Relaxing partition admissibility in Cluster-DAGs: a causal calculus with arbitrary variable clustering

TL;DR

This paper extends Cluster DAGs to support arbitrary variable clustering, including cyclic structures, by relaxing admissibility constraints, and develops a sound, complete causal calculus for these generalized models.

Contribution

It introduces a novel relaxation of admissibility constraints in C-DAGs, enabling cyclic cluster representations and extending causal reasoning tools.

Findings

Supports cyclic C-DAGs by relaxing admissibility

Extends d-separation and causal calculus to cyclic models

Ensures soundness and completeness of the new calculus

Abstract

Cluster DAGs (C-DAGs) provide an abstraction of causal graphs in which nodes represent clusters of variables, and edges encode both cluster-level causal relationships and dependencies arisen from unobserved confounding. C-DAGs define an equivalence class of acyclic causal graphs that agree on cluster-level relationships, enabling causal reasoning at a higher level of abstraction. However, when the chosen clustering induces cycles in the resulting C-DAG, the partition is deemed inadmissible under conventional C-DAG semantics. In this work, we extend the C-DAG framework to support arbitrary variable clusterings by relaxing the partition admissibility constraint, thereby allowing cyclic C-DAG representations. We extend the notions of d-separation and causal calculus to this setting, significantly broadening the scope of causal reasoning across clusters and enabling the application of C-DAGs in previously intractable scenarios. Our calculus is both sound and atomically complete with respect to the do-calculus: all valid interventional queries at the cluster level can be derived using our rules, each corresponding to a primitive do-calculus step.