Semiparametric counterfactual density estimation

Causal effects are often characterized with averages, which can give an incomplete picture of the underlying counterfactual distributions. Here we consider estimating the entire counterfactual density and generic functionals thereof. We focus on two kinds of target parameters: density approximations and the distance between counterfactual densities. We study nonparametric efficiency bounds, giving results for smooth but otherwise generic models and distances. Importantly, we show how these bounds connect to means of particular nontrivial functions of counterfactuals, linking the problems of density and mean estimation. We propose doubly robust-style estimators, and study their rates of convergence, showing that they can be optimally efficient in large nonparametric models. We also give analogous methods for model selection and aggregation, when many models may be available and of interest. Our results all hold for generic models and distances, but we highlight results for L2 projections on linear models and Kullbach–Leibler projections on exponential families. Finally, we illustrate our method by estimating the density of the CD4 count among patients with HIV, had all been treated with combination therapy versus zidovudine alone, as well as a density effect. Our methods are implemented in the R package npcausal on GitHub.