Aligning Graphical and Functional Causal Abstractions

TL;DR

This paper establishes a formal connection between graphical and functional causal abstractions, showing their equivalence and extending the frameworks with Partial Cluster DAGs to enhance expressivity.

Contribution

It aligns the notions of graphical and functional causal abstractions, proving their equivalence and introducing Partial Cluster DAGs for greater expressivity.

Findings

Proves equivalence between Cluster DAGs and certain $ au$-abstractions.

Introduces Partial Cluster DAGs to extend graphical abstraction expressivity.

Provides a rigorous theoretical bridge between different causal abstraction frameworks.

Abstract

Causal abstractions allow us to relate causal models on different levels of granularity. To ensure that the models agree on cause and effect, frameworks for causal abstractions define notions of consistency. Two distinct methods for causal abstraction are common in the literature: (i) graphical abstractions, such as Cluster DAGs, which relate models on a structural level, and (ii) functional abstractions, like $\alpha$-abstractions, which relate models by maps between variables and their ranges. In this paper we will align the notions of graphical and functional consistency and show an equivalence between the class of Cluster DAGs, consistent $\alpha$-abstractions with the range of abstracted variables mapped bijectively, and constructive $\tau$-abstractions. Furthermore, we extend this alignment and the expressivity of graphical abstractions by introducing Partial Cluster DAGs. Our results provide a rigorous bridge between the functional and graphical frameworks and allow for adoption and transfer of results between them.