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.

翻译：许多因果推断中的统计问题涉及一个与实际观测数据不同的概率分布；更复杂的是，感兴趣的对象往往是这个其他概率分布的边际量。即使问题是非参数可识别的，这也会给统计推断带来许多实际困难。特别是，很难进行基于似然的推断，甚至难以以通用方式从模型中模拟数据。我们引入"节俭参数化"方法，将感兴趣的因果效应置于中心位置，然后围绕它构建模型的其余部分。我们通过这种方式提供了一种使用感兴趣的因果量构造正规非冗余参数化的方法。对于离散变量，我们可以使用优势比来完成参数化；对于连续变量，copula 是自然选择；同时还将讨论其他可能性。我们的方法允许我们从参数化指定的因果分布构建和模拟模型，并使用基于似然的方法（包括完全贝叶斯方法）进行拟合。我们的方案涵盖了平均因果效应、处理组处理效应的参数化，以及其他感兴趣的因果量。