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The Causal Chain Problem

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Abstract

This paper addresses a problem that arises when it comes to inferring deterministic causal chains from pertinent empirical data. It will be shown that to every deterministic chain there exists an empirically equivalent common cause structure. Thus, our overall conviction that deterministic chains are one of the most ubiquitous (macroscopic) causal structures is underdetermined by empirical data. It will be argued that even though the chain and its associated common cause model are empirically equivalent there exists an important asymmetry between the two models with respect to model expansions. This asymmetry might constitute a basis on which to disambiguate corresponding causal inferences on non-empirical grounds.

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Notes

  1. As regards the notion of a Bayesian network cf. e.g. (Pearl 1985).

  2. Cf. (Ragin 1987), (Ragin 2000), or (Mahoney 2000).

  3. Cf. e.g. (Spirtes et al. 2000), pp. 53–57, (Glymour 2007). The causal faithfulness assumption is also briefly reviewed in Sect. 2 below.

  4. Cf. e.g. (Spirtes et al. 2000), pp. 29-31, (Glymour 1997, 2007). While constraint based BN-algorithms categorically rule out unfaithful structures, metric procedures merely rank them low (Cf. (Neapolitan 2004)).

  5. Both Boolean and BN-methodologies are designed to uncover acyclic structures only. The causal structures considered in the following are hence implicitly assumed not to feature feedbacks.

  6. Cf. e.g. (Pearl 2000), pp. 19, 145, (Verma and Pearl 1991).

  7. Cf. also (Spirtes et al. 2000) and (Glymour 1997). Two vertices V 1 and V 2 are said to be adjacent in a graph G iff there is an edge between V 1 and V 2 in G. In a directed edge from a vertex V 1 to a vertex V 2, V 1 is called the tail and V 2 the head. Two edges collide at a vertex V 1 iff V 1 is the head of both edges.

  8. Cf. e.g. (Suppes 1970).

  9. Cf. (Spirtes et al. 2000), ch. 4, (Pearl 2000), or (Woodward 2003), ch. 3.

  10. Cf. e.g. (Spirtes et al. 2000), pp. 53-57, or (Glymour 2007).

  11. For details Cf. (Glymour 2007), p. 236.

  12. Cf. e.g. (Quine 1952) and (Quine 1959).

  13. The restriction to binary variables primarily serves conceptual simplicity. It allows for a straightforward implementation of Boolean optimization procedures, which are of great relevance to the uncovering of deterministic structures. Nonetheless, the restriction to binary variables implies that structures involving multi-valued variables must be encoded in binary terms before they can be treated by Boolean procedures.

  14. For more details on the notion of coincidence cf. (Baumgartner 2008a), Appendix A.

  15. Coincidences correspond to what (Ragin 1987) calls configurations.

  16. For further details cf. (Baumgartner 2008b) and (Baumgartner forthcoming).

  17. Defining a minimally sufficient condition in terms of proper parts and not—as might be expected—in terms of proper subsets that correspond to reductions of sufficient conditions by one or more conjuncts allows for a simpler procedure to identify minimally sufficient conditions. For if a sufficient condition has no sufficient proper parts, it does not have sufficient proper subsets either. Hence, in order to show that a sufficient condition A 1 A 2…A n is minimally sufficient it suffices to establish that A 1 A 2…A n has no proper parts—establishing that it has no sufficient proper subsets is unnecessary.

  18. Of course, C and E are moreover each minimally sufficient for themselves. However, as self-causation is normally excluded, these reflexive dependencies are not amenable to a causal interpretation to begin with. Reflexive dependencies are therefore neglected in the context at hand.

  19. Dependencies as recorded in \(({\mathsf{R}})\) are not as straightforwardly causally interpretable as might be suggested here. Mackie’s (1974) famous Manchester Factory Hooters example demonstrates that minimally sufficient conditions are not directly amenable to a causal interpretation. In Baumgartner (2008b) I indicate what additional constraints have to be met in order to warrantably causally interpret dependencies as in \(({\mathsf{R}})\). For the context at hand, however, we can ignore these complications.

  20. In the following, I interchangeably speak of common cause structures and epiphenomena. Note that this terminology differs from the notion of an epiphenomenon used in the literature on mental causation. In the latter context an epiphenomenon is a physically caused mental side effect which itself is causally inert. Here “epiphenomenon” just describes a causal structure featuring at least one cause with at least two parallel effects (cf. e.g. graphs (c), (c1) or (c2) of Fig. 1 or (e) of Fig. 2). Nothing with respect to a causal impotence of these parallel effects is implied by referring to such a structure as being epiphenomenal.

  21. Cf. (Baumgartner 2008b).

  22. Further exemplary reductions of chains to epiphenomena can be found in Figs. 4–6 below.

  23. Most of the suggestions as to how to disambiguate inferences to complex causal structures considered in the following have also been discussed in the context of resolving probabilistic ambiguities (Cf. e.g. (Suppes 1970), (Spirtes et al. 2000), (Hausman 1998), or (Woodward 2003)).

  24. Cf. e.g. (Lewis 1979), (Brand 1980) or (Huemer and Kovitz 2003).

  25. For a modern variant of this account see e.g. (Dowe 2000).

  26. For a condensed presentation of the pros and cons of a transference theory of causation cf. (Dowe 2007). For more details cf. (Dowe 2000) or (Kistler 2001).

  27. Cf. (Salmon 1994).

  28. In the social science literature there is a related methodology called process-tracing that aims to establish the existence of a causal mechanism between two investigated variables Y 1 and Y 2 by successively filling in intermediate variables on the path \(\langle Y_1, Y_2\rangle\) (Cf. e.g. (Mahoney 2000) or (George and Bennett 2005), ch. 10). Some authors interested in social mechanisms argue that such mechanisms are unobservable primitive entities (Cf. e.g. (Steinmetz 1998)). As such they could not be treated on a par with ordinary causal variables as done in the graphs of Fig. 4. However, unobservable mechanisms, apparently, are of no avail when it comes to distinguishing between structures (d) and (e) on empirical grounds.

  29. For details on this graphical notation cf. (Baumgartner 2006), ch. 2.

  30. Similarly Pearl (2000).

  31. Cf. (Woodward 2003), p. 101.

  32. This is just one consequence of the fact that the notions of causation and intervention are directly interdefined in the interventionist framework of (Woodward 2003). For further consequences of this conceptual interdependence Cf. (Baumgartner unpublished).

  33. Cf. Cartwright’s famous dictum “no causes in, no causes out”, (Cartwright 1983).

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Acknowledgements

My particular thanks go to Timm Lampert for countless indispensable discussions about deterministic causal structures. Moreover, I am grateful to Delphine Chapuis-Schmitz, Mehmet Elgin, Clark Glymour, Gerd Grasshoff, John Norton, Richard Scheines, Daniel Steel, Jim Woodward, and an anonymous referee for this journal for very helpful discussions and comments on earlier drafts. Finally, I thank the Swiss National Science Foundation for generous support of this work (grant 101311-103988/1).

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    Michael Baumgartner

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Baumgartner, M. The Causal Chain Problem. Erkenn 69, 201–226 (2008). https://doi.org/10.1007/s10670-008-9113-2

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