Many statistical problems in causal inference involve a probability distribution other than the one from which data
are actually observed; as an additional complication, the object of interest is often a marginal quantity of this other
probability distribution. This creates many practical complications for statistical inference, even where the
problem is non-parametrically identified. In particular, it is difficult to perform likelihood-based inference, or
even to simulate from the model in a general way. We introduce the ‘frugal parameterization’, which places
the causal effect of interest at its centre, and then builds the rest of the model around it. We do this in a way
that provides a recipe for constructing a regular, non-redundant parameterization using causal quantities of
interest. In the case of discrete variables, we can use odds ratios to complete the parameterization, while in
the continuous case copulas are the natural choice; other possibilities are also discussed. Our methods allow
us to construct and simulate from models with parametrically specified causal distributions, and fit them
using likelihood-based methods, including fully Bayesian approaches. Our proposal includes parameterizations
for the average causal effect and effect of treatment on the treated, as well as other causal quantities of interest.