A Distributional Perspective on Pearl's Causal Hierarchy: From Marginal to Joint and Individualized Potential Outcomes

Abstract

Pearl's causal hierarchy is a foundational lens for formulating causal questions and is most often discussed within the framework of structural causal models. We recast the hierarchy in potential outcomes language and make its information ordering operational at the level of causal estimands. Specifically, we classify an estimand according to whether it requires marginal potential outcome distributions, their joint distribution or nested cross-world quantities, or individual-level counterfactual outcomes. We apply this criterion systematically to a broad range of estimands, including cases whose classification depends on the formulation of the scientific question. We then clarify what additional assumptions are needed when an estimand depends not only on the marginal distributions of potential outcomes, but also on their unobserved joint distribution. In particular, randomization identifies the marginals, whereas monotonicity, copula restrictions, rank preservation, and partial identification restrict or characterize the remaining uncertainty about the joint distribution. The resulting framework provides a practical map from a scientific question to an estimand, the probabilistic object it requires, and the assumptions needed for identification.