Abstract
The causal Bayes net framework specifies a set of axioms for causal discovery. This article explores the set of causal variables that function as relata in these axioms. Spirtes (2007) showed how a causal system can be equivalently described by two different sets of variables that stand in a non-trivial translation-relation to each other, suggesting that there is no “correct” set of causal variables. I extend Spirtes’ result to the general framework of linear structural equation models and then explore to what extent the possibility to intervene or a preference for simpler causal systems may help in selecting among sets of causal variables.
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Notes
“We advocate no definition of causation, but [...] we try to make our usage systematic, and to make explicit our assumptions connecting causal structure with probability, counterfactuals and manipulations.” (Spirtes et al. 1993); “...regard ‘X is a direct cause of Y with respect to variables V’ as an unanalyzed primitive relation,...” (Glymour 2004)
\(x_1\) and \(x_2\) correspond to \(P_1\) and \(P_2\) in Spirtes’ paper. Similarly, \(y_1\) and \(y_2\), used below, correspond to C and D in his paper. I changed notation to make it consistent throughout this presentation.
One may object that \(y_3\) is a dubious causal variable in this model, since it basically extinguishes after its initial action. I will not pursue this line of argument, since the more common class of models discussed in the next section shows that the indistinguishability that appears somewhat forced in this example, is in fact quite general.
Of course, there are other classes of models, in which, for example, the error term is re-sampled at each time step (VAR models). In this case a somewhat different analysis from what follows is needed and I do not know whether the same results hold.
This is trivially guaranteed for acyclic models.
I have restricted the confounding to background conditions, since, as a reviewer pointed out, it is not obvious that confounding due to a common cause, which varies at the rate of the observed variables, can be represented implicitly in the error covariance matrix of a model where the errors are only re-sampled once per sample and remain constant over the equilibrating process.
In order for such an equilibrium to exist we have to assume that the eigenvalues of \(\mathbf{U}_{\mathbf{{x}}}\mathbf{B}\) (the manipulated direct effects matrix) are all smaller than 1 in absolute value as well. This is again trivially guaranteed for acyclic models, but not at all obvious for models with feedback. For the sake of argument here, however, this assumption is rhetorical: If we find that despite the fact that all interventional distributions lead to an equilibrium, we still have no constraints that identify the variables, then in cases where the system diverges under intervention this type of equilibrium based approach to identifying causal variables does not get off the ground in the first place.
\(\mathbf{A}\) has the same eigenvalues as \(\mathbf{B}\), so the existence of an equilibrium is guaranteed if it is guaranteed for \(\mathbf{B}\).
This point can be made analogously for the deterministic models Spirtes considered.
Recall that the diagonal elements are standardized to zero in LSEMs.
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Acknowledgments
I am very grateful for the detailed, constructive and insightful comments and the correction of mistakes by three anonymous reviewers.
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Eberhardt, F. Green and grue causal variables. Synthese 193, 1029–1046 (2016). https://doi.org/10.1007/s11229-015-0832-z
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DOI: https://doi.org/10.1007/s11229-015-0832-z