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--in fact, a causal graph is 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.

翻译：因果干预是因果推断中的关键工具。在结构因果模型框架下，它通过do-演算规则得到公理化。我们为概率分布族提供了简洁的公理化体系，使其能够刻画不同类型干预分布的特征。这些公理化自然地构建了一套简明且清晰的因果理论，具有以下优势：无需依赖结构因果模型所施加的建模假设；仅需基于单变量干预；涵盖包含潜变量和因果环的大多数情形；更重要的是，不假设存在潜在的真实因果图——事实上，因果图是我们理论的副产品。我们证明，在此公理化框架下，干预分布相对于所定义的干预因果图具有马尔可夫性，而观测联合概率分布相对于所得因果图具有马尔可夫性；这些结果与结构因果模型的情形一致，因此现有因果推断理论仍然适用。我们还证明了一类广泛的结构因果模型满足本文提出的理论。