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：因果推断中的许多统计问题涉及一个与实际观测数据来源不同的概率分布；更复杂的是，目标对象往往是该另一概率分布的边缘量。即使问题在非参数条件下可识别，这仍为统计推断带来诸多实际困难。具体而言，我们难以执行基于似然的推断，甚至难以通过通用方式从该模型中进行模拟。我们引入"节俭参数化"方法，该方法将目标因果效应置于核心位置，并围绕其构建模型的其他部分。我们提供了一种通过目标因果量构建正则、非冗余参数化的系统性方案。针对离散变量，可使用比值比完成参数化；针对连续变量，连接函数是自然选择；文中还讨论了其他可能性。该方法允许我们构建并模拟具有参数化指定因果分布的模型，并通过包括完全贝叶斯方法在内的基于似然的方法进行拟合。我们的方案涵盖了平均因果效应、处理组平均处理效应及其他目标因果量的参数化方法。