A clarification on the links between potential outcomes and do-interventions

TL;DR

This paper clarifies the mathematical and conceptual differences between Pearl's structural causal models and Rubin's potential-outcome framework, highlighting conditions under which their counterfactuals align or diverge.

Contribution

It provides a formal comparison of the two causal models, specifying conditions for their counterfactual outcomes to be equivalent or not, and clarifies common misconceptions in causal inference.

Findings

Structural and potential-outcome models do not always produce equivalent counterfactuals.

Conditions under which the models' counterfactuals coincide are explicitly characterized.

Real-world scenarios may prevent the models from aligning, challenging assumptions in causal inference.

Abstract

Most of the scientific literature on causal modeling considers the structural framework of Pearl and the potential-outcome framework of Rubin to be formally equivalent, and therefore interchangeably uses do-interventions and the potential-outcome subscript notation to write counterfactual outcomes. In this paper, we agnostically superimpose the two causal models to specify under which mathematical conditions structural counterfactual outcomes and potential outcomes need to, do not need to, can, or cannot be equal (almost surely or law). Our comparison reminds that a structural causal model and a Rubin causal model compatible with the same observations do not have to coincide, and highlights real-world problems where they even cannot correspond. Then, we examine common claims and practices from the causal-inference literature in the light of these results. In doing so, we aim at clarifying the relationship between the two causal frameworks, and the interpretation of their respective counterfactuals.