Title: Causal Discovery over Clusters of Variables in Markovian Systems
Venue: NeurIPS 2025
Accepted: yes
Keywords: Causal Inference
Abstract: Causal discovery methods are powerful tools for uncovering the structure of relationships among variables, yet they face significant challenges in scalability and interpretability, especially in high-dimensional settings. In many domains, researchers are not only interested in causal links between individual variables, but also in relationships among sets or clusters of variables. Learning causal structure at the cluster level can both reveal higher-order relationships of interest and improve scalability. In this work, we introduce an approach for causal discovery over clusters in Markov causal systems. We propose a new graphical model that encodes knowledge of relationships between user-defined clusters while fully representing independencies and dependencies over clusters, faithful to a given distribution. We then define and characterize a graphical equivalence class of these models that share cluster-level independence information. Lastly, we present a sound and complete algorithm for causal discovery to represent learnable causal relationships between clusters of variables.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a new graphical model, the αC-DAG, designed to learn causal relationships between predefined clusters of variables from observational data. It incorporates independence arcs, separation marks, and connection marks to represent conditional independence patterns among clusters and defines a custom d-separation criterion. The authors also propose αC-CPDAGs to capture equivalence classes of these graphs, along with a corresponding learning algorithm called CLOC. They prove the soundness of the framework and show that CLOC performs more efficiently than a PC-then-cluster baseline on synthetic data.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n1. The paper introduces a novel graphical model, the αC-DAG, and its equivalence class representation, the αC-CPDAG, specifically designed to capture causal relationships between predefined clusters of variables. The framework adds expressive symbolic elements—independence arcs, separation marks, and connection marks—that allow for a more nuanced encoding of conditional independencies at the cluster level.\n2. The authors establish sound and complete d-separation semantics for αC-DAGs, define a precise notion of cluster-level Markov equivalence, and prove the correctness of their orientation rules and learning algorithm.\n4. The experiments demonstrate meaningful comparisons against a standard PC-then-cluster baseline. Results show that the proposed approach achieves comparable accuracy.\n\n\nWeaknesses\n1. The method assumes a provided clustering of variables that is both acyclic and accurate. This assumption directly encodes structural information, which contradicts the premise of discovering causal structure from observational data. As a result, the learning process becomes more of a structural summarization under a fixed partition than genuine discovery.\n2. The graphical formalism of αC-DAG blends causal relationships and statistical independencies into a single representation. While this increases expressive power, it also blurs the distinction between mechanistic causality and observed conditional dependence, potentially complicating the interpretation and soundness of orientation rules derived from such hybrid semantics.\n3. The model relies heavily on conditional independence tests to annotate arcs and guide structure learning. However, multivariate CI tests are known to be statistically unstable, especially with small samples or non-Gaussian distributions, and the framework does not incorporate any robustness measures or uncertainty modeling.\n\nQUESTIONS:\n1. How sensitive is the method to small errors or noise in the cluster partition? Would a few misclassified variables significantly distort the learned structure?\n2. Can the approach be extended to support overlapping or hierarchical clusters, which are common in domains like text modeling?\n3. In cases where conditional independence tests produce conflicting results (due to statistical noise or small samples), how are contradictory arc annotations resolved?\n4. Does the graphical formalism still maintain interpretability and scalability when applied to systems with dozens or hundreds of clusters?\n\nLIMITATIONS:\n1. The approach assumes causal sufficiency at the cluster level. These assumptions are strong and often unrealistic in practical settings, particularly when latent confounding or weak interactions exist across clusters.\n2. The algorithm requires the cluster partition to be given and acyclic, which may not always be feasible. No method is proposed to validate, refine, or learn the partition itself.\n3. The symbolic structure of αC-DAGs (e.g., independence arcs, separation marks, connection marks) adds considerable complexity. The usability of such graphs in practical decision-making or downstream tasks like intervention planning is not demonstrated.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper tackles the problem of learning Cluster-DAGs (C-DAGs) from observational data, where a C-DAG is defined as the clustering of a low-level probabilistic causal model. The authors highlight how the connection between $d$-separations and conditional independence relations does not typically hold when considering clusters of variables. To this end, they introduce the novel notion of *independence arcs*, i.e., an arc connecting two edges of a C-DAG. The authors show how conditional independences between clusters can be expressed as a function of paths composed of independence arcs. Using this result, they design a constraint-based method — named CLOC — to recover C-DAGs from data, up to an equivalence class analogous to the Markov Equivalence Class for DAGs.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The paper well-motivates the introduction of their method, showing how naively applying causal discovery algorithms, such as PC, might not return the ground-truth C-DAG.\n- The introduction of independence arcs is interesting, novel, and correctly captures conditional independences among clusters.\n- Under the commonly adopted faithfulness assumption, the CLOC algorithm provably returns the equivalence class of the ground-truth C-DAG with the corresponding independence arcs, i.e., an $\\alpha$ CP-DAG in the terminology of the authors.\n\nWeaknesses:\n- The work shares the limitation of existing works on C-DAGs, by assuming that the clusters are provided. A practitioner does not know how variables are causally related, but must know how to cluster them.\n- The notation of independence arcs is quite confusing, and figures such as Figure 3 are difficult to interpret. For instance, overlapping symbols in Figure 3d — see for instance between clusters $\\mathbf{W}, \\mathbf{Y}$ — make it quite unreadable.\n\nQUESTIONS:\nThe scope and the contributions of the work are sufficiently clear, and I do not have particular questions for the authors.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors introduce αC-DAGs, a novel graphical representation that augments cluster DAGs (C-DAGs) with three annotations: independence arcs, separation marks, and connection marks. These additions allow αC-DAGs to fully encode conditional independence relationships over clusters, which is not possible with standard C-DAGs. The paper establishes soundness and completeness of a new d-separation criterion for αC-DAGs (Theorem 1). It then defines αC-CPDAGs as a representation for Markov equivalence classes of αC-DAGs. The CLOC algorithm is proposed for learning αC-CPDAGs from observational data, leveraging constraint-based methods and five orientation rules (R0-R5), which are proven sound. Experiments on synthetic data (up to 256 variables and 8 clusters) show that CLOC matches the cluster-level accuracy of the standard PC algorithm applied to the full variable set and then aggregated, while requiring fewer conditional independence tests and less computational time.\n\nSTRENGTHS AND WEAKNESSES:\nThe αC-DAG representation and the associated d-separation criterion are, to the best of my knowledge, new contributions. The paper has formal definitions and theorems (e.g., soundness and completeness of d-separation in αC-DAGs) and proofs (which I have not read). They also provided the CLOC algorithm, which is well-specified, and the orientation rules are motivated and proven sound.\n\nWeakness:\n\nThe approach relies on user-provided clusters or partitions that must form a DAG. The paper does not address how such a partition is obtained in practice. If the partition is misspecified (e.g., clusters are not acyclic), the method fails, right? This limits practicality, especially since cluster-level relationships might be unknown a priori, and the user has no idea if there is a cycle or not.\n\nTheorem 3 shows soundness but leaves open whether the αC-CPDAG always captures the full equivalence class. This is a critical gap because without completeness, the learned graph might be less informative than possible. Right?\n\nExperimental comparison overlooks recent methods for cluster-level causal discovery techniques; why is no other comparison provided?\nAlso, there is no real-world validation. I also expected to see an obligation study on misspecified clusters or partitions.\n\nThe presentation was a bit hard to follow for me with many notations and definitions. I am not sure if you can relax some notations to improve readability.\n\nQUESTIONS:\n-Why are clusters of nodes ultimately integrated into a graph over variables representing a class of DAGs? Why is there no other structure or cycle?\n\n-How sensitive is CLOC to misspecified partitions? You should have done an obligation study.\n\n-How did you perform multivariate CI tests?\n\nLIMITATIONS:\nAs the authors mentioned, users are required to have knowledge of a partition of variables that does not induce a cycle, which is very limiting in practice. \n\nThere is an experimental or theoretical analysis of how high-dimensional CI tests impact CLOC performance.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

comments: SUMMARY:
This paper addresses the challenge of causal discovery in high-dimensional settings by proposing an approach based on clustered variables. The authors introduce Cluster Directed Acyclic Graphs (C-DAGs), where nodes represent clusters of variables rather than individual variables, simplifying causal discovery in complex systems.

STRENGTHS AND WEAKNESSES:
​​Strengths:​​

- The paper introduces a novel approach to handling causal relationships between variable clusters, improving the scalability for high-dimensional data.
- The authors provide soundness/completeness for d-separation in the cluster of variables in Markovian systems.
- By reducing complexity through clustering, the method could make causal discovery more feasible in domains with many variables.

​​Weaknesses:​​

-  One of my concerns is that while clustering improves scalability, it might introduce statistical challenges. As the number of conditional independence tests grows (with increasing clusters or conditioning sets), the risk of the statistical hypothesis errors​​ escalates, and the conditional independence on the small conditional set should be more trustworthy than the one with have larger one. The paper does not sufficiently address how their method mitigates these errors, which could undermine the reliability of learned structures.
-  Moreover, can you compare the proposed method with the PC algorithm and other causal discovery algorithms without using the cluster by, for example, learn the full causal structure and then inferring the causal graph.
- The approach assumes a predefined partition of variables. Can you provide some real-world examples?

QUESTIONS:
See the weaknesses above.

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-02