[Paper Review] Comparing Causal Frameworks: Potential Outcomes, Structural Models, Graphs, and Abstractions

Duligur Ibeling, Thomas Icard|arXiv (Cornell University)|Jun 25, 2023

Bayesian Modeling and Causal Inference4 citations

TL;DR

This paper establishes a formal theoretical relationship between the Rubin Causal Model (RCM) and Structural Causal Models (SCM), showing that every RCM can be viewed as an abstraction of a representable RCM that satisfies composition and reversibility principles. A key result proves that all RCMs—even those violating SCM algebraic constraints—emerge as abstractions of representable ones, reconciling the two frameworks through a neutral logical language and completeness results for counterfactual reasoning.

ABSTRACT

The aim of this paper is to make clear and precise the relationship between the Rubin causal model (RCM) and structural causal model (SCM) frameworks for causal inference. Adopting a neutral logical perspective, and drawing on previous work, we show what is required for an RCM to be representable by an SCM. A key result then shows that every RCM -- including those that violate algebraic principles implied by the SCM framework -- emerges as an abstraction of some representable RCM. Finally, we illustrate the power of this conciliatory perspective by pinpointing an important role for SCM principles in classic applications of RCMs; conversely, we offer a characterization of the algebraic constraints implied by a graph, helping to substantiate further comparisons between the two frameworks.

Motivation & Objective

• To clarify the theoretical relationship between the Rubin Causal Model (RCM) and Structural Causal Models (SCM), resolving long-standing debates about their equivalence.
• To identify the precise conditions under which an RCM can be represented by an SCM, focusing on composition and reversibility principles.
• To demonstrate that RCMs violating SCM algebraic constraints arise as abstractions of representable RCMs, thereby reconciling conceptual and formal differences.
• To develop a framework-neutral logical language for counterfactual probabilities, enabling direct comparison between RCM and SCM assumptions.
• To provide completeness results for RCMs and representable RCMs, and to characterize algebraic constraints implied by graphs, advancing theoretical synthesis of the two frameworks.

Proposed method

• Adopt a neutral logical perspective, using a formal language for counterfactual probabilities to compare RCM and SCM frameworks without prior bias.
• Define representable RCMs as those satisfying composition and reversibility principles, which are necessary and sufficient for embedding into an SCM.
• Introduce the concept of causal abstraction to explain why some RCMs fail to satisfy SCM algebraic constraints—namely, when low-level causal details are elided.
• Prove Theorem 1: every RCM is a constructive abstraction of a representable RCM, showing that non-representable RCMs arise from abstraction of valid SCM-embeddable models.
• Establish completeness results: Theorem 2 shows the language is complete for all RCMs; Corollary 2 shows completeness for representable RCMs.
• Characterize graph-implied algebraic constraints via Theorem 3, linking graphical separation (d-separation) in SWIGs to conditional independence in counterfactual distributions.

Experimental results

Research questions

• RQ1 Under what conditions can an RCM be represented as an SCM, and what principles (e.g., composition, reversibility) are required?
• RQ2 How can RCMs that violate SCM algebraic constraints still be meaningfully related to SCMs, and what role does abstraction play?
• RQ3 What is the precise relationship between graphical assumptions (e.g., d-separation in SWIGs) and algebraic constraints in counterfactual distributions?
• RQ4 Can a common formal language unify reasoning across RCM and SCM frameworks, and is it possible to achieve completeness in this language?
• RQ5 What are the minimal principles needed to fully characterize the algebraic constraints implied by a causal graph?

Key findings

• Theorem 1 establishes that every RCM is a constructive abstraction of a representable RCM that satisfies composition and reversibility, resolving concerns about incompatibility.
• The paper shows that SUTVA assumptions can be interpreted as conditions on good variable abstractions, linking classical RCM assumptions to modern abstraction theory.
• Theorem 2 proves that the framework-neutral language is sound and complete for all RCMs, enabling full logical reasoning about counterfactuals.
• Corollary 2 establishes completeness for the subclass of representable RCMs, showing that SCM principles fully capture their logical structure.
• Theorem 3 provides a partial characterization of algebraic constraints implied by a graph, linking d-separation in SWIGs to conditional independence in counterfactual distributions.
• Theorem 4 shows that single-world intervention graphs (SWIGs) yield a completeness result for the same logical language, reinforcing the compatibility of RCM and SCM perspectives.