Causal intervention is an essential tool in causal inference. It is axiomatized under the rules of do-calculus in the case of structure causal models. We provide simple axiomatizations for families of probability distributions to be different types of interventional distributions. Our axiomatizations neatly lead to a simple and clear theory of causality that has several advantages: it does not need to make use of any modeling assumptions such as those imposed by structural causal models; it only relies on interventions on single variables; it includes most cases with latent variables and causal cycles; and more importantly, it does not assume the existence of an underlying true causal graph as we do not take it as the primitive object--in fact, a causal graph is derived as a by-product of our theory. We show that, under our axiomatizations, the intervened distributions are Markovian to the defined intervened causal graphs, and an observed joint probability distribution is Markovian to the obtained causal graph; these results are consistent with the case of structural causal models, and as a result, the existing theory of causal inference applies. We also show that a large class of natural structural causal models satisfy the theory presented here. We note that the aim of this paper is axiomatization of interventional families, which is subtly different from "causal modeling."

翻译：因果干预是因果推断中的重要工具。在结构因果模型框架下，它通过do-演算规则得以公理化。本文为概率分布族成为不同类型干预分布提供了简洁的公理化体系。这些公理化自然地导出了一个简洁清晰的因果理论，具有多项优势：无需依赖结构因果模型等建模假设；仅需对单变量进行干预；涵盖包含隐变量和因果循环的大部分情形；更重要的是，不假定潜在真实因果图的存在，因为我们不将其作为原始对象——事实上，因果图是作为该理论的副产品推导得出的。我们证明，在该公理化体系下，干预分布关于所定义的干预因果图满足马尔可夫性，且观测联合概率分布关于所得因果图满足马尔可夫性；这些结果与结构因果模型情形一致，因此现有因果推断理论仍然适用。我们还证明，一大类自然的结构因果模型满足本文所述理论。需要指出的是，本文目的在于干预族的公理化，这与"因果建模"存在细微差别。