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

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 framework to define counterfactual outcomes. In this article, we agnostically superimpose a structural causal model and a Rubin causal model compatible with the same observations to specify the mathematical conditions under which counterfactual outcomes obtained via do-interventions and potential outcomes need to, do not need to, can, or cannot be equal (almost surely or in law). Our comparison builds upon the fact that such causal models do not have to produce the same counterfactual outcomes and highlights real-world problems where they generally cannot correspond under classical causal-inference assumptions. Then, we examine common claims and practices from the causality literature in the light of this comparison. In doing so, we aim at clarifying the links between the two causal frameworks and the interpretation of their respective counterfactuals.