Relaxing partition admissibility in Cluster-DAGs: a causal calculus with arbitrary variable clustering
This work addresses a limitation in causal reasoning for researchers and practitioners by extending the C-DAG framework to support arbitrary clusterings, though it is incremental as it builds on existing C-DAG semantics.
The paper tackles the problem of handling cycles in Cluster DAGs (C-DAGs) by relaxing the partition admissibility constraint, allowing arbitrary variable clusterings and enabling causal reasoning in previously intractable scenarios, with the result being a sound and atomically complete calculus for all valid interventional queries at the cluster level.
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.