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

arXiv: https://arxiv.org/abs/2511.01396 · HTML: https://mlsanity.com/paper/2511.01396/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 20 references that are in the map.
- First posted on arXiv 3 Nov 2025.

## Builds on (most cited first, 9 of 9)

- [Causal Consistency of Structural Equation Models](https://mlsanity.com/paper/1707.00819/) (2017)
- [Abstracting Causal Models](https://mlsanity.com/paper/1812.03789/) (2018)
- [Visual Causal Feature Learning](https://mlsanity.com/paper/1412.2309/) (2014)
- [Causal Effect Identification in Cluster DAGs](https://mlsanity.com/paper/2202.12263/) (2022)
- [Multi-Level Cause-Effect Systems](https://mlsanity.com/paper/1512.07942/) (2015)
- [Foundations of Causal Discovery on Groups of Variables](https://mlsanity.com/paper/2306.07047/) (2023)
- [Identifiability of Direct Effects from Summary Causal Graphs](https://mlsanity.com/paper/2306.16958/) (2023)
- [Vector causal inference between two groups of variables](https://mlsanity.com/paper/2209.14283/) (2022)
- [Identifying Macro Conditional Independencies and Macro Total Effects in Summary Causal Graphs with Latent Confounding](https://mlsanity.com/paper/2407.07934/) (2024)

## Built on by (most cited first, 5 of 5)

- [Cluster-Dags as Powerful Background Knowledge For Causal Discovery](https://mlsanity.com/paper/2512.10032/) (2025)
- [Fairness under Graph Uncertainty: Achieving Interventional Fairness with Partially Known Causal Graphs over Clusters of Variables](https://mlsanity.com/paper/2602.23611/) (2026)
- [From Local to Cluster: A Unified Framework for Causal Discovery with Latent Variables](https://mlsanity.com/paper/2604.22416/) (2026)
- [Invariance of Clustering Operations in Causal Effect Identification](https://mlsanity.com/paper/2610.03101/) (2026)
- [Causal Effect Identifiability in Causal Abstractions: A Strict Hierarchy of Criteria](https://mlsanity.com/paper/2610.13520/) (2026)

## Abstract (arXiv)

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.