A Modeling Approach for Mechanisms Featuring Causal Cycles University of Groningen A Modeling Approach for Mechanisms Featuring Causal Cycles Gebharter, Alexander; Schurz, Gerhard Published in: Philosophy of Science DOI: 10.1086/687876 IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2016 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Gebharter, A., & Schurz, G. (2016). A Modeling Approach for Mechanisms Featuring Causal Cycles. 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Download date: 06-04-2021 https://doi.org/10.1086/687876 https://research.rug.nl/en/publications/a-modeling-approach-for-mechanisms-featuring-causal-cycles(baf8a541-4636-4433-af52-b5931f0e7fe1).html https://doi.org/10.1086/687876 All u A Modeling Approach for Mechanisms Featuring Causal Cycles Alexander Gebharter and Gerhard Schurz*y Mechanisms play an important role in many sciences when it comes to questions con- cerning explanation, prediction, and control. Answering such questions in a quantitative way requires a formal representation of mechanisms. Gebharter’s “A Formal Framework for Representing Mechanisms?” suggests to represent mechanisms by means of arrows in an acyclic causal net. In this article we show how this approach can be extended in such a way that it can also be fruitfully applied to mechanisms featuring causal feedback. 1. Introduction. Questions concerning explanation, prediction, and control in the sciences are oftentimes answered by pointing at the system of inter- est’s underlying mechanism and showing how causal interactions of this mechanism’s parts bring about the phenomenon of interest. Mechanisms are typically characterized qualitatively. Glennan (1996), for example, defines a mechanism underlying a behavior as a “complex system which produces that behavior by of the interaction of a number of parts according to direct causal laws” (52). (For other prominent characterizations see, e.g., Machamer, Darden, and Craver [2000, 3] or Bechtel and Abrahamsen [2005, 423]). For providing quantitatively precise mechanistic explanation/prediction and answering questions concerning the results of manipulations, however, a formal representation of mechanisms is required. Casini et al. (2011) suggest to model mechanisms by means of recursive Bayesian networks (RBNs). Gebharter (2014) highlights two problems with Casini et al.’s approach and *To contact the authors, please write to: Alexander Gebharter, Düsseldorf Center for Logic and Philosophy of Science (DCLPS), Heinrich Heine University Düsseldorf, Universi- tätsstraße 1, 40225 Düsseldorf, Germany; e-mail: alexander.gebharter@gmail.com. yThis work was supported by Deutsche Forschungsgemeinschaft (DFG), research unit CausationFLawsFDispositionsFExplanation (FOR 1063). We thank Lorenzo Casini, David Danks, Christian J. Feldbacher, Clark Glymour, Marie I. Kaiser, Daniel Koch, Marcel Weber, and Naftali Weinberger for helpful remarks and discussions. Philosophy of Science, 83 (December 2016) pp. 934–945. 0031-8248/2016/8305-0025$10.00 Copyright 2016 by the Philosophy of Science Association. All rights reserved. 934 This content downloaded from 129.125.019.061 on October 29, 2018 04:19:48 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). MODELING FEATURING CAUSAL CYCLES 935 suggests the multilevel causal model (MLCM) approach as an alternative.1 While Casini et al. represent mechanisms by a special kind of node of a Bayesian network (BN), Gebharter represents them by directed or bidirected causal arrows. The latter seems promising; it suggests, for example, to de- velop new methods for discovering submechanisms, that is, the causal struc- ture inside the causal arrows (see, e.g., Murray-Watters and Glymour 2015). In addition, many results from the statistics and machine learning literature can be directly applied to models of mechanisms. Zhang (2008), for exam- ple, shows how the effects of interventions can be computed in models fea- turing bidirected arrows, and Richardson (2009) develops a factorization criterion equivalent to the d-connection condition for such models. One of the shortcomings the RBN and the MLCM approach share is that they presuppose acyclicity, and thus, do not allow for a representation of mechanisms featuring feedback.2 Clarke, Leuridan, and Williamson (2014) further develop the RBN approach in such a way that it can be applied to mechanisms featuring causal cycles. They distinguish between static and dy- namic problems. Static problems are “situations in which a specific cycle reaches equilibrium . . . and where the equilibrium itself is of interest, rather than the process of reaching equilibrium.” A dynamic problem is a “situation in which it is the change in the values of variables over time that is of inter- est” (sec. 6). Clarke et al. suggest to solve static problems on the basis of the notion of d-separation (Pearl 2000, sec. 1.2.3) and dynamic problems by means of dynamic Bayesian networks (DBNs). In this article, we follow their exam- ple and demonstrate how the MLCM approach for representing mechanisms can be modified and extended in a similar way. The article is structured as follows: in section 2 we introduce the causal modeling framework used in the article. In section 3 we give an overview of the MLCM approach. In sections 4.1 and 4.2 we demonstrate by means of a simple toy mechanism how the MLCM approach can be modified in such a way that it can be applied to static and dynamic problems, respectively. Both modifications mirror Clarke et al.’s (2014) suggestions for solving static and dynamic problems without sharing certain shortcomings. 2. Causal Nets. A causal net (or model) is a triple hV, E, Pi, where hV, Ei is a directed graph providing causal information about the elements of V, V is a set of random variables, and E is a binary relation on V that is interpreted as direct causal connection relative to V. Set V’s elements are called the graph’s 1. For an attempt to defend the RBN approach against the objections made in Gebharter (2014), see Casini (2016). For another problem with the RBN approach, see Gebharter (2016). 2. For other problems that may arise in general for attempts to model mechanisms by means of BNs, see Kaiser (2016) and Weber (2016). This content downloaded from 129.125.019.061 on October 29, 2018 04:19:48 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 936 ALEXANDER GEBHARTER AND GERHARD SCHURZ All u vertices, E’s elements its edges. And P is a probability distribution over V representing regularities produced by the causal structure underlying V. Causal connections between variables are represented by directed and bidirected arrows: “X → Y” means that X is a direct cause of Y, and “X ↔ Y ” means that X and Y are effects of a common cause not included in V. Causal models are assumed to not feature self-edges X → X or X ↔ X. The set of Y’s parents is Par(Y) 5 fX ∈ V : X → Yg. A chain of n ≥ 1 edges (of any kind) connecting two variables X and Y is called a path between X and Y if it does not go through any variable more often than once. (Note that p’s being a path between X and Y allows that X 5 Y.) A path X → ::: → Y is called a di- rected path from X to Y; X is called a cause of Y and Y an effect of X. A var- iable Z lying on a path X → ::: → Z → ::: → Y is called an intermediate cause lying on this path. A path X ← ::: ← Z → ::: → Y is called a common cause path with Z as a common cause of X and Y lying on this path. A path connecting X and Y containing a subpath Zi @ → Zj ← @ Zk is called a col- lider path connecting X and Y, and Zj is called a collider lying on this path. 3 A path between X and Y indicates a common cause path if it either is a common cause path or a collider-free path that contains a bidirected edge. A path X → ::: → X is called a causal cycle. A graph is called cyclic if it features causal cycles; it is called acyclic otherwise. Likewise for causal models. For now we only require the causal net approach’s most central axiom, the causal Markov condition (CMC). A model hV, E, Pi satisfies CMC if and only if (iff) every X ∈ V is probabilistically independent of its noneffects conditional on its direct causes (Spirtes, Glymour, and Scheines 1993, 54). If an acyclic causal model satisfies CMC, then its graph determines the fol- lowing Markov factorization (54): P(X1,:::, Xn) 5 Yn i51 P(Xi �� Par(Xi)): (1) 3. The Multilevel Causal Model Approach. TheMLCMapproachisbased on the simple idea that mechanisms are devices bringing about certain input-output behaviors (cf. Bechtel 2007, sec. 3; Craver 2007, 145). This suggests a representation of mechanisms by a causal model’s arrows. The variables at the arrows’ tails stand for the mechanism’s input, the variables at the arrows’ heads stand for the mechanism’s output, and the arrows rep- resent the not-further-specified mechanism. A graph describing such a mech- anism can be supplemented by a probability distribution P that quantitatively describes the system’s behavior. Mechanistic explanation requires investigating how a mechanism pro- duces the phenomenon of interest; it requires a more detailed description of 3. The metasymbol “@” stands for an arrowtail or an arrowhead. This content downloaded from 129.125.019.061 on October 29, 2018 04:19:48 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). MODELING FEATURING CAUSAL CYCLES 937 the underlying causal structure producing that phenomenon. One can give such an explanation by supplementing a causal model M, whose graph’s arrows represent a mechanism, by another causal model M 0 that contains new variables describing the behaviors of some parts of the mechanism. So, metaphorically speaking, we are “zooming” into the device represented by the arrows. However, it must be assured that the more detailed causal model M 0 fits to the original model M with respect to its causal structure and its probability distribution. The following notion of a restriction states condi- tions for such a fit (Gebharter 2014, 147):4 M 5 hV, E, Pi is a restriction of M 0 5hV 0, E0, P0i iff V ⊂ V 0, P 0↑V 5P,5 the following two conditions hold for all X , Y ∈ V, and no path not implied by these conditions is in hV, Ei: 1. If there is a path from X to Y in hV0, E0i and no vertex on this path different from X and Y is in V, then X → Y in hV, Ei. 2. If X and Y are connected by a path p in hV 0, E0i indicating a com- mon cause path and no vertex on p different from X and Y is in V, then X ↔ Y in hV, Ei. This notion tells us which causal models M 0 are candidates for mecha- nistically explaining phenomena described by a less detailed model M. It also tells us how we can marginalize out variables from M 0 while preserving information about the causal and probabilistic relationships among vari- ables in V provided by M 0. For a detailed motivation of this notion of a restriction, see Gebharter (2014, 147–48). We can now define an MLCM as a structure hM1,:::, Mni such that every causal model Mi with i > 1 is a restriction of M1, while M1 satisfies CMC (Gebharter 2014, 148). The latter condition reflects a basic assumption of the causal nets approach, that is, that for explaining a probability distribu- tion P, reference to an underlying causal structure satisfying CMC is re- quired (cf. Spirtes et al. 1993, sec. 6.1).6 Let us briefly illustrate by means of figure 1 how MLCMs can be used for modeling mechanisms. Model M2 describes the mechanism’s top level. The mechanism has two input variables (X1, X2) and three output variables (Y1, Y2, Y3). The arrows stand for the not-further-specified mechanism. Mecha- nistic explanation of a certain phenomenon, for example, of an input-output behavior P( y1, y2, y3 j x1, x2), requires a more detailed story about what is 4. This definition is inspired by Steel (2005, 12). We thank Clark Glymour for pointing out that the marginalization method this definition provides is essentially a “slim” ver- sion of the mixed ancestral graph representation developed by Richardson and Spirtes (2002) for latent variable models. 5. Here, P0 ↑ V is the restriction of P 0 to V. 6. Note that CMC will typically be violated by models featuring bidirected arrows. This content downloaded from 129.125.019.061 on October 29, 2018 04:19:48 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 938 ALEXANDER GEBHARTER AND GERHARD SCHURZ All u happening within the mechanism, that is, within the system represented by the arrows. This story is told by M1. Model M1 features three new variables (Z1, Z2, Z3) describing parts of the mechanism. Model M1’s causal structure tells us over which causal paths through the mechanism X1 and X2 cause Y1, Y2, and Y3. Model M2 is a restriction of M1. And M1 is assumed to satisfy CMC. 4. Modeling Mechanisms with Causal Cycles. We introduce the following toy mechanism for investigating the question of how to model mechanisms featuring causal cycles within the MLCM approach: a simple temperature regulation system, where OT stands for the outside temperature, IT for the inside temperature, and CK for a control knob. The behavior of interest is that IT is relatively insensitive to OT when CK 5 on, that is, that P(it j ot, CK 5 on) ≈ P(it j CK 5 on) for arbitrary OT and IT values. A simple input-output representation of this mechanism would be a causal model M2 with the graphical structure OT → IT ← CK. A mechanistic ex- planation of P(it j ot, CK 5 on) ≈ P(it j CK 5 on) by means of an MLCM would require connecting M2 to a more detailed model M1 satisfying CMC. Since the system represented is a self-regulatory system, M1 is expected to feature a cycle IT → ::: → IT. But cyclic causal models do have some prob- lems with CMC. While CMC can, in principle, be applied to cyclic causal models, it turns out to be inadequate. Let us illustrate this by means of the following example borrowed from Spirtes et al. (1993, 359): Suppose a causal model with the structure X1 → X2 → X3 → X4 → X1 satisfies CMC. Then CMC implies no probabilistic independence. But since {X2, X4} blocks all causal paths connecting X1 and X3 and correlations are assumed to arise only because of causal connections, no probabilistic influence from X1 should reach X3 when X2’s and X4’s values are fixed. So conditionalizing on {X2, X4} should render X1 and X3 probabilistically independent. The remainder of this section shows by means of the exemplary mech- anism introduced above how the MLCM approach can be modified in such a way that it can be used to model mechanisms featuring causal cycles. To this end, as already mentioned, we have to distinguish between static and Figure 1. This content downloaded from 129.125.019.061 on October 29, 2018 04:19:48 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). MODELING FEATURING CAUSAL CYCLES 939 dynamic problems. Solving the static problem requires a model capable of explaining P(it j ot, CK 5 on) ≈ P(it j CK 5 on) when the underlying cy- cle IT → ::: → IT has reached equilibrium. Solving the dynamic problem requires a model that allows for an explanation of how IT → ::: → IT pro- duces P(it j ot, CK 5 on) ≈ P(it j CK 5 on) over a period of time. 4.1. Solving the Static Problem. To solve the static problem, we have to modify the definition of an MLCM: instead of requiring that the most detailed causal model M1 of the MLCM satisfies CMC, we rather require M1 to satisfy the d-connection condition. A model hV, E, Pi satisfies the d- connection condition iff for every dependence of variables X and Y given some Z ⊆ fX , Yg there is a d-connection between X and Y given Z (Schurz and Gebharter 2016). Variables X and Y are d-connected given Z iff there is a path p connecting X and Y such that no intermediate or common cause on p is in Z, while every collider on p is in Z or has an effect in Z. Variables X and Y are d-separated by Z otherwise. The d-connection condition is equivalent to CMC for acyclic causal models (Lauritzen et al. 1990). This equivalence reveals the full content of CMC: whenever a causal model satisfies CMC, then every dependence can be explained by some causal connection in the model, and every indepen- dence can be explained by missing causal connections in the model. The d-connection condition’s clear advantage over CMC is that it implies the independencies to be expected when applied to causal cycles (Spirtes 1995; Pearl and Dechter 1996). To demonstrate this, assume that the causal model X1 → X2 → X3 → X4 → X1 discussed earlier in section 4 satisfies the d-connection condition. As we saw in section 4, CMC implies no independ- encies for this causal model. But since X1 and X3 are d-separated by {X2, X4}, the d-connection condition implies the expected independence of X1 and X3 given {X2, X4}. Let us now see how the static problem can be solved for our exem- plary mechanism within the modified MLCM approach. The static problem concerns our exemplary mechanism when it has reached equilibrium. The system can be represented by the two-stage MLCM depicted in figure 2. Model M2 represents the system at the top level. Variables OT and CK are directly causally relevant to IT. Model M1 provides more detailed information about what is happening within the mechanism: the inside temperature is Figure 2. This content downloaded from 129.125.019.061 on October 29, 2018 04:19:48 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 940 ALEXANDER GEBHARTER AND GERHARD SCHURZ All u measured by a temperature sensor (S), which is directly causally relevant to an air conditioner (AC), which, in turn, is under direct causal influence of CK. Model M1 is assumed to satisfy the d-connection condition, and M2 is a restriction of M1. The MLCM mechanistically explains why OT is relatively insensitive to IT when the cycle IT → S → AC → IT has reached equilib- rium and CK 5 on, that is, why P(itjot, CK 5 on) ≈ P(itjCK 5 on) holds. If CK is off, then AC is off, and there is no self-regulation due to the causal cycle IT → S → AC → IT. Thus, OT will have an influence on IT. But when CK is set to one of its on values, then AC responds to S according to CK’s adjustment. Since AC → IT overwrites OT → IT when CK 5 on, IT’s value is robust with respect to changes of OT’s value when CK 5 on. This overwriting property of AC → IT is represented by the bold arrow in figure 2. Let us finally mention some open problems. First, cyclic models possibly featuring bidirected arrows do not admit the Markov factorization. Since we assume the d-connection condition to hold, they do, however, factor accord- ing to the following equation: P(X1,:::, Xn) 5 Yn i51 P(Xi �� dSep(Xi)): (2) The set dSep(Xi) is constructed as follows: Let Pred(Xi) be the set of Xi’s predecessors in the ordering X1,:::, Xn. Now search for sets dPred(Xi) ⊆ Pred(Xi) such that U 5 Pred(Xi)ndPred(Xi) d-separates Xi from all ele- ments of dPred(Xi). (Note that U may be empty.) If there are no such sets dPred(Xi), then identify dSep(Xi) with Xi’s predecessors Pred(Xi). If there are such sets dPred(Xi), then take one of the largest of these sets and iden- tify dSep(Xi) with the corresponding separator set U 5 Pred(Xi)ndPred(Xi). For the ordering P(OT, CK, IT, AC, S), for example, the joint distribution of M1 factors as P(OT) � P(CK) � P(IT jOT, CK) � P(AC j OT, CK, IT) � P(S j CK, IT, AC). Equation (2) has two disadvantages. First, it depends on an ordering of variables. Second, a probability distribution that factors according to equa- tion (2) may not imply all independencies implied by the d-connection con- dition. For example, it does not imply an independence between OT and S conditional on {IT, AC}, although OT and S are d-separated by {IT, AC}. One open problem is to find out whether there is an order-independent fac- torization criterion equivalent with the d-connection condition. Another open problem is search. Causal discovery of the latent structure inside a mechanism’s causal arrows in the possible presence of feedback loops can be expected to be an even harder problem than discovery without feedback (cf. Murray-Watters and Glymour 2015). We conjecture that effects of interventions for cyclic graphs possibly fea- turing bidirected arrows can be computed as usual. To compute postinter- vention probabilities P(z j x̂) for an instantiation z of a set of variables Z, one This content downloaded from 129.125.019.061 on October 29, 2018 04:19:48 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). MODELING FEATURING CAUSAL CYCLES 941 needs, first, to delete all the arrows with an arrowhead pointing at X from the graph.7 Second, use d-separation information provided by the manipulated graph to compute P(z j x̂). Before we take a look at how to solve the dynamic problem, let us briefly discuss the relationship of the solution to the static problem suggested above with Clarke et al.’s (2014) solution. Although both approaches use Pearl’s (2000) notion of d-separation instead of CMC to account for cycles, the structures used for probabilistic reasoning differ in the two approaches. Clarke et al. use the “true” cyclic graph to construct an equilibrium network, that is, a BN that is then used “to model the probability distribution of the equilibrium solution” (2014, sec. 6.1). In our view, this move has at least two shortcomings: i) Independencies implied by the d-connection condition and the origi- nal cyclic causal structure may not be implied by the equilibrium net- work. We illustrate this by means of our model M1, whose equilib- rium network could be the one depicted in figure 3. (See Clarke et al. [2014], sec. 6.1, for details on how to construct equilibrium net- works.) Now note that OT and CK, for example, are not d-separated in the equilibrium network. So the equilibrium network’s graph does not capture the independence between OT and CK implied by the d- connection condition and the fact that OT and CK are d-separated in M1’s graph. ii) Since the arrows of the equilibrium network do not capture the “true” causal relations anymore, it cannot be used for predicting the effects of interventions. To illustrate this, assume we are interested in the postintervention probability P(sj bck) in our model M1. In case we use M1’s graph for computing this probability, we arrive at P(sj bck) 5 P(sjck). If we use the equilibrium network’s graph, how- ever, we arrive at P(sj bck) 5 P(s). But since the control knob is caus- ally relevant for the sensor, P(sFck) will not equal P(s) when inter- vening on CK. 4.2. Solving the Dynamic Problem. Solving the dynamic problem re- quires an extension of the MLCM approach that allows for representing the system’s behavior over a period of time. Clarke et al. (2014) model such behavior by means of DBNs (cf. Murphy 2002). The basic idea behind this move is to roll out the causal cycles over time. We use dynamic causal models (DCMs) that also allow for bidirected arrows. A DCM M is a quadruple hV, E, P, t : V → N1i, where V is a set of infinitely many variables X1,1, ::: , Xn,1, X1,2, ::: , Xn,2, :::. The variables Xi,1 (with 1 ≤ i ≤ n) describe the system at its initial state (stage 1), the variables 7. Here, “x̂” is shorthand for “X is forced to take value x by intervention.” This content downloaded from 129.125.019.061 on October 29, 2018 04:19:48 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 942 ALEXANDER GEBHARTER AND GERHARD SCHURZ All u Xi,t11 (with 1 ≤ i ≤ n), the system at later stages t 1 1. The DCMs we consider involve some idealization: directed arrows only connect variables at dif- ferent stages, and if there is a directed arrow going from a variable Xi,t to a variable Xj,t1u for some stage t, then for every stage t there is such a directed arrow going from Xi,t to Xj,t1u. So the pattern of directed arrows between stages t and t 1 u is always the same. What one ideally wants is a DCM hV, E, P, ti with the following addi- tional properties: (i) arrows do not skip stages, (ii) bidirected arrows occur only between variables of one and the same stage, (iii) every two stages ti, tj (with i, j > 1) share the same pattern of bidirected arrows, and (iv) P(Xi,t j Par(Xi,t)) 5 P(Xi, t11 j Par(Xi, t11)) holds for all Xi,t ∈ V with t > 1. For a finite segment of such an “ideal” DCM, see figure 4. The depicted graph’s first stage features more bidirected arrows than later stages. These additional bi- directed arrows account for correlations between X1,1 and X2,1, X2,1 and X3,1, and X1,1 and X3,1 because of not-represented past common causes (of the kind described by variables in V). Let us now come back to the question of how the dynamic problem can be solved within the MLCM approach. The phenomenon we are interested in is that IT is relatively robust to variations of OT over a period of time when CK 5 on. Our simple temperature regulation system can be modeled by a two-stage MLCM hM1, M2i (see fig. 5 for a finite segment). The mech- anism’s top level is represented by M2, which is a restriction of M1. Model M1, which is assumed to satisfy the d-connection condition, provides more detailed information about the mechanism bringing about the phenomenon of interest. When adding new intermediate causes, we will typically also add new stages. In our example, we added two new variables (S and AC) and two new stages between consecutive stages of M2 arriving at ITt* → St*11 → Figure 4. Figure 3. This content downloaded from 129.125.019.061 on October 29, 2018 04:19:48 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). MODELING FEATURING CAUSAL CYCLES 943 ACt*12 in M1. We assume the intervals between M1’s stages to correspond to the time the causal processes we are interested in require to bring about their effects. To guarantee that M1’s and M2’s probability distributions fit together, we require t* 5 t, t* 1 1 5 t 1 1=3, t* 1 2 5 t 1 2=3, t* 1 3 5 t 1 3=3, and so on (where t stands for M2’s and t* for M1’s stages). Now M1 provides information about the causal structure within the tem- perature regulation system. Variables OT and IT are directly causally rele- vant to themselves at the next stage. Variable IT also causally depends on OT and AC, while S depends only on IT, AC only on S and CK, and CK on no variable in the model. The bidirected arrows at stage 1 account for dependencies to be expected because of not-represented common causes. Here we assumed that model M1 is especially nice, that is, that it satisfies (i)–(iv) discussed a few paragraphs above. Unfortunately, model M2 is not that nice. Since we marginalized out S and AC and there were directed paths from OTt* to ITt*16 and from ITt* to ITt*16 all going through St*13 or ACt*13 in M1, M2 features directed arrows OTt → ITt12 and ITt → ITt12 skipping stages. Since there were paths indicating a common cause path between OTt* and ITt*16 going through St*13 or ACt*13 in M1, M2 features bidirected arrows OTt ↔ ITt12. Note that there are also bidirected arrows between OTt and ITt11 and between ITt and ITt11. Now the MLCM mechanistically explains why IT is relatively robust with respect to OT changes when CK 5 on over a period of time. If CK is off over several stages, then also AC is off, and there is no regulation of IT over paths ITt* → St*11 → ACt*12 → ITt*13; IT’s value will increase and decrease (with a slight time lag) with OT’s value. If, however, CK is fixed to one of its on values over several stages, then over several stages ACt*11 responds to St* according to CKt*’s adjustment. Now the crucial control mechanism consists of ITt*11 and its parents OTt*, ITt*, and ACt*. The bold Figure 5. This content downloaded from 129.125.019.061 on October 29, 2018 04:19:48 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 944 ALEXANDER GEBHARTER AND GERHARD SCHURZ All u arrows ACt* → ITt*11 in figure 5 overwrite OTt* → ITt*11 and ITt* → ITt*11 when CKt* 5 on; that is, PCK t*5on (itt*11 j act*, ut*) ≈ PCK t*5on (itt*11 j act*) holds, where Ut* ⊆ fOTt*, ITt*g. This control mechanism will, after a short period of time, cancel deviations of IT’s value from CK’s adjustment brought about by OT’s influence. Here are some possible open problems: first, some of the arrows in M2 may seem to misrepresent the “true” causal processes going on inside the temperature regulation system. There is, for example, a directed arrow going from CKt to ITt11 but no directed arrow from CKt to ACt11, although CK can clearly influence IT only through AC. This is a typical problem arising for dynamic models. One can, however, learn something about M1’s structure from M2: the (direct or indirect) cause-effect relationships among variables in M2 will also hold for M1. Another problem is, again, search. For solutions of several discovery problems involving time series, see, for example, Danks and Plis (2014). Finally, factorization and interventions: since our DCMs do not feature feedback loops, we conjecture that Richardson’s (2009) factorization criterion and Zhang’s (2008) results about how to compute the effects of inter- ventions in models with bidirected arrows can be fruitfully applied to DCMs. Let us finally have a look at how our solution to the dynamic problem relates to the one suggested by Clarke et al. (2014). Both modeling strat- egies use the same basic idea, that is, to roll out the cycles over time. While the arrows of the DCMs we use are intended to capture the “true” causal re- lations between variables of interest, the directed arrows in Clarke et al.’s DBNssurprisinglyarenotintendedtorepresentthe“true”causalrelationships (cf.sec.6.2).Thus,theirmodelsshareproblem(ii)discussedattheendofsec- tion 4.1 with the equilibrium network they use for solving static problems: the model cannot be used to compute the effects of interventions. 5. Conclusion. Clarke et al. (2014) have extended Casini et al.’s (2011) RBN approach for modeling mechanisms in such a way that it can be ap- plied to mechanisms featuring causal feedback. In this article we followed their example and showed how the MLCM approach can be modified in a similar way. Like Clarke et al. we distinguish between static and dynamic problems when it comes to modeling mechanisms with causal cycles. Our solutions to both problems within the MLCM approach mirror Clarke et al.’s solutions for the RBN approach while avoidingseveral problems. The MLCM approach can be used for modeling mechanisms whose causal cycles have reached equilibrium (i.e., static problems) by introducing the requirement that the most detailed causal model M1 has to satisfy the d-connection condition instead of CMC. The dynamic problem, which concerns the development of the system over a period of time, can be solved within the MLCM approach by using DCMs. Both solutions, however, come with new challenges, whose investigation we leave to future research. This content downloaded from 129.125.019.061 on October 29, 2018 04:19:48 AM se subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). MODELING FEATURING CAUSAL CYCLES 945 REFERENCES Bechtel, W. 2007. “Reducing Psychology While Maintaining Its Autonomy via Mechanistic Expla- nation.” In The Matter of the Mind: Philosophical Essays on Psychology, Neuroscience, and Reduction, ed. M. Schouton and H. L. de Jong, 172–98. Oxford: Blackwell. Bechtel, W., and A. Abrahamsen. 2005. “Explanation: A Mechanist Alternative.” Studies in History and Philosophy of Biological and Biomedical Sciences 36:421–41. Casini, L. 2016. “How to Model Mechanistic Hierarchies.” Philosophy of Science, in this issue. Casini, L., P. M. lllari, F. Russo, and J. Williamson. 2011. “Models for Prediction, Explanation and Control: Recursive Bayesian Networks.” Theoria 26 (70): 5–33. Clarke, B., B. Leuridan, and J. Williamson. 2014. “Modelling Mechanisms with Causal Cycles.” Synthese 191 (8): 1651–81. Craver, C. 2007. Explaining the Brain. Oxford: Clarendon. Danks, D., and S. Plis. 2014. “Learning Causal Structure from Undersampled Time Series.” Paper presented at the NIPS 2013 Workshop on Causality. Gebharter, A. 2014. “A Formal Framework for Representing Mechanisms?” Philosophy of Science 81 (1): 138–53. ———. 2016. “Another Problem with RBN Models of Mechanisms.” Theoria 31 (2): 177–88. Glennan, S. 1996. “Mechanisms and the Nature of Causation.” Erkenntnis 44 (1): 49–71. Kaiser, M. I. 2016. “On the Limits of Causal Modeling: Spatially-Structurally Complex Biological Phenomena.” Philosophy of Science, in this issue. Lauritzen, S. L., A. P. Dawid, B. N. Larsen, and H. G. Leimer. 1990. “Independence Properties of Directed Markov Fields.” Networks 20 (5): 491–505. Machamer, P., L. Darden, and C. Craver. 2000. “Thinking about Mechanisms.” Philosophy of Sci- ence 67 (1): 1–25. Murphy, K. P. 2002. Dynamic Bayesian Networks. Berkeley: University of California Press. Murray-Watters, A., and C. Glymour. 2015. “What’s Going on Inside the Arrows? Discovering the Hidden Springs in Causal Models.” Philosophy of Science 82 (4): 556–86. Pearl, J. 2000. Causality. 1st ed. Cambridge: Cambridge University Press. Pearl, J., and R. Dechter. 1996. “Identifying Independencies in Causal Graphs with Feedback.” In Proceedings of the 12th International Conference on Uncertainty in Artificial Intelligence, 420–26. San Francisco: Kaufmann. Richardson, T. 2009. “A Factorization Criterion for Acyclic Directed Mixed Graphs.” In Pro- ceedings of the 25th Conference on Uncertainty in Artificial Intelligence, ed. J. Bilmes and A. Ng, 462–70. Arlington, VA: Association for Uncertainty in Artificial Intelligence. Richardson, T., and P. Spirtes. 2002. “Ancestral Graph Markov Models.” Annals of Statistics 30 (4): 962–1030. Schurz, G., and A. Gebharter. 2016. “Causality as a Theoretical Concept: Explanatory Warrant and Empirical Content of the Theory of Causal Nets.” Synthese 193 (4): 1073–1103. Spirtes, P. 1995. “Directed Cyclic Graphical Representations of Feedback Models.” In Proceed- ings of the 11th Conference on Uncertainty in Artificial Intelligence, 491–98. San Francisco: Kaufmann. Spirtes, P., C. Glymour, and R. Scheines. 1993. Causation, Prediction, and Search. 1st ed. Dor- drecht: Springer. Steel, D. 2005. “Indeterminism and the Causal Markov Condition.” British Journal for the Philos- ophy of Science 56 (1): 3–26. Weber, M. 2016. “On the Incompatibility of Dynamical Biological Mechanisms and Causal Graphs.” Philosophy of Science, in this issue. Zhang, J. 2008. “Causal Reasoning with Ancestral Graphs.” Journal of Machine Learning Research 9:1437–74. 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