Robins' causal inference theory assumes existence of treatment specific counterfactual variables so that the observed data augmented by the counterfactual data will satisfy a consistency and a randomization assumption. In this paper we provide an explicit function that maps the observed data into a counterfactual variable which satisfies the consistency and randomization assumptions. This offers a practically useful imputation method for counterfactuals. Gill & Robins 's construction of counterfactuals can be used as an imputation method in principle, but it is very hard to implement in practice. Robins  shows that the counterfactual distribution can be identified from the observed data distribution by a G-computation formula under an additional identifiability assumption. He proves this for discrete variables. Gill & Robins  prove the G-computation formula for continuous variables under some continuity assumptions and reformulation of the consistency and the randomization assumptions. We prove that if treatment is discrete (which deals with a less general case compared with Gill & Robins , then Robins' G-computation formula holds under the original consistency, randomization assumptions and a generalized version of the identifiability assumption.
Statistical Methodology | Statistical Theory
Yu, Zhuo and van der Laan, Mark J., "A Note on the Construction of Counterfactuals and the G-computation Formula" (June 2005). U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 181.