Recent developments enable the quantification of causal control given a structural causal model (SCM). This has been accomplished by introducing quantities which encode changes in the entropy of one variable when intervening on another. These measures, named causal entropy and causal information gain, aim to address limitations in existing information theoretical approaches for machine learning tasks where causality plays a crucial role. They have not yet been properly mathematically studied. Our research contributes to the formal understanding of the notions of causal entropy and causal information gain by establishing and analyzing fundamental properties of these concepts, including bounds and chain rules. Furthermore, we elucidate the relationship between causal entropy and stochastic interventions. We also propose definitions for causal conditional entropy and causal conditional information gain. Overall, this exploration paves the way for enhancing causal machine learning tasks through the study of recently-proposed information theoretic quantities grounded in considerations about causality.

通过引入编码在干预一个变量时另一个变量熵的变化的量，最近的研究实现了给定结构因果模型（SCM）的因果控制的量化。我们的研究通过建立和分析这些概念的基本性质，包括界限和链规则，对因果熵和因果信息增益的概念的形式理解做出了贡献。此外，我们阐明了因果熵与随机干预之间的关系，并提出了因果条件熵和因果条件信息增益的定义。总体而言，这种探索为通过研究最近提出的在因果性考虑方面扎根的信息论量来增强因果机器学习任务铺平了道路。

因果熵和信息增益的基本属性