Score-Based Learning of Cluster DAGs from Interventions

arXiv:2610.11947 · stat.ML, cs.LG · Submitted 2026-10-08 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: I'm Tom, and with me are Jane, Lu, senior AI researcher at Tsinghua, Meng, lead engineer at a mysterious AI startup and Lalam, the in-house Large Language Model.

Jane: Today's paper: "Score-Based Learning of Cluster DAGs from Interventions".

Tom: The gist The score-based method coarse introduces a first score-based method for learning an interventional coarsening from heterogeneous data,

Jane: First, who's behind it and why it matters.

Paper summary: Jane: So looking at "Score-Based Learning of Cluster DAGs from Interventions," what's the big picture here for causal modeling? The authors are essentially proposing a new way to learn that coarse structure without relying on the traditional constraint-based methods that have those reliability issues >

Tom: They’re taking the two-phase approach—clusters first, then edges—but they’re replacing the edge recovery with something score-based, which is much more robust when you can assume linear Gaussian conditions hold >

Lu: I think this shifts the focus from brute-force testing to using data summaries. The interventions give you a causal order over the blocks, and then a simple score comparison lets you find those edges in a single local search per cluster >

Meng: From an engineering standpoint, that makes sense because we're talking about polynomial time complexity, and they prove consistency if our initial identification of which variables are affected by the intervention is correct >

Lalam: And as an AI, I see this as a way to make causal abstraction more scalable for complex systems. If we can do this faster and more reliably with heterogeneous data, it could help us build better cultural models or whatever we're modeling >

Jane: It really seems like they’ve managed to handle both the unknown macrovariables and the out-of-reach fine DAG simultaneously when heterogeneity is available >

Tom: The authors are showing that by using the intervention signatures to define what variables can be merged, you can read the grouping and causal order off those shifts alone, leaving only a small skeleton search for the edges >

Conclusion: Tom: So, we’re wrapping up this look at "Score-Based Learning of Cluster DAGs from Interventions." Basically, these authors are showing how to get a smaller, high-level map of a huge causal network by just looking at how things change when you intervene in them.

Jane: Right. They take all those messy variables and group them into clusters that behave similarly under intervention signatures—that’s the core idea here.

Lu: What’s really interesting is they keep the two-phase structure, but instead of using some hard-to-find constraint testing for the edges, they use a score comparison method.

Meng: So if we think about this practically, it means we don't have to run these super slow tests every single time we want to find those connections between the clusters.

Lalam: From my perspective on how models learn structure, this is significant because it suggests that grouping variables based on intervention effects alone can be a very powerful and efficient way to build a model skeleton.

Tom: Exactly. The paper proves that this whole procedure runs in polynomial time, which is huge for real-world data where things get really big fast. They show it’s consistent too, meaning if you test the variables right, you get the right result in the long run.

Jane: It really boils down to this: interventions don't just tell us *what* we can group together; they actually tell us the causal order between those groups as well.

Lu: And they show that once you have that order fixed, learning the final connections between the clusters becomes a much simpler task—just one local search per cluster using some pooled score.

Meng: That speed improvement in finding edges is what makes this method practical for scaling up to hundreds of nodes in dense graphs. It’s not just theoretically sound; they tested it on real interventional data and it matched the best current edge recovery methods.

Tom: It sounds like the main implication is that we can tackle those massive, out-of-reach fine DAG problems without getting completely bogged down by impossible computational searches.

Jane: So what this means for us looking at complex systems, whether it’s biology or economics, is that if we have intervention data but don't know the whole graph structure, we can start building a simplified map much faster.

Lu: It opens up a new way to handle heterogeneity—where variables aren't all treated equally—by using those signatures as the fundamental organizing principle.

Meng: It’s promising for any engineer who is trying to build scalable causal models where the underlying structure is too complex to fully map out at once. We need to see how this fits into our existing pipelines.

Tom: The next step, of course, is seeing how this translates from theory into a working system that can handle the messy details of actual real-world measurements.

Informatics Institute, University of Amsterdam · Department of Mathematical Sciences, University of Copenhagen

stat.ML, cs.LG

Submitted: 2026-10-08

Updated: 2026-10-08

Code: https://github.com/tedescoG/coarse

License: http://creativecommons.org/licenses/by-nc-sa/4.0/

Importance score: 85/100

The gist: The gist The score-based method coarse introduces a first score-based method for learning an interventional coarsening from heterogeneous data, which merges variables with equal intervention

Key concepts

Interventional Coarsening (GE)
This is the goal: creating a smaller DAG where nodes are groups of original variables that cannot be distinguished by interventions. Variables sharing the same intervention signature are merged into one node in this coarse DAG.
Intervention Signature ($\sigma(v)$)
The signature of a variable is defined by which environments cause its marginal distribution to shift. If two variables have the same set of environments that change their marginals, they share the same signature and should be grouped together.
Pooled BIC Score
This score is used to determine the best edges between clusters. It combines individual BIC scores for each cluster, factoring in how many environments affect each cluster, allowing for a single local search per cluster instead of complex constraint-based searches.

Terminology

Summary

The gist The score-based method coarse introduces a first score-based method for learning an interventional coarsening from heterogeneous data, which merges variables with equal intervention signatures and is identifiable from marginal shifts alone

Interventional Coarsening and Identification

The paper extends the coarsening framework of Madaleno et al. [39] to unknown, multi-target interventions, resulting in an interventional coarsening G E that merges variables with equal intervention signatures and is identifiable from marginal shifts alone The signature of a node v is defined as the set of environments that shift its marginal, σ(v):= for e ∈ E excluding 0: f(e)(Xv) ≠ f(0)(Xv) This leads to the interventional coarsening G E, which merges nodes sharing a common image under χ such that χ(v) = χ(w) if and only if σ(v) = σ(w) Under Assumption 28, where each environment is a single-target intervention, the signature-based coarsening ΠE coincides with the intervention-ancestor based coarsening ΠI

Causal Order Determination

The interventions themselves identify a causal order over the blocks By Lemma 10, if u → w in G, then σ(u) ⊆ σ(w) This allows comparing supports to order the blocks consistently with causation Theorem 13 states that supports encode the causal order, showing that if πa and πb are adjacent in G E, then πa → πb in G if and only if πa <supp πb This removes the need for a search over orderings

Scoring Coarse Parents

The procedure keeps the two-phase template—clusters first, edges second—and replaces the CI-tested edge phase with score comparisons The fine structural equation model induces a Gaussian block-level model that factorizes over blocks The pooled BIC is defined as BIC(H, D):= for H a coarse DAG on ΠE with parent sets Paj: Xqj=1 BICj (πj,Paj) = -ne log Σb(e)j − λ log(ne) dj This score is decomposable, with BIC(e)j depending on H only through (πj,Paj)

Consistency and Performance

The procedure runs in polynomial time and is consistent, provided the variables affected by each intervention are correctly identified Theorem 19 proves that under oracle tests, as n → ∞ the output of Alg. 4 satisfies P(Gˆ = G E) → 1 On synthetic and real-world interventional data, coarse matches state-of-the-art edge recovery given enough samples, while its edge phase runs up to two orders of magnitude faster

Limitations and Variants

The design carries four limitations, including the pipeline being hybrid and sequential Finite-sample variants like CV-coarse select α by cross-validation to control false shifts, while kPC-coarse replaces each block with its leading principal components to reduce dimension The analysis shows that the loss in performance stems from the partition estimation stage, and the choice of penalty multiplier λ matters only at small ne The method recovers the block structure of the physical system in a fraction of a second on real data

Conclusion

The coarse method successfully reads the grouping and order off the shifts, leaving only a small, polynomial skeleton search This approach addresses both unknown macrovariables and out-of-reach fine DAGs when heterogeneity is available The system that motivates causal abstraction—high-dimensional measurements of cells, brains, and images—face two obstacles at once: the right macrovariables are unknown, and the fine DAG is out of reach When heterogeneity is available, coarse addresses both The interventions themselves determine not only the identifiability target but also the search order

How it works

The procedure learns the interventional coarsening in two phases, first reading blocks and a causal order over them off marginal shifts alone Second, with the order fixed, it runs a single score-based local search per block The pool for each block π is pa⋆(π):= for π' ∈ ΠE: supp([π') ⊊ [supp(π) This pool satisfies the property that it contains all parents and no descendants

Scoring Coarse Parents

The local score used is the pooled BIC of Def. 17, which is defined as BIC(e)j(πj,Paj):= for H a coarse DAG on ΠE with parent sets Paj: -ne log Σb(e)j − λ log(ne) dj The maximum likelihood estimate (MLE) is obtained by maximizing each l(e)j separately, one block regression at a time

Estimation and Guarantees

The partition estimation relies on the interventional descendant matrix M, where Mv,e = 1 if pv,e < α Algorithm 3 refines the partition by picking a block πa and its successor πb based on M The final coarse DAG Gˆ is constructed by running a single GrowShrink call over each parent pool pac⋆(πj) Theorem 20 proves the worst-case complexity is O(d 2n + E q 3d 3)

Comparison against RePaRe

Coarse and RePaRe recover the true partition (ARI=1) and the same coarse DAG, with 8 of the 10 true coarse edges and no false edge Coarse runs faster in every configuration, by up to 510× under the oracle partition The pooled BIC is score equivalent and, by construction, decomposable

Additional Experiments

The experiments on ER graphs show that constraint-based partition recovery is the bottleneck of the procedure The results are robust to noise distributions, as the Gaussian score is insensitive to the noise family in the linear model The method successfully recovers block structure of physical systems in a fraction of a second on real data The interventions can resolve resolution lost to multi-target interventions, from ARI 0.56 at m = 10 to 0.86 at m = 40 for k = 2

Summary

Coarse is the first score-based method for learning a coarsened causal model from heterogeneous data Intervention signatures fix the partition and a causal order over the blocks, and a BIC pooled over environments drives one local search per block The target, the interventional coarsening G E, extends that of Madaleno et al.

Improvements for AI systems

  1. Bold header: Score-based learning of cluster DAGs from heterogeneous data. This system can learn a coarse DAG by grouping variables based on identical intervention signatures (merges variables with equal intervention signatures) and then recovering the causal edges using a score comparison rather than CI tests, achieving an edge phase up to two orders of magnitude faster.

  2. Bold header: Identification of causal order from interventions. The system uses intervention signatures to identify a causal order over the blocks, which eliminates the need for searching over topological orders, as it proves that every (and only) descendant of an intervention target records a shift.

  3. Bold header: Robust inference under unknown multi-target interventions. The framework extends interventional coarsening to unknown targets, where the resulting coarsenings merges variables with equal intervention signatures, and the procedure remains identifiable from marginal shifts alone, even when single-target versions are not.

Abstract

Graphical approaches to causal abstraction transform a low-level causal directed acyclic graph (DAG) over many measured variables into a smaller, high-level DAG whose nodes cluster the original variables and whose edges summarize the causal relations between clusters. Such cluster DAGs are easier to interpret, but learning them requires finding the clusters and recovering the edges between them. Madaleno et al. (2026) learn the interventional coarsening (the cluster DAG that merges variables the interventions cannot distinguish) in two constraint-based phases: first the clusters, then the edges. We introduce COARSE, the first score-based method for this task: it keeps the two-phase structure but, under linear Gaussian assumptions, swaps the constraint-based edge phase for a score-based one. We show that the interventions themselves identify a causal order over the clusters, and learning the edges reduces to a single local search per cluster under a cluster-level BIC score. We prove that the procedure runs in polynomial time and, provided the variables affected by each intervention are correctly identified, that it is consistent. On synthetic and real-world interventional data, COARSE matches state-of-the-art edge recovery given enough samples, with an edge phase up to two orders of magnitude faster, including on dense graphs with hundreds of nodes.

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