyttinen et al. 2011Þ. I refer to the parameterized model corresponding to the first structure as PM1 and that for the second structure as PM2. As can be verified from appendix B, PM1 and PM2 have identical passive observational distributions, identical manipulated distributions for an experiment intervening only on x, an experiment intervening only on y, and ðunsurprisinglyÞ for an experiment intervening only on z. That is, the two parameterized models are not only indistinguishable on the basis of independence and dependence tests for any single-intervention experiment or passive observation. They are indistinguishable in principle, that is, for any statistical tool, given only single-intervention experiments ðand passive observationÞ, because those ðexperimentalÞ distributions are identical for the two models. This underdetermination exists despite the fact that all ðexperimentalÞ distributions are faithful to the underlying causal structure. The models are, however, distinguishable in a double-intervention experi- ment intervening on x and y simultaneously. Only for such an experiment do the experimental distributions differ so that the presence of the x → z edge in PM1 is in principle detectable. I do not know, but conjecture, that this in- principle-underdetermination ðrather than just the underdetermination based on the ½in-�dependence structure, as shown in app. AÞ can be generalized to arbitrary numbers of variables and will hold for any set of experiments that at most intervene on N 2 2 variables. The example shows that in order to identify the causal structure by single- intervention experiments some additional parametric assumption beyond Markov, faithfulness, and acyclicity is necessary. Alternatively, without ad- ditional assumptions, causal discovery requires a large set of very demand- ing experiments, each intervening on a large number of variables simulta- neously. For many fields of study it is not clear that such experiments are feasible, let alone affordable or ethically acceptable. It is unclear how com- mon cases like PM1 and PM2 are. It is possible that in practice such cases are quite rare. When the assumption of faithfulness was subject to philosophi- cal scrutiny, one argument in its defense was that a failure of faithfulness was for certain types of parameterizations a measure-zero event ðSpirtes et al. 2000, theorem 3.2Þ. While this defense of faithfulness has not received much philosophical sympathy, such assessments of the likelihood of trouble are of interest when one is willing or forced to make the antecedent parametric as- EXPERIMENTAL INDISTINGUISHABILITY OF CAUSAL STRUCTURES 689 This content downloaded from 131.215.71.79 on Thu, 27 Feb 2014 10:44:28 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp sumptions anyway. The example here does not involve a violation of faith- fulness, but a similar analysis of the likelihood of underdetermination de- spite experimentation is possible. PM1 and PM2 cast a rather dark shadow on the hopes that experiments on their own can provide a gold standard for causal discovery. They suggest that causal discovery, whether experimental or observational, depends cru- cially on the assumptions one makes about the true causal model. As the earlier examples show, assumptions interact with each other and with the available experiments to yield insights about the underlying causal struc- ture. Different sets of assumptions and different sets of experiments result in different degrees of insight and underdetermination, but there is no clear hierarchy either within the set of possible assumptions, or between experi- ments and assumptions about the model space or parameterization. 3. Interventionism. On the interventionist account of causation, “X is a direct cause of Y with respect to some variable set V if and only if there is a possible intervention on X that will change Y ðor the probability distribution of YÞ when all other variables in V besides X and Y are held fixed at some value by interventions” ðWoodward 2003Þ. The intuition is easy enough: in figure 2, x is a direct cause of z because x and z are dependent in the double- intervention experiment intervening on x and y simultaneously. According to this definition of a direct cause it is true by definition that N experiments each intervening on N 2 1 variables are sufficient to identify the causal structure among a set of N variables even when causal sufficiency does not hold. ðAbove I only discussed necessary conditions.Þ If each of the N experiments leaves out a different variable from its intervention set, then each experiment can be used to determine the presence of the direct effects from the N 2 1 intervened variables to the one nonintervened one. Together the experiments determine the entire causal structure. An interventionist should therefore have no problem with the results discussed so far, since the cases of experimental underdetermination that I have considered were all restricted to experiments intervening on at most N 2 2 variables. The causal structures could always be distinguished by an experiment intervening on all but one variable. But there are unusual cases. In appendix C, I provide another parame- terization ðPM3Þ for the first causal structure in figure 2 ðthe one with the extra x → z edgeÞ. The example and its implications are discussed more thoroughly than can be done here in Eberhardt ð2013Þ. PM3 is very similar to PM1 and PM2. In fact, for a passive observation and a single intervention on x, y, or z they all imply the exact same distributions. However, PM3 is also indistinguishable from PM2 for a double-intervention experiment on x and y ðand similarly, of course, for all other double-intervention experi- mentsÞ. That is, PM3 and PM2 differ in their causal structure with regard to 690 FREDERICK EBERHARDT This content downloaded from 131.215.71.79 on Thu, 27 Feb 2014 10:44:28 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp the x → z edge but are experimentally indistinguishable for all possible experiments on the observed variables. In what sense, then, is the direct arrow from x → z in PM3 justified? After all, in a double-intervention experiment on x and y, x will appear indepen- dent of z. Given Woodward’s definition of a direct cause, x is not a direct cause relative to the set of observed variables fx, y, zg. However, if one in- cluded u and v as well, x would become a direct cause of z, since x changes the probability distribution of z in an experiment that changes x and holds y, u, and v fixed. So, the interventionist can avoid the apparent contradiction. The defi- nition of a direct cause is protected from the implications of PM3 since it is relativized to the set of variables under consideration. But one may find a certain level of discomfort that this interventionist definition permits the possibility that a variable ðx hereÞ ðiÞ is not a direct cause relative to V 5 fx, y, zg, ðiiÞ is not even an indirect cause when y is subject to intervention and V 5 fx, y, zg, ðiiiÞ but is a direct cause relative to V* 5 fx, y, z, u, vg. Unlike PM1, PM3 violates the assumption of faithfulness in the double- intervention distribution when x and y are manipulated simultaneously: in PM3 x is independent of z despite being ðdirectlyÞ causally connected. Violations of faithfulness have been recognized to cause problems for the interventionist account ðStrevens 2008Þ. In particular, when there are two causal pathways between a variable p and a variable q that cancel each other out exactly, then an intervention on p will leave p and q independent despite the ðdoubleÞ causal connection. But this case here is different: in the double-intervention distribution intervening on x and y that is crucial to determining whether x is a direct cause of z, there is only one pathway be- tween x and z. Thus, we are faced here with a violation of faithfulness that does not follow the well-understood case of canceling pathways. But like those cases, it shows that the interventionist account of causation either misses certain causal relations or implicitly depends on additional assump- tions about the underlying causal model. The interventionist need not as- sume faithfulness. As indicated earlier the assumption of linearity guaran- tees identifiability using only single-intervention experiments even if we do not assume faithfulness. In other words, a linear parameterization of struc- ture 1 cannot be made indistinguishable from a linear parameterization of structure 2. Part of the appeal of the interventionist account is its sensitivity to the set of variables under consideration when defining causal relations. This helped enormously to disentangle direct from total and contributing causes. Ex- EXPERIMENTAL INDISTINGUISHABILITY OF CAUSAL STRUCTURES 691 This content downloaded from 131.215.71.79 on Thu, 27 Feb 2014 10:44:28 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp amples like PM3 suggest that the relativity may be too general for defini- tional purposes unless one makes additional assumptions: I may measure one set of variables in an experiment and say there is no causal connection between two variables. You may measure a strict superset of my variables and intervene on a strict superset of my intervened variables and come to the conclusion that the same pair of variables stand in a direct causal relation. Moreover, the claim would hold when all the interventions were success- fully surgical, that is, breaking causal connections. The other part of the interventionist appeal was the apparent indepen- dence of the interventionist account from substantive assumptions such as faithfulness that have received little sympathy despite their wide applica- tion. This article suggests that you cannot have both. Appendix A Theorem: Assuming only causal Markov, faithfulness, and acyclicity, n experiments are in the worst case necessary to discover the causal struc- ture among n variables. Proof: Suppose that every pair of variables in V is subject to confound- ing. Consequently, independence tests conditional on any nonintervened variable will always return a dependence, since they open causal connec- tions via the unmeasured variables. Without loss of generality we can assume that the following about the causal hierarchy over the variables is known: ðx1; x2Þ > x3 > : : : > xn: In words: the causal order between x1 and x2 is unknown, but they are both higher in the order than any other variable. To satisfy the order, there must ðat leastÞ be a path x3 → x4 → : : : → xn21 → xn: Let an experiment E 5 ðJ, UÞ be defined as a partition of the variables in V into a set J and U 5 V \J, where the variables in J are subject to a surgical intervention simultaneously and independently, and the variables in U are not. Now note the following: the only experiment that establishes whether x2 → x1 are experiments with x2 in J1 and x1 not in J1. That is, x2 is subject to an intervention ðwith possibly other variablesÞ and x1 is not. Select any one such experiment and call it E1 5 ðJ1, U1Þ. 692 FREDERICK EBERHARDT This content downloaded from 131.215.71.79 on Thu, 27 Feb 2014 10:44:28 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Suppose that experiment E1 showed that x2 and x1 were independent, such that the ordering between x1 and x2 remains underdetermined. The only experiment that establishes whether x1 → x2 are experiments E2 with x1 in J2 and x2 not in J2. Experiments E1 and E2 resolve the order between x1 and x2, suppose without loss of generality that it is x1 → x2. In the worst case this required two experiments. Now for the remainder: the only experiment that establishes whether x1 → x3 are experiments E3 with x1 and x2 in J3 and x3 not in J3. Note that none of the previous experiments could have been an E3. The only experiment that establishes whether x1 → x4 are experiments E4 with x1, x2, x3 in J4 and x4 not in J4. None of the previous experi- ments could have been an E4. . . . The only experiment that establishes whether x1 → xn is an experiment En with x1, . . ., xn–1 in Jn and xn not in Jn. None of the previous experi- ments could have been an En. It follows that n experiments are in the worst case necessary to discover the causal structure. QED The above proof shows that in the worst case a sequence of n experi- ments is necessary that have intervention sets that intervene on at least n 2 i variables simultaneously for each integer i in 1 < i < n. Appendix B Parameterization PM1 for structure 1 in figure 2 ðall variables are binaryÞ: pðu 5 1Þ 5 :5 pðz 5 1ju 5 1; v 5 1; x 5 1; y 5 1Þ 5 :8 pðz 5 1ju 5 1; v 5 1; x 5 1; y 5 0Þ 5 :8 pðz 5 1ju 5 1; v 5 1; x 5 0; y 5 1Þ 5 :84 pðv 5 1Þ 5 :5 pðz 5 1ju 5 1; v 5 1; x 5 0; y 5 0Þ 5 :8 pðz 5 1ju 5 1; v 5 0; x 5 1; y 5 1Þ 5 :8 pðz 5 1ju 5 1; v 5 0; x 5 1; y 5 0Þ 5 :8 pðx 5 1ju 5 1Þ 5 :8 pðz 5 1ju 5 1; v 5 0; x 5 0; y 5 1Þ 5 :64 pðx 5 1ju 5 0Þ 5 :2 pðz 5 1ju 5 1; v 5 0; x 5 0; y 5 0Þ 5 :8 pðz 5 1ju 5 0; v 5 1; x 5 1; y 5 1Þ 5 :8 pðz 5 1ju 5 0; v 5 1; x 5 1; y 5 0Þ 5 :8 pðy 5 1jv 5 1; x 5 1Þ 5 :8 pðz 5 1ju 5 0; v 5 1; x 5 0; y 5 1Þ 5 :79 pðy 5 1jv 5 1; x 5 0Þ 5 :8 pðz 5 1ju 5 0; v 5 1; x 5 0; y 5 0Þ 5 :8 pðy 5 1jv 5 0; x 5 1Þ 5 :8 pðz 5 1ju 5 0; v 5 0; x 5 1; y 5 1Þ 5 :8 pðy 5 1jv 5 0; x 5 0Þ 5 :2 pðz 5 1ju 5 0; v 5 0; x 5 1; y 5 0Þ 5 :2 pðz 5 1ju 5 0; v 5 0; x 5 0; y 5 1Þ 5 :84 pðz 5 1ju 5 0; v 5 0; x 5 0; y 5 0Þ 5 :2 EXPERIMENTAL INDISTINGUISHABILITY OF CAUSAL STRUCTURES 693 This content downloaded from 131.215.71.79 on Thu, 27 Feb 2014 10:44:28 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Parameterization PM2 for structure 2 in figure 2: pðu 5 1Þ 5 :5 pðv 5 1Þ 5 :5 pðz 5 1ju 5 1; v 5 1; y 5 1Þ 5 :8 pðz 5 1ju 5 1; v 5 1; y 5 0Þ 5 :8 pðx 5 1ju 5 1Þ 5 :8 pðz 5 1ju 5 1; v 5 0; y 5 1Þ 5 :8 pðx 5 1ju 5 0Þ 5 :2 pðz 5 1ju 5 1; v 5 0; y 5 0Þ 5 :8 pðz 5 1ju 5 0; v 5 1; y 5 1Þ 5 :8 pðy 5 1jv 5 1; x 5 1Þ 5 :8 pðz 5 1ju 5 0; v 5 1; y 5 0Þ 5 :8 pðy 5 1jv 5 1; x 5 0Þ 5 :8 pðz 5 1ju 5 0; v 5 0; y 5 1Þ 5 :8 pðy 5 1jv 5 0; x 5 1Þ 5 :8 pðz 5 1ju 5 0; v 5 0; y 5 0Þ 5 :2 pðy 5 1jv 5 0; x 5 0Þ 5 :2 Passive observational distribution: PM1: PðX; Y; ZÞ 5 o uv PðUÞPðVÞPðX jUÞPðYjV; XÞPðZjU; V; X; YÞ PM2: PðX; Y; ZÞ 5 o uv PðUÞPðVÞPðX jUÞPðYjV; XÞPðZjU; V; YÞ Experimental distribution when x is subject to an intervention ðI write PðAjBjjBÞ to mean the conditional probability of A given B in an experiment where B has been subject to a surgical interventionÞ: PM1: PðY; ZjX jjXÞ 5 o uv PðUÞPðVÞPðYjV; XÞPðZjU; V; X; YÞ PM2: PðY; ZjX jjXÞ 5 o uv PðUÞPðVÞPðYjV; XÞPðZjU; V; YÞ Experimental distribution when y is subject to an intervention: PM1: PðX; ZjYjjYÞ 5 o uv PðUÞPðVÞPðX jUÞPðZjU; V; X; YÞ PM2: PðX; ZjYjjYÞ 5 o uv PðUÞPðVÞPðX jUÞPðZjU; V; YÞ Experimental distribution when z is subject to an intervention: PM1: PðX; YjZjjZÞ 5 o uv PðUÞPðVÞPðX jUÞPðYjV; XÞ PM2: PðX; YjZjjZÞ 5 o uv PðUÞPðVÞPðX jUÞPðYjV; XÞ By substituting the terms of PM1 and PM2 in the above equations it can be verified that PM1 and PM2 have identical passive observational and single-intervention distributions but that they differ for the following double-intervention distribution on x and y. 694 FREDERICK EBERHARDT This content downloaded from 131.215.71.79 on Thu, 27 Feb 2014 10:44:28 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp Experimental distribution when x and y are subject to an intervention: PM1: PðZjX; YjjX; YÞ 5 o uv PðUÞPðVÞPðZjU; V; X; YÞ PM2: PðZjX; YjjX; YÞ 5 o uv PðUÞPðVÞPðZjU; V; YÞ PM1 and PM2 ðunsurprisinglyÞ have identical distributions for the other two double-intervention distributions, since the x → z edge is broken and the remaining parameters are identical in the parameterizations: Experimental distribution when x and z are subject to an intervention: PM1: PðYjX; ZjjX; ZÞ 5 o v PðVÞPðYjV; XÞ PM2: PðYjX; ZjjX; ZÞ 5 o v PðVÞPðYjV; XÞ Experimental distribution when y and z are subject to an intervention: PM1: PðX jY; ZjjY; ZÞ 5 o u PðUÞPðX jUÞ PM2: PðX jY; ZjjY; ZÞ 5 o u PðUÞPðX jUÞ Appendix C Parameterization PM3 for structure 1 in figure 2: pðu 5 1Þ 5 :5 pðz 5 1ju 5 1; v 5 1; x 5 1; y 5 1Þ 5 :825 pðz 5 1ju 5 1; v 5 1; x 5 1; y 5 0Þ 5 :8 pðz 5 1ju 5 1; v 5 1; x 5 0; y 5 1Þ 5 :8 pðv 5 1Þ 5 :5 pðz 5 1ju 5 1; v 5 1; x 5 0; y 5 0Þ 5 :8 pðz 5 1ju 5 1; v 5 0; x 5 1; y 5 1Þ 5 :775 pðz 5 1ju 5 1; v 5 0; x 5 1; y 5 0Þ 5 :8 pðx 5 1ju 5 1Þ 5 :8 pðz 5 1ju 5 1; v 5 0; x 5 0; y 5 1Þ 5 :8 pðx 5 1ju 5 0Þ 5 :2 pðz 5 1ju 5 1; v 5 0; x 5 0; y 5 0Þ 5 :8 pðz 5 1ju 5 0; v 5 1; x 5 1; y 5 1Þ 5 :7 pðz 5 1ju 5 0; v 5 1; x 5 1; y 5 0Þ 5 :8 pðy 5 1jv 5 1; x 5 1Þ 5 :8 pðz 5 1ju 5 0; v 5 1; x 5 0; y 5 1Þ 5 :8 pðy 5 1jv 5 1; x 5 0Þ 5 :8 pðz 5 1ju 5 0; v 5 1; x 5 0; y 5 0Þ 5 :8 pðy 5 1jv 5 0; x 5 1Þ 5 :8 pðz 5 1ju 5 0; v 5 0; x 5 1; y 5 1Þ 5 :9 pðy 5 1jv 5 0; x 5 0Þ 5 :2 pðz 5 1ju 5 0; v 5 0; x 5 1; y 5 0Þ 5 :2 pðz 5 1ju 5 0; v 5 0; x 5 0; y 5 1Þ 5 :8 pðz 5 1ju 5 0; v 5 0; x 5 0; y 5 0Þ 5 :2 Substituting the parameters of PM3 in the equations for the passive obser- vational or any experimental distributions of PM1 in appendix 2, it can be EXPERIMENTAL INDISTINGUISHABILITY OF CAUSAL STRUCTURES 695 This content downloaded from 131.215.71.79 on Thu, 27 Feb 2014 10:44:28 AM All use subject to JSTOR Terms and Conditions http://www.jstor.org/page/info/about/policies/terms.jsp verified that PM2 and PM3 are experimentally indistinguishable for all pos- sible experiments on fx, y, zg. 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