Abstract
Mechanistic philosophy of science views a large part of scientific activity as engaged in modelling mechanisms. While science textbooks tend to offer qualitative models of mechanisms, there is increasing demand for models from which one can draw quantitative predictions and explanations. Casini et al. (Theoria 26(1):5–33, 2011) put forward the Recursive Bayesian Networks (RBN) formalism as well suited to this end. The RBN formalism is an extension of the standard Bayesian net formalism, an extension that allows for modelling the hierarchical nature of mechanisms. Like the standard Bayesian net formalism, it models causal relationships using directed acyclic graphs. Given this appeal to acyclicity, causal cycles pose a prima facie problem for the RBN approach. This paper argues that the problem is a significant one given the ubiquity of causal cycles in mechanisms, but that the problem can be solved by combining two sorts of solution strategy in a judicious way.
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Notes
Note that, as with all models, a RBN model only models some aspects of a mechanism. The main goal is to model the hierarchical structure of the mechanism together with the causal structure at each level of the hierarchy, in such a way that the model can be used to draw quantitative inferences. See Casini et al. (2011) for a fuller presentation of the motivation behind this sort of model, and Sect. 7 of this paper for pointers to possible limitations of the RBN approach.
Woodward’s concept of ‘mechanism’, or more precisely: of ‘mechanistic model’, is not explicitly multi-level or hierarchical, in contrast to those on which we focus in this paper. In the next section, the hierarchical nature will serve as one of the main reasons to adopt the RBN approach to mechanistic modelling.
As such, his account of causality can also form the starting point for causal Bayes net accounts of the structure of scientific theories (see Leuridan 2014).
For mentions of causal feedback in Explaining the Brain, see e.g. pp. 81 and 178–180. Several of Craver’s figures also contain cycles: see Fig. 4.1 (p. 115) and Fig. 4.6 (p. 121) and relatedly 5.7 (p. 189) and 5.8 (p. 194). Moreover, Fig. 3.2 (p. 71) and relatedly 4.1 (p. 166), leave open the possibility of causal feedback.
This property of negative feedback systems to tend toward equilibrium is the case when there is no significant delay in the system. When delay is present, as it is in many biological systems, oscillations will tend to arise. We would like to thank Mike Joffe for pressing us on this issue.
However, this is not always the case. For instance, various positive feedback loops in pregnancy serve to appropriately maintain hormone levels.
These kinds of contingent-feedback cycles are, in other words, more sensitive to background conditions than the other two kinds. This makes them unstable, in Mitchell’s sense of stability as describing the sensitivity of relations to their background conditions (Mitchell 2009, p. 56). This should be discriminated from robustness, which describes the degree to which a function is maintained when one or more constitutive elements are disrupted (Mitchell 2009, pp. 69–73).
This cycle is more complicated than suggested above. For example, the thyroid secretes two hormones—T3 and T4—which can be interconverted, and feedback occurs at various intermediate points in the cycle. But this simple version is adequate for our discussion.
Similar examples of neurological oscillators are also discussed by (Bechtel (2011), pp. 548–549).
Structural equation modelling had previously been put forward for this purpose. But structural equation models attempt to model deterministic relationships between cause and effect (with error terms which are usually assumed to be independent and normally distributed), while Bayesian networks seek to represent the probability distribution of the variables in question. In general it is harder to devise an accurate model of deterministic relationships than it is to determine probabilistic relationships between cause and effect.
Note that RBNs are not the only hierarchical extension of Bayesian nets. See (Williamson (2005), §10.2) for a comparison between RBNs and other related formalisms.
Although Bayesian nets have been extended to handle certain continuous cases, we restrict attention to discrete variables in this paper.
This corresponds to the notion of ‘bottoming-out’ in the mechanistic literature (see Sect. 2).
While these arrows would not normally be interpreted causally, the question arises as to whether they might be if Craver’s views of mutual manipulability and causality are endorsed. See footnote 25.
In this definition, \(C \in Descendants (C)\) by convention.
The paper by Pearl and Dechter (1996) extends previous results by Spirtes (1995) and Koster (1996) who have shown that the \(d\)-separation test is valid for cyclic graphs with linear equations and normal distributions over the error terms. Given that we restrict attention to discrete variables in this paper (see Sect. 4), we will only discuss Pearl and Dechter.
Disturbances are not explicitly mentioned in Sect. 4, but they can easily be introduced. Disturbances are variables that represent errors due to omitted factors (see Pearl 2000, p. 27). The assumption of uncorrelated disturbances is not a severe restriction. Given a graph \(G\) and associated probability distribution \(P\), such that not all disturbances are independent, one may construct an augmented graph \(G'\) in which all disturbances are independent. The augmented graph \(G'\) is obtained by adding, for each pair of dependent disturbances, a dummy root node as a common cause of the disturbances (Pearl and Dechter 1996, p. 422).
Unlike in the acyclic case, however, ‘the joint distribution of feedback systems cannot be written as a product of the conditional distributions of each child variable, given its parents’ (Pearl and Dechter 1996, p. 420). Hence factorization, on which we relied in Sect. 4, cannot be applied to DCGs. This can be shown by means of a simple example by (Spirtes et al. (2000), §12.1.2). Applying the factorization to the graph \(X \leftrightarrows Y\), would lead to \(P(X, Y) = P(X \mid Y) \times P(Y \mid X)\), which would mean that \(X\) and \(Y\) are independent, contrary to what the graph suggests.
Simultaneous causation and backwards causation do not fit this picture. However, such cases rarely if ever occur in models of mechanisms, so we set them aside in this paper.
Separation in undirected graphs (such as moral graphs) is defined as follows: let \(G\) be an undirected graph with vertex set \(V\), then two sets of vertices \(X, Y \subseteq V\) are separated by \(Z \subseteq V\) if and only if every path (sequence of undirected edges) from each vertex in \(X\) to each vertex in \(Y\) contains some vertex in \(Z\).
Prima facie, this approach runs counter to Bechtel’s appeal not to model mechanisms sequentially (see Sect. 2). By means of the transition network, however, the cyclic organization is captured as well. We would like to thank Michael Wilde for pointing out this seeming incongruity.
To take a concrete example, suppose that variables \(A\) and \(B\) directly cause \(C\) which directly causes \(D\) which in turn directly causes each of \(A\) and \(B\). A causal DAG at time \(0\) obtained by unwinding this cycle might have arrows from \(A_0\) and \(B_0\) to \(C_0\) and an arrow from \(C_0\) to \(D_0\). The Markov condition requires that \(A_0\) and \(B_0\) be probabilistically independent. But these two variables have a common cause not represented in the graph—the previous instance of \(D\)—which can render them probabilistically dependent. Thus the Markov condition can fail in this causal graph.
For a detailed account of mutual manipulability, see (Craver (2007), pp. 152–160). For a recent critical discussion of Craver’s claim that interlevel constitutive relations cannot be causal, and whether this claim is compatible with his mutual manipulability account of constitutive relevance, see Leuridan (2012).
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Acknowledgments
We would like to thank Lorenzo Casini, George Darby, Phyllis Illari, Mike Joffe, Federica Russo, and Michael Wilde for their helpful comments on this paper. Jon Williamson’s research is supported by the UK Arts and Humanities Research Council. Bert Leuridan is Postdoctoral Fellow of the Research Foundation—Flanders (FWO).
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Appendix: Transforming a moral graph into a DAG
Appendix: Transforming a moral graph into a DAG
Here we present the algorithm of (Williamson (2005), §5.7) for transforming an undirected graph \({\mathcal {G}}\) into a directed acyclic graph \({\mathcal {H}}\) which preserves the required independencies (Williamson 2005, Theorem 5.3): if \(Z\) \(d\)-separates \(X\) from \(Y\) in the DAG \({\mathcal {H}}\) then \(X\) and \(Y\) are separated by \(Z\) in the undirected graph \({\mathcal {G}}\); this separation in \({\mathcal {G}}\) implies that \(X\) and \(Y\) are probabilistically independent conditional on \(Z\); hence, \(d\)-separation in \({\mathcal {H}}\) implies that \(X\) and \(Y\) are probabilistically independent conditional on \(Z\). Thus \({\mathcal {H}}\) can be used as the graph of a Bayesian network.
An undirected graph is triangulated if for every cycle involving four or more vertices there is an edge in the graph between two vertices that are non-adjacent in the cycle. The first step of the procedure is to construct a triangulated graph \({\mathcal {G}}^T\) from the undirected graph \({\mathcal {G}}\). One of a number of standard triangulation algorithms can be applied to construct \({\mathcal {G}}^T\) (see, e.g., Neapolitan 1990, §3.2.3; Cowell et al. 1999, §4.4.1).
Next, re-order the variables in \(V\) according to maximum cardinality search with respect to \({\mathcal {G}}^T\): choose an arbitrary vertex as \(V_{1}\); at each step select the vertex which is adjacent to the largest number of previously numbered vertices, breaking ties arbitrarily. Let \(D_{1},\ldots ,D_{l}\) be the cliques (i.e., maximal complete subgraphs) of \({\mathcal {G}}^T\), ordered according to highest labelled vertex. Let \(E_{j} = D_{j}\cap (\bigcup _{i=1}^{j-1} D_{i})\) and \(F_{j} = D_{j} {\setminus } E_{j}\), for \(j=1,\ldots ,l\).
Finally, construct a DAG \({\mathcal {H}}\) as follows. Take variables in \(V\) as vertices. Step 1: add an arrow from each vertex in \(E_{j}\) to each vertex in \(F_{j}\), for \(j=1,\ldots ,l\). Step 2: add further arrows to ensure that there is an arrow between each pair of vertices in \(D_{j}, j=1,\ldots ,l\), taking care that no cycles are introduced (there is always some orientation of an added arrow which will not yield a cycle).
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Clarke, B., Leuridan, B. & Williamson, J. Modelling mechanisms with causal cycles. Synthese 191, 1651–1681 (2014). https://doi.org/10.1007/s11229-013-0360-7
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DOI: https://doi.org/10.1007/s11229-013-0360-7