  We consider the problem of causal inference based on observational data (or
the related missing data problem) with a binary or discrete treatment variable.
In that context we study counterfactual density estimation, which provides more
nuanced information than counterfactual mean estimation (i.e., the average
treatment effect). We impose the shape-constraint of log-concavity (a
unimodality constraint) on the counterfactual densities, and then develop
doubly robust estimators of the log-concave counterfactual density (based on an
augmented inverse-probability weighted pseudo-outcome), and show the
consistency in various global metrics of that estimator. Based on that
estimator we also develop asymptotically valid pointwise confidence intervals
for the counterfactual density.
