Abstract
Some explanations appeal to facts about the causal structure of a system in order to shed light on a particular phenomenon; these are explanations which do more than cite the causes X and Y of some state-of-affairs Z, but rather appeal to “macro-level” causal features—for example the fact that A causes B as well as C, or perhaps that D is a strong inhibitor of E—in order to explain Z. Appeals to these kinds of “macro-level” causal features appear in a wide variety of social scientific and biological research; statements about features such as “patriarchy,” “healthcare infrastructure,” and “functioning DNA repair mechanism,” for instance, can be understood as claims about what would be different (with respect to some target phenomenon) in a system with a different causal structure. I suggest interpreting counterfactual questions involving structural features as questions about alternative parameter settings of causal models, and propose an extension of the usual interventionist framework for causal explanation which enables scientists to explore the consequences of interventions on “macro-level” structure.
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Notes
This relates to a point raised by Cartwright (1999, pp. 16–17) and others: policies may produce violations of faithfulness (Spirtes et al. 2000). That is because policies which engineer structural parameters on converging paths to exactly cancel make variables appear statistically independent even when they are in fact causally dependent. This can pose problems for applying causal search algorithms that rely on faithfulness as an assumption.
Note that Hoover would object to my characterization here, because we use different terminology, viz., different definitions of “parameter,” “variable,” and “structure.” I will return to Hoover’s alternative view later in the paper, after my own view has been more formally spelled out. In this discussion, I use “variable” and “parameter” as is common in classical statistics and the literature on causal modeling (e.g., Pearl, Woodward, Spirtes et al.): variables are measured quantities and parameters are population-level quantities inferred or estimated from the data.
If we leave the world of SEMs and instead model a system with a parameterized, discrete-valued Bayes net, then the variables can each take on a finite number of “states” and parameters are just state transition probabilities.
\(\varepsilon \) is inside the function because the authors do not assume that the causal mechanism is necessarily additive in the noise variable. They also allow for causal feedback; more on that below.
Note that some variables may not be causally related and some may have no causes at all among the measured variables, only error terms. This is all easy to accomodate with functions defined accordingly: functions can be independent of some (or all) of their arguments, and so we can assume every variable is an argument of every function to be maximally general.
Steel’s focus in (2006) is invariance and explanation across macro/micro descriptions of social systems. He is not concerned with learning the truth-values of structure-altering counterfactual claims, so he does not provide any procedure for doing so. The examples he uses to motivate his definition of structure-altering interventions may be quite plausible, and it would be illuminating to work out a technique for predicting the outcomes of structure-altering interventions to compare with the proposal in the next section.
See Bright et al. (2016). Note that some counterfactual questions may be ambiguous, especially in the context of non-linear models. For example, to ask “What would be the distribution of Y if the causal effect of \(X_1\) was twice as strong?” relative to the model \(Y = 0.4 X_1 -1.3 X_2 + 0.2 X_1X_2\), is ambiguous because I could have in mind the coefficient on \(X_1\), the coefficient on the interaction term \(X_1 X_2\), or some combination of these. Each option corresponds to a different intervention.
I omit equations for distance and the other exogenous variables from the model. Of course it is quite likely that a better model is not linear and includes many more factors, but this model is for illustrative purposes only. See Bhargava et al. (2005) for an actual empirical study which partially inspired this example.
I am indebted to Greg Gandenberger for raising this point.
More technically, I am not assuming that researchers will have a joint probability distribution over all the variables in the expanded model which includes P, nor even that such a joint probability distribution would be well-defined in complex, non-recursive cases. My only claim is that one can always write down an expanded system of equations with moderator variables which is equivalent to the actual observed system when the policy is “off” and which is describes the counterfactual system when the policy is “on.”
The eigenvalues of the coefficient matrix of the non-recursive structural equations must have modulus less than or equal to 1.
If the quantity of interest is only the post-manipulation mean of Y, and not its entire density function, then it may be prudent to simply add the mean of the relevant error variable at each step. The errors are assumed to be Gaussian with mean zero in many typical models, in which case this is trivial. However, more generally if one is interested in the distribution of Y and the error variables are not normally distributed, i.e., if there is important information in the error distribution, then simply adding the mean of \(\varepsilon \) will not suffice. When the errors are significantly non-Gaussian, then the full distribution of \(\varepsilon \) is likely relevant. Thanks to Jonathan Livengood for raising this issue.
Thanks to Conor Mayo-Wilson for discussion on this point.
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Acknowledgements
I would like to thank Liam Kofi Bright, David Danks, Greg Gandenberger, Clark Glymour, Kevin Hoover, Jonathan Livengood, Conor Mayo-Wilson, Jim Woodward, and two anonymous referees for comments on earlier drafts.
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Malinsky, D. Intervening on structure. Synthese 195, 2295–2312 (2018). https://doi.org/10.1007/s11229-017-1341-z
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DOI: https://doi.org/10.1007/s11229-017-1341-z