Fundamental Properties of Causal Entropy and Information Gain
This foundational work addresses a gap in causal machine learning by providing rigorous mathematical underpinnings for recently proposed information-theoretic measures, potentially enhancing tasks where causality is crucial.
The paper tackles the lack of formal mathematical study of causal entropy and information gain, which quantify causal control in structural causal models, by establishing fundamental properties like bounds and chain rules, and proposing definitions for causal conditional entropy and information gain.
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.