Title: Relaxing partition admissibility in Cluster-DAGs: a causal calculus with arbitrary variable clustering
Venue: NeurIPS 2025
Accepted: yes
Keywords: Causality

Causal Inference

Cluster-DAGs

Do-Calculus for Higher-Level abstractions

Identification
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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper addresses causal effect identification problem in abstract causal graphs, in which each macro-level node is a cluster of the micro-level variables of some underlying causal mixed DAG. The authors extend the existing results in Anand et al. (2023) to allow cycles in the abstract graphs with a reformulation of d-separation and do-calculus rules that are provable sound and complete.\n\nSTRENGTHS AND WEAKNESSES:\n### **Strengths:**\n\nThe paper is well-written and easy to follow, with clear illustrative examples. The authors present solid theoretical results that address the problem at hand. The authors should it is sufficient to search for a compatible graph of size-3 clusters to efficiently apply do-calculus in practice. The method does not require knowledge of the micro-level causal structure but instead assumes known structure at the macro-level. Moreover, the proposed rules are agnostic to specific clustering types.\n\n### **Weaknesses:**\n* The paper supports causal reasoning up to L2-inference, whereas Anand et al. (2023) additionally support counterfactual inference. \n\n* A major limitation is the absence of numerical experiments to validate the theoretical findings. It would significantly strengthen the paper if the authors included a numerical demonstration showing that, given a valid clustering, one can derive the same  conclusions of interventional inference from the abstract DAGs as those drawn from the micro-level DAGs. \n\n* Additionally, it would be beneficial for the authors to further characterize the algorithm’s computational complexity.\n\nQUESTIONS:\nAlthough I appreciate the theoretical contributions, I remain uncertain about their practical implications. While I am fully aware of the scope of the work, my questions relate primarily to causal discovery, which may help clarify the potential application of the theory.\n1. The results suggest that if the d-separation rules do not hold in a C-DAG, there exists a compatible micro-level graph where they do not hold. What is the implication of this? Does it mean that the current clustering is invalid, and we should instead seek one where do-calculus consistently holds across all compatible graphs?\n2. The paper assumes known abstract structure. In practice, in what scenarios is the macro-level structure known while the micro-level structure remains unknown?\n3. Given a clustering of macro-level variables, there may be multiple compatible micro-level DAGs. Does the true underlying DAG among micro-level variables always belong to this set of compatible graphs?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-21"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn a lot of complex real-world scenarios, access to a fine-grained causal graph is difficult, and often we have to deal with a coarser representation of these graphs. This paper studies one such abstraction of fine-grained causal graph called Cluster-DAG (CDAG) and extends the limitation of previous work, which needed an acyclicity constraint on the C-DAG. They also extend the notion of d-separation and do-calculus for the CDAG.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n- The paper is well written, and the presentation is clear with supporting examples throughout to give intuition.\n\n- The authors provide atomically complete do-calculus for the C-DAG (Theorem 2), which can then be used to answer both Level 1 (observational) and Level 2 (interventional) queries in Pearl’s hierarchy.\n\n- They provide a compact representation of the search space for the microlevel DAGs compatible with the C-DAG in the form of an unfolded graph (Definition 6) instead of enumerating over the whole search space. Their definition of a compatible graph (Definition 5) is also constructive, which helps create instances to test for a particular d-separation (Proposition 3).\n\n\n**Weaknesses**\n\n- (Major) This work assumes that the cardinality of the cluster variable is known, which is then used to create the compatible or unfolded graph. There is a discussion to limit the number of variables to 3 in Theorem 4, showing that this preserves all the relevant dependencies. But, I am not sure if the result will hold if the cardinality is misspecified, which is very likely in the real-world scenarios where we only have access to the coarse C-DAG. \n\n- (Minor) This work assumes that when an intervention is performed on a cluster node all the micro-level nodes are also intervened on (the author also mentions this in Line 355). This assumption might be difficult to achieve in practice, where an intervention on such a large number of nodes simultaneously might be infeasible.\n\nQUESTIONS:\n- In Definition 5 (Line 193), when adding the arrows for V^{C} -> W^{C}, why do we only add one arrow between these two clusters of variables? Why not add as many as possible so that the compatible graph remains acyclic? I understand that this might lead to a question of which edges to choose, and we are adding all those edges anyway in the unfolded graph. Please clarify / give intuition for this.\n \n- Theorem 4 is interesting since it shows that using a C-DAG with cluster size at most 3 leads to the same do-calculus results. Please add some intuition on why a cluster size of 3 was necessary and what information is theoretically lost (if any) when we reduce the size of the cluster up to 3.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper extends cluster DAGs (Anand2023) by providing a do-calculus in the presence of cycles, which one would expect to be present between clusters in many applications. The calculus is based on the assumption of an underlying acyclic graph, and the interventions considered act on all members of a cluster. On the basis of a novel d-separation criterion, they show that it is possible to determine whether a causal query would fail in any underlying graph that is compatible with a given cluster DAG in a sound and atomically complete fashion. Moreover, they find that that the result does not change if all clusters of size greater than 3 are reduced to size 3, which greatly improves the computational efficiency of the calculus.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\n\n- Their results provide a do-calculus for what's arguably the most practically relevant setting: cyclic clusters.\n- Exceptionally clear, despite the great volume of theoretical results.\n- Strong result on how to improve the computational efficiency of applying their calculus.\n\n**Weaknesses**\n\n- Some notational issues (see \"Questions\").\n- Rather short discussion of limitations and implications of their results.\n\n---\n\nThis is the best paper I have read in a while.\n\nQUESTIONS:\nMain points:\n- line 117: what do you do if the topological ordering is not unique? (also, what is $V$ without index here?)\n- Definition 4: should it have a name? $\\sigma$ is a tuple of sets but treated as a single set in some observations; please clarify. Line 163: should it say \"ancestors of or in\"?\n- Theorem 2: In R2, shouldn't it be $\\bar{X}$? In R3, why the brackets in the union?\n- line 257: What is $X^m(Z^m)$?\n- Suggestion: add main-text pointers to corresponding proofs in appendix\n- Suggestion: rename _structure of interest_ to something less generic\n\nDetails:\n- line 28: representations don't have to be learnt\n- line 80: $Z$ not yet defined\n- Line 166: what do you mean by \"under\"?\n- Line 223: what do you mean by \"generally\"?\n- Footnote 1: what is the $\\sim$ relation?\n\nLIMITATIONS:\nA more extensive discussion on limitations and implications of your work would be welcome.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper generalizes Anand et al.'s (2023) result on do-calculus for Cluster-DAGs (C-DAGs), a kind of graphical structure derived from partitioning the vertex set of an underlying acyclic directed mixed graph and adding appropriate types of edge between cells of the partition according to the edges in the underlying graph. While Anand et al. (2023) only considers C-DAGs that do not contain any directed cycle and so effectively rules out certain partitions of vertices, the current work allows C-DAGs that contains cycles and as a result takes all partitions as admissible. In addition to presenting a do-calculus for the less constrained C-DAGs and demonstrating its soundness and atomic completeness, the paper also establishes that for the purpose of checking the applicability of a do-calculus rule to a given C-DAG, it suffices to check it in a simplified C-DAG in which all the clusters of size greater than three (i.e., contains more than three micro-vertices) are treated as if they contain only three micro-vertices.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. Despite being a highly theoretical paper with dense details on graphical definitions and manipulations, it is clearly written and fairly readable.\n\n2. The central results are elegant. \n\n3. Although it builds on Anand et al.'s (2023), there is at least one non-trivial innovation (the introduction of the canonical compatible graph), and one interesting finding on the possibility of reducing the computational cost, so the improvement is not merely incremental.  \n\nWeaknesses:\n\n1. No real examples are given to illustrate the utility of the theoretical constructs. As a result, it is hard to estimate how much impact it will make.\n\n2. The authors often use the terms \"micro-variables\" and \"macro-variables\" to describe the contrast between variables represented by the vertices in the underlying graph and those represented by clusters of vertices. However, it is not entirely clear to me that this terminology is appropriate. If the state space of a cluster-variable is merely the Cartesian product of the state spaces of its constituent variables (i.e., there is no coarse-graining of the state space), then referring to it as a macro-variable seems misleading, at least from my perspective.\n\nQUESTIONS:\nMinor questions:\n\n1. I have not tracked recent usage of the term \"root\" in graph theory, but what are called \"roots\" in this paper seem to me more appropriately called \"leaves\".\n\n2. On p. 3, line 6 from the bottom, what does \"*the* topological order\" mean? Topological ordering isn't unique in general, is it?\n\n3. The informal remark about a \"structure of interest\" reminds me of the notion of a \"d-connecting walk\", which is also a kind of combination of a d-connecting path and paths from colliders to conditioning vertices. Is there any connection worth noting here between the two notions?\n\n4. On p. 5, last line, why is the qualifier \"V_v != W_w\" needed? \n\n5. In the statement of Theorem 2, the notion of \"density compatible with [a] compatible graph\" is not explained previously. Moreover, I have a vague impression that Pearl's do-calculus requires positive density. Do I remember it wrong?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}