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：近期研究进展使得在给定结构因果模型（SCM）的基础上量化因果控制成为可能。这一突破通过引入描述干预某一变量时另一变量熵值变化的量得以实现。这些被称为因果熵和因果信息增益的度量，旨在解决现有信息理论方法在因果性起关键作用的机器学习任务中存在的局限性。目前对这些度量尚未进行严格的数学研究。本研究通过建立并分析因果熵和因果信息增益的基本性质（包括边界和链式法则），深化了对这两个概念的数学理解。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，本探索通过研究基于因果性考量提出的新型信息论量度，为提升因果机器学习任务性能开辟了道路。