  We propose a general Bayesian nonparametric (BNP) approach to causal
inference in the point treatment setting. The joint distribution of the
observed data (outcome, treatment, and confounders) is modeled using an
enriched Dirichlet process. The combination of the observed data model and
causal assumptions allows us to identify any type of causal effect -
differences, ratios, or quantile effects, either marginally or for
subpopulations of interest. The proposed BNP model is well-suited for causal
inference problems, as it does not require parametric assumptions about the
distribution of confounders and naturally leads to a computationally efficient
Gibbs sampling algorithm. By flexibly modeling the joint distribution, we are
also able to impute (via data augmentation) values for missing covariates
within the algorithm under an assumption of ignorable missingness, obviating
the need to create separate imputed data sets. This approach for imputing the
missing covariates has the additional advantage of guaranteeing congeniality
between the imputation model and the analysis model, and because we use a BNP
approach, parametric models are avoided for imputation. The performance of the
method is assessed using simulation studies. The method is applied to data from
a cohort study of human immunodeficiency virus/hepatitis C virus co-infected
patients.
