Structural causal models (SCMs), also known as (nonparametric) structural equation models (SEMs), are widely used for causal modeling purposes. In particular, acyclic SCMs, also known as recursive SEMs, form a well-studied subclass of SCMs that generalize causal Bayesian networks to allow for latent confounders. In this paper, we investigate SCMs in a more general setting, allowing for the presence of both latent confounders and cycles. We show that in the presence of cycles, many of the convenient properties of acyclic SCMs do not hold in general: they do not always have a solution; they do not always induce unique observational, interventional and counterfactual distributions; a marginalization does not always exist, and if it exists the marginal model does not always respect the latent projection; they do not always satisfy a Markov property; and their graphs are not always consistent with their causal semantics. We prove that for SCMs in general each of these properties does hold under certain solvability conditions. Our work generalizes results for SCMs with cycles that were only known for certain special cases so far. We introduce the class of simple SCMs that extends the class of acyclic SCMs to the cyclic setting, while preserving many of the convenient properties of acyclic SCMs. With this paper, we aim to provide the foundations for a general theory of statistical causal modeling with SCMs.
S. Bongers and J.M. Mooij are supported in part by NWO, the Netherlands Organization for Scientific Research (VIDI grant 639.072.410 and VENI grant 639.031.036). P. Forré and J.M. Mooij are supported in part by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement n 639466). J. Peters is supported by research grants from VILLUM FONDEN (18968) and the Carlsberg Foundation.
The authors are grateful to Bernhard Schölkopf and Robin Evans for stimulating discussions, and to Noud de Kroon, Tineke Blom and Alexander Ly for providing helpful comments on earlier drafts. We thank two anonymous reviewers and the Associate Editor for helpful comments.
"Foundations of structural causal models with cycles and latent variables." Ann. Statist. 49 (5) 2885 - 2915, October 2021. https://doi.org/10.1214/21-AOS2064