A Unifying Framework for Causal Modeling with Infinitely Many Variables

Abstract

Structural-equations models (SEMs) are perhaps the most commonly used framework for modeling causality, but they do not capture all domains of interest. For example, dynamical systems that evolve in continuous time are an important class of domains that are not (naturally) captured by SEMs. A wide variety of approaches have been proposed to fill the gap, including dynamical structural causal models (Bongers, Blom and Mooij 2018), causal constraints models (Blom, Bongers and Mooij 2019), and counterfactual resimulation (Laurent, Yang, and Fontana 2018). These models complement common-sense causal interpretations of specific dynamical systems, such as systems of ODEs. All these approaches look quite different from each other and from SEMs. They are hard to compare, and concepts developed for one approach may not make sense for another. But they are capturing the same notion of causality as SEMs do, in the sense that interventions map to outcomes. We propose a class of models that are, in a certain natural sense, the most expressive generalization of SEMs. Our generalized SEMs (GSEMs) can be viewed as a unifying framework that recovers structural dynamical causal models, causal constraints models, counterfactual resimulation, and common-sense causal interpretations of systems of ODEs and hybrid automata (Alur et al. 1992) as special cases. The input-output behavior, or “interface”, of GSEMs is exactly that of SEMs, which means that definitions of concepts like actual cause, responsibility, blame, and explanation, can be immediately lifted from SEMs to GSEMs. The generality of GSEMs also makes them ideally suited to studying causality in the abstract; for example, they have been used to establish independence relationships among Halpern’s axioms for SEMs (Peters and Halpern 2022).