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.

翻译：近期进展使得在给定结构因果模型（SCM）下量化因果控制成为可能。这通过引入量化干预某个变量时另一个变量熵值变化的度量实现。这些被称为因果熵和因果信息增益的度量，旨在解决现有信息论方法在因果性起关键作用的机器学习任务中的局限性。目前尚未对这些度量进行严格的数学研究。本研究通过建立并分析这些概念的基本性质（包括界和链式法则），为因果熵和因果信息增益的形式化理解做出贡献。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，本研究通过探究基于因果性考量而新近提出的信息论量度，为提升因果机器学习任务铺平了道路。