Score-Based Learning of Cluster DAGs from Interventions

Gaetano Tedesco; Alex Markham

stat.MLOct 8, 2026

Score-Based Learning of Cluster DAGs from Interventions

Authors: Gaetano Tedesco, Alex Markham

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

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.

Figures & tables

Panel of Figure 1 — Figure 1: Full coarse pipeline on ER graphs, $\rho=0.2$ , $m=5$ . Both recovery metrics improve with $n_{e}$ at every $d$ .

F-score $\uparrow$ , $n_{e}=10^{3}$
$d$	10	20	50	100	200	300	500
coarse	1.00 $\pm$ 0.00	1.00 $\pm$ 0.02	0.99 $\pm$ 0.02	0.98 $\pm$ 0.02	0.95 $\pm$ 0.05	0.84 $\pm$ 0.08	0.71 $\pm$ 0.05
RePaRe	1.00 $\pm$ 0.01	0.96 $\pm$ 0.03	0.95 $\pm$ 0.05	0.93 $\pm$ 0.07	0.95 $\pm$ 0.08	0.95 $\pm$ 0.10	0.97 $\pm$ 0.04
F-score $\uparrow$ , $n_{e}=10^{5}$
coarse	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00
RePaRe	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	0.99 $\pm$ 0.01	1.00 $\pm$ 0.01

Table 1: Edge recovery under the oracle partition, ER graphs,

\rho=0.2

m=5

. F-score as mean

\pm

sample standard deviation over 20 seeds; speed-up is the ratio of mean RePaRe to mean coarse runtime. Bold marks the better method where the two differ at the reported precision.

Method	ARI	F1 $\uparrow$	Null	F1 $\uparrow$	Null	Fit(s)	Search(s)
		Variable level		Block level
coarse	1.00	0.74	0.13 [0.00, 0.38]	0.89	0.17 [0.00, 0.44]	0.02	0.40
RePaRe	1.00	0.74	0.12 [0.00, 0.35]	0.89	0.16 [0.00, 0.44]	0.10	1.54
GIES	–	0.62	0.10 [0.02, 0.21]	0.80	0.27 [0.12, 0.45]	1.83	–
GnIES	–	0.02	0.12 [0.02, 0.23]	0.07	0.27 [0.11, 0.46]	376	–
UT-IGSP	–	0.44	0.12 [0.02, 0.23]	0.39	0.28 [0.11, 0.45]	0.19	2.69

Table 2: Light tunnel, ungrouped mode. Directed F-score at the variable and block level, each against its negative control (mean and 95% interval over

1000

random DAGs with the same number of edges). Fit and Search are wall-clock seconds of the final fit and hyperparameter search.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 2 — Figure 2: Full coarse pipeline on ER graphs, $d=10$ ; one curve per density $\rho$ . Denser graphs and more environments delay convergence, but every setting converges.

Panel of Figure 3 — Figure 3: As Fig. 2 , on scale-free graphs.

F-score $\uparrow$ , coarse
$d$	10	20	50	100	200	300	500
$n_{e}=10^{3}$	1.00 $\pm$ 0.00	1.00 $\pm$ 0.02	0.99 $\pm$ 0.02	0.98 $\pm$ 0.02	0.95 $\pm$ 0.05	0.84 $\pm$ 0.08	0.71 $\pm$ 0.05
$n_{e}=5\times 10^{3}$	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.01	1.00 $\pm$ 0.00	1.00 $\pm$ 0.01	1.00 $\pm$ 0.01	1.00 $\pm$ 0.01
$n_{e}=10^{4}$	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.01	1.00 $\pm$ 0.00	1.00 $\pm$ 0.01
$n_{e}=5\times 10^{4}$	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00
$n_{e}=10^{5}$	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00	1.00 $\pm$ 0.00

Appendix

Table 3: Edge recovery under the oracle partition, ER graphs,

\rho=0.2

m=5

, all sample sizes. F-score as mean

\pm

sample standard deviation over 20 seeds; speed-up is the ratio of mean RePaRe to mean coarse runtime. Bold marks the better method where the two differ at the reported precision.

Panel of Figure 4 — Figure 4: Edge F-score under the oracle partition against the BIC penalty multiplier, ER graphs, $d=50$ , $m=5$ ; one curve per $n_{e}$ , dashed line at the standard BIC. Precision stays above $0.99$ for $\lambda\geq 0.5$ , so the loss at large $\lambda$ and small $n_{e}$ is recall.

Figure 5: Partition ARI against the per-test cutoff, ER graphs,

d=250

m=5

. Grey: fixed

\alpha

(median and interquartile range); markers: cutoff and ARI reached by each rule (medians with interquartile ranges). Every data-driven rule but Bonferroni lands in the flat region of the curve; CV reaches the oracle without ground truth.

Figure 6: ARI of each rule minus the oracle ARI on the same dataset, over 20 seeds.

d

stands for density.

Figure 7:

\Pi^{\mathcal{E}}

against

\Pi^{\mathcal{I}}

at the population level, native design, ER graphs,

d=50

;

k

targets per environment, one curve per number of environments

m

(mean and 95% CI over 200 seeds). Resolution is lost as soon as

k>1

; more environments refine

\Pi^{\mathcal{I}}

faster than

\Pi^{\mathcal{E}}

but hide fewer edges.

Figure 8: As Fig. 7 , overlap design: all targets come from a pool of

20

nodes and

m

grows; one curve per

k

, dashed line at the single-target value. Overlapping environments recover the resolution lost to multi-target interventions.

Figure 9: Partition estimated by coarse with cross-validated

\alpha

against

\Pi^{\mathcal{E}}

(top) and

\Pi^{\mathcal{I}}

(bottom), ER graphs,

d=50

m=10

; one curve per

n_{e}

, dashed: population

\mathrm{ARI}(\Pi^{\mathcal{E}},\Pi^{\mathcal{I}})

. The estimate tracks

\Pi^{\mathcal{E}}

at every

k

; its distance to

\Pi^{\mathcal{I}}

is the population gap.

Figure 10: Edge recovery under the oracle partition for coarse and its reduced-rank variants, ER graphs,

d=100

\rho=0.1

; dashed line at

n_{e}=d

. The variants score where coarse cannot (

n_{e}

below the family size) and lose precision where it can.

Figure 11: Per-block parent recovery under the oracle partition for coarse and its reduced-rank variants, ER graphs,

d=100

\rho=0.1

: exact parent set (top) and parent recall (bottom) against

n_{e}/F_{j}

, with

F_{j}

the number of variables in the true family of

\pi_{j}

for every method; blocks binned on a

\log_{2}

scale, dashed line at

n_{e}=F_{j}

. coarse near 0 recovery below the line stems from the non-existence of the MLE; the variants recover part of the parents below it but add spurious ones above it.

Figure 12: Share of blocks whose family size, in each method’s own units, is below

n_{e}

, same setting as Fig. 10 . The crossover of Fig. 10 coincides with the point where every family of coarse becomes feasible.

Panel of Figure 13 — Figure 13: Edge F-score under the oracle partition for five noise families, ER graphs, $\rho=0.2$ , $m=5$ . The Gaussian score is insensitive to the noise family in the linear model.

Ungrouped ( $m=5$ )
Method	Sel.	ARI $\uparrow$	Prec. $\uparrow$	Rec. $\uparrow$	F1 $\uparrow$	Fit(s)	Search(s)
coarse	score	1.00	1.00	0.80	0.89	0.02	0.40
	oracle	1.00	1.00	0.80	0.89	0.01	–
RePaRe	score	1.00	1.00	0.80	0.89	0.10	1.54
	oracle	1.00	1.00	0.80	0.89	0.10	–
GIES	–	–	0.63	0.62	0.62	1.83	–

Appendix

Table 4: Light tunnel, all methods and selections. Score rows select hyperparameters without ground truth ( coarse : cross-validation; RePaRe: GnIES score; UT-IGSP: BIC), oracle rows by maximum ARI, then F-score; GIES and GnIES have no hyperparameters. Metrics refer to the coarse DAG for coarse and RePaRe and to the variable-level graph for the others; GnIES is scored on the skeleton. Fit and Search are wall-clock seconds of the final fit and of the hyperparameter search.

Method	Prec. $\uparrow$	Rec. $\uparrow$	F1 $\uparrow$	Null F1	$\hat{m}$
Variable level
coarse	0.86	0.64	0.74	0.18 [0.06, 0.29]	29
RePaRe	0.86	0.64	0.74	0.18 [0.06, 0.29]	29
GIES	0.63	0.62	0.62	0.20 [0.10, 0.31]	38
GnIES	0.41	0.56	0.47	0.24 [0.13, 0.34]	54
UT-IGSP	0.45	0.64	0.53	0.24 [0.13, 0.34]	55

Appendix

Table 5: Light tunnel, ungrouped mode: adjacency (undirected) metrics against the exact negative control. Under the null hypothesis, the number of true adjacencies among the

\hat{m}

estimated ones is hypergeometric. Null F1 is its mean and 95% interval.

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Learning causal graphs from interventional data is a challenging problem with broad applications. In molecular biology, for example, a central goal is to uncover gene regulatory networks from large-scale perturbation data. An ideal algorithm for this task should scale to thousands of nodes, incorporate interventions even when their targets are unknown, quantify uncertainty, and provide identifiability guarantees. However, existing approaches---e.g. approaches using score-based optimization or approximate Bayesian inference---often fail to meet all of these criteria. To address these limitations, we develop Amortized Bayesian Causal Discovery of Extended Factor Graphs (ABCDEFG). Our method guarantees exact acyclicity, scales to graphs with thousands of nodes, and naturally handles interventions even when their targets are unknown. Additionally, ABCDEFG estimates a posterior distribution whose maximum a posteriori estimate provably identifies the true causal graph up to an equivalence class. On simulated datasets, ABCDEFG achieves state-of-the-art accuracy, producing a well-calibrated posterior distribution while outperforming previous score-based and approximate Bayesian methods. Applied to large-scale single-cell perturbation data, ABCDEFG identifies both established and novel gene targets of growth factors.

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