Many statistical problems in causal inference involve probability distributions other than those in which the data are actually observed. To make matters worse, the objects of interest are often the marginal quantities of other probability distributions. This introduces a lot of practical complexity to statistical inference, even when the problem is identified nonparametrically. In particular, it is difficult to perform likelihood-based inference, or even simulate from models in a general way.
We introduce a “frugal parameterization” that places the causal effect of interest at its center and builds the rest of the model around it. We do this in a way that uses the causal quantity of interest to provide a recipe for constructing a regular non-redundant parameterization. For discrete variables, odds ratios can be used to complete the parameterization, but for continuous variables copulas are a natural choice. Other possibilities are also discussed.
Our method allows us to build and simulate models with parametrically specified causal distributions and fit them using likelihood-based methods, including full Bayesian approaches. Our proposal includes a parameterization of average causal effects and effects of treatment on treatment and other causal quantities of interest.