Title: Identifying Macro Causal Effects in C-DMGs over DMGs
Venue: NeurIPS 2025
Accepted: yes
Keywords: Causality

Cluster graph

do-calculus

sigma-separation
Abstract: The do-calculus is a sound and complete tool for identifying causal effects in acyclic directed mixed graphs (ADMGs) induced by structural causal models (SCMs). However, in many real-world applications, especially in high-dimensional settings, constructing a fully specified ADMG is often infeasible. This limitation has led to growing interest in partially specified causal representations, particularly through cluster-directed mixed graphs (C-DMGs), which group variables into clusters and offer a more abstract yet practical view of causal dependencies. While these representations can include cycles, recent work has shown that the do-calculus remains sound and complete for identifying macro-level causal effects in C-DMGs over ADMGs under the assumption that all clusters sizes are greater than 1. Nevertheless, real-world systems often exhibit cyclic causal dynamics at the structural level. To account for this, input-output structural causal models (ioSCMs) have been introduced as a generalization of SCMs that allow for cycles. ioSCMs induce another type of graph structure known as a directed mixed graph (DMG). Analogous to the ADMG setting, one can define C-DMGs over DMGs as high-level representations of causal relations among clusters of variables. In this paper, we prove that, unlike in the ADMG setting, the do-calculus is unconditionally sound and complete for identifying macro causal effects in C-DMGs over DMGs. Furthermore, we show that the graphical criteria for non-identifiability of macro causal effects previously established C-DMGs over ADMGs naturally extends to a subset of C-DMGs over DMGs.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper tackles causal‐effect identification when the on a cluster-level abstraction called a C-DMG (nodes = variable clusters; edges = aggregate causal/confounding relations). Previous theory covered the acyclic case (ADMG); however, real systems often have feedback loops, motivating the cyclic DMG setting. The authors prove that, unlike ADMG setting, the do-calculus is unconditionally sound and complete for identifying macro causal effects in C-DMGs over DMGs. They also prove graphical criteria for non-identifiability of macro causal effects established in C-DMGs over ADMGs naturally extends to a subset of C-DMGs over DMGs. The work builds on input-output structural causal models (ioSCMs) which generalize SCMs to allow for cycles, providing a theoretical framework for causal identification in partially specified representations of cyclic systems.\n\nSTRENGTHS AND WEAKNESSES:\nI found the paper’s narrative easy to follow and well organized. Although I had only passing familiarity with C-DMGs and ioSCMs, the exposition—especially the step-by-step illustrative examples—made the main ideas accessible. The identification / non-identification examples were particularly helpful. Overall, the work extends prior research in a meaningful direction and consistently provides the background needed to understand each new result. I should mention that I haven't read the proofs.\n\nWeaknesses:\n\n-The “completeness” theorem is relative: it connects C-DMGs to compatible underlying DMGs but does not establish absolute completeness for the cyclic setting, as the author acknowledged.\n\n-The paper is entirely theoretical. No algorithmic implementation or empirical demonstration is offered. Can the authors design even a small synthetic experiment to showcase how the derived formulas recover interventional quantities in practice?\n\n-The practical motivation for using C-DMGs over DMGs, rather than the acyclic C-DMGs over ADMGs studied earlier, is not fully articulated. Concrete real-world scenarios where true feedback cycles matter would strengthen the case for this extension.\n\n-I think including intuition or proof sketches would be helpful in the main body.\n\nQUESTIONS:\n-I think in Theorem 3 it's better to recall what is c^w.\n\n-Definition 7 is hard to follow without any explanation for each case.\n\n-Have you considered developing an algorithm for identifying macro causal effects in C-DMGs over DMGs to help practitioners apply these results?\n\nCould you elaborate on why clusters in cycles must have size greater than 1 for Theorem 5, and whether this requirement could be relaxed?\n\nLIMITATIONS:\nThe authors adequately address the limitations of their work, particularly regarding the completeness of do-calculus and the soundness (but not proven completeness) of the SC-hedge criterion. They are transparent about the current boundaries of their theoretical results.\nNo guidance on how to learn C-DMGs from data, nor on the computational cost of the identification search. and no empirical validation or real-world case study.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proves soundness and completeness of the do calculus in a setting with macroscopic variables which are related by a possibly cyclic causal graph (resulting from coarse-graining a possibly cyclic graph with confounders into clusters of variables). The paper therefore extends known results for cluster graphs obtained from acyclic directed mixed graphs and generalizes them to cluster graphs obtained from directed mixed graphs.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n\nThe paper contains solid math, it is carefully written. The paper seems fair with respect to citing other work, it is very transparent about its contributions. \n\n\nWeakness:\n\nThe relevance of aggregating variables into clusters of variables without dimensionality reduction is limited. Coarse-grained causal models in all applications I know work with aggregations that reduce the dimension, which raises problems that are conceptually harder than the problems solved here because interventions are no longer well-defined. I would appreciate a better motivation of clustering by convincing real-life examples. \n\n\nGiven that the results have already shown for C-DMG over ADMGs, extension to C-DMG over DMGs appears to incremental, given the concept of sigma-separation, which the paper builds upon. \n\nI know that reviewers are often too fast with judging substantial work as incremental. Sometimes because results look straightforward in hindsight only and subtleties are no longer visible once the solution is there. The authors can try to convince me that I’m falling into the same fallacy by explaining subtleties and conceptual problems they needed to solve.\n\nQUESTIONS:\n-\tPlease comment more on why generalization from C-DMG over ADMGs to C-DMG over DMGs raises serious challenges. \n\n-\t“However, Forré and Mooij [2020] showed that replacing d-separation by σ-separation in the three rules allows the do-calculus to be applicable on DMGs induced by ioSCMs.” I assume that it is crucial that the DMG is generated by an ioSCM because the semantics of interventions are not described a priori without specifying the generating model behind the DMG, right? Isn’t there just an ioSCM for the vector of cluster variables?   \n\n-\t“which may contain true cyclic causal relations”: what is a non-true cyclic causla relation? \n\n\n-\tInclomplete references: Philip Boeken and Joris M. Mooij. Dynamic structural causal models, 2024, and Patrick Forré and Joris M. Mooij. Constraint-based causal… \n\n\n-\tAbstract: “under the assumption that all clusters size are greater than 1.” Does the paper explain why this condition is necessary?\n\nLIMITATIONS:\nI like that the discussion of limitations mentions the following subtlety: “The first limitation is that the completeness result in Theorem 4 does not take into account that there might exist different sequences of rules of the do-calculus in different DMGs that can give the same final identification of the causal effect”\n\nA more serious limitation is my concern regarding relevance.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-01"}

comments: SUMMARY:
Authors prove that do-calculus is sound and complete for C-DMGs over DMGs.

STRENGTHS AND WEAKNESSES:
- Strengths
    - Proving the do-calculus to this class of models makes them useful to identify the so called "macro" causal effects.
- Weaknesses
    - Notation layers stratify one on top the other (e.g. E→ = E+ ∣ Vobs ∪ J, while there is an already defined V = Vobs ∪ J). I don't think the current notation choice helps the reader to follow the content of this dense paper.
    - Some references appears to be non-peer reviewed (e.g. either arXiv or "unpublished notes" references).

QUESTIONS:
- One of my major concerns is the actual instantiation of distributions for Example 4 and 5, especially given the presence of cycles in Example 5. In the introduction it is said that "cluster graphs allow for cycles, which can arise naturally in feedback systems or time-dependent processes, complicating the analysis compared to traditional ADMGs", but there is not explicit discussion on time-related effects, leaving the actual computation of the proposed identified effects dubious. I get the value of this contribution is to prove the correctness and completeness of the do-calculus for this class of models, but how can we use the output of the identification process in practice?

LIMITATIONS:
- Discussed in the conclusions, but I would prefer to have a dedicated section to better highlight the current limits of the proposed approach.
- For instance, what are the actual implications of the statement: "there might exist different sequences of rules of the do-calculus in different DMGs that can give the same final identification of the causal effect"? Is it a limitation in the sense of "minimality" of sequence of rules application?
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nCausal identification from partially specified causal structures is a fundamental problem that is receiving increasing attention in recent years.  The paper introduces the problem of identification of macro causal effects from C-DMGs against compatible DMGs.  Below are the main takeaways from the paper:\n\n1) $\\sigma$-separation in C-DMGs implies corresponding $\\sigma$-separation in all compatible DMGs.\n\n2) Non-$\\sigma$-separation in C-DMGs implies existence of a compatible DMG where the corresponding variables are non-$\\sigma$- separated.\n\n3) The do-calculus (with $\\sigma$ separation) is sound for C-DMGs.\n\n4) Non-applicability of a do-calculus rule implies non-applicability of the same rule in some compatible DMG.\n\n5) An application of SC-Hedge for C-DMGs over DMGs, for a restricted class of graphs.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:  The manuscript is well-written and I really enjoyed reading the paper with nice running examples.  The technical contributions appear sound and advance the state of the art in causal identification from partial causal graphs.\n\nWeaknesses:  The theorem statements are nice, my only issue is that the proofs are relatively straightforward and most of the concepts extend from previous work on ADMGs.  Proving one of the following results: do-calculus completeness for non-identifiability, or a more general graphical condition for non-identiability would have greatly improved the contributions from a technical/novelty standpoint.  The proofs in Appendix are hard to understand and they rely on two important concepts: the extended graphs and compatible graphs, whose formal definitions are missing in the manuscript.\n\nQUESTIONS:\nThe notion of `completeness of do-calculus' originated from the early works of Tian and Pearl, which is different from the one discussed here.  I would suggest clarifying the definition of completeness over DMGs earlier in the manuscript, possibly even in the abstract.\n\nI am trying to understand if the non-applicability of a set of do-calculus rules $\\mathcal{R}$ on a given C-DMG would imply existence of a single compatible DMG where none of the rules in $\\mathcal{R}$ are applicable?  Such a claim, if true would make Theorem 4 much stronger.\n\nBelow are some minor suggestions / typos:\n\n1. I suggest using Pa, An, De, Scc and other abbreviations in non-math mode.\n\n2. Figure 1:  Two ADMGs → Two DMGs\n\n3. Some typos in lines 160 – 164 (due to its, lead to cycle between clusters)\n\n4. Please define compatible graphs and extended graphs – the core components of the proof.\n\n5. Line 228: in in\n\n6. Line 266:  conditioned → intervened\n\n7. Please proofread Appendix.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-14"}