In this paper we provide a theoretical analysis of counterfactual invariance. We present a variety of existing definitions, study how they relate to each other and what their graphical implications are. We then turn to the current major question surrounding counterfactual invariance, how does it relate to conditional independence? We show that whilst counterfactual invariance implies conditional independence, conditional independence does not give any implications about the degree or likelihood of satisfying counterfactual invariance. Furthermore, we show that for discrete causal models counterfactually invariant functions are often constrained to be functions of particular variables, or even constant.

翻译：本文对反事实不变性进行了理论分析。我们介绍了多种现有定义，研究了它们之间的相互关系及其在图模型中的含义。随后，我们转向当前围绕反事实不变性的主要问题：它与条件独立性有何关联？我们证明，虽然反事实不变性蕴含条件独立性，但条件独立性并不能对反事实不变性的满足程度或可能性提供任何暗示。此外，我们指出，在离散因果模型中，反事实不变函数通常被约束为特定变量的函数，甚至可能是常数函数。