  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. The first is a density
approximation, defined by a projection onto a finite-dimensional model using a
generalized distance metric, which includes f-divergences as well as $L_p$
norms. The second is the distance between counterfactual densities, which can
be used as a more nuanced effect measure than the mean difference, and as a
tool for model selection. We study nonparametric efficiency bounds for these
targets, giving results for smooth but otherwise generic models and distances.
Importantly, we show how these bounds connect to means of particular
non-trivial functions of counterfactuals, linking the problems of density and
mean estimation. We go on to propose doubly robust-style estimators for the
density approximations and distances, and study their rates of convergence,
showing 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 throughout we highlight what happens for particular choices,
such as $L_2$ projections on linear models, and KL projections on exponential
families. Finally we illustrate by estimating the density of CD4 count among
patients with HIV, had all been treated with combination therapy versus
zidovudine alone, as well as a density effect. Our results suggest combination
therapy may have increased CD4 count most for high-risk patients. Our methods
are implemented in the freely available R package npcausal on GitHub.
