A Distinction between Causal Effects in Structural and Rubin Causal Models

Structural Causal Models define causal effects in terms of a single Data Generating Process (DGP), and the Rubin Causal Model defines causal effects in terms of a model that can represent counterfactuals from many DGPs. Under these different definitions, notationally similar causal effects make distinct claims about the results of interventions to the system under investigation: Structural equations imply conditional independencies in the data that potential outcomes do not. One implication is that the DAG of a Rubin Causal Model is different from the DAG of a Structural Causal Model. Another is that Pearl's do-calculus does not apply to potential outcomes and the Rubin Causal Model.