Willem Schooltink

Willem Schooltink, Fabio Massimo Zennaro

Abstractions of causal models allow for the coarsening of models such that relations of cause and effect are preserved. Whereas abstractions focus on the relation between two models, in this paper we study a framework for causal embeddings which enable multiple detailed models to be mapped into sub-systems of a coarser causal model. We define causal embeddings as a generalization of abstraction, and present a generalized notion of consistency. By defining a multi-resolution marginal problem, we showcase the relevance of causal embeddings for both the statistical marginal problem and the causal marginal problem; furthermore, we illustrate its practical use in merging datasets coming from models with different representations.

Willem Schooltink, Fabio Massimo Zennaro

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 $α$-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 $α$-abstractions with the range of abstracted variables mapped bijectively, and constructive $τ$-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.