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arxiv: 2605.18633 · v1 · pith:DN3ARTYQnew · submitted 2026-05-18 · 📊 stat.ME · stat.ML

Stable Causal Discovery via Directed Acyclic Graph Aggregation

Yunan Wu , Yue Wang , Chunlin Li , Chenglong Ye This is my paper

Pith reviewed 2026-05-20 08:23 UTC · model grok-4.3

classification 📊 stat.ME stat.ML
keywords causal discoverydirected acyclic graphsmodel averagingDAG aggregationacyclicity preservationfinite-sample boundsedge selection consistencypredictive likelihood weighting
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The pith

Aggregating multiple candidate DAGs weighted by out-of-sample predictive likelihood produces stable acyclic causal graphs with finite-sample guarantees.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes a framework that combines several candidate directed acyclic graphs learned from data by weighting each one according to how well it predicts on held-out portions from repeated splits. This addresses the instability that arises when searching a huge space of possible graphs with only finite samples, which often yields unreliable causal structures. A simple threshold is then applied to the combined edge scores to force the final output to contain no cycles. The authors prove a finite-sample risk bound for the procedure, show that acyclicity is automatically preserved, and establish that edge selection becomes consistent when the weights obey mild conditions. Simulations on random, hub, and chain graphs plus a real protein-signaling dataset indicate that the aggregated result performs at least as well as the single best candidate and better than bootstrap alternatives on standard recovery measures.

Core claim

DAGgr aggregates multiple candidate DAGs into one stable representation by weighting each graph with its out-of-sample predictive likelihood across repeated data splits. A thresholding rule on the resulting edge-importance scores guarantees that the final graph remains acyclic. The method comes with a finite-sample risk bound and proves consistent edge selection under mild conditions on the weights.

What carries the argument

Weighting of candidate DAGs by out-of-sample predictive likelihood across repeated splits, followed by thresholding on edge-importance scores to enforce acyclicity.

If this is right

  • The aggregated graph matches or exceeds the best individual candidate in structural recovery metrics.
  • It consistently outperforms bootstrap-aggregation baselines across structural recovery metrics.
  • Edge selection is consistent under mild conditions on the weights.
  • The thresholding rule ensures the output graph is acyclic by construction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same weighting-plus-thresholding idea could be tested for stabilizing other combinatorial searches such as variable selection in high dimensions.
  • If predictive likelihood on splits correlates with true causal accuracy, the approach might reduce erroneous causal claims in domains with model uncertainty.
  • Varying the number of data splits or the threshold value on new synthetic examples would provide a direct check on how sensitive the consistency result is to those choices.

Load-bearing premise

The weighting scheme based on out-of-sample predictive likelihood across repeated data splits produces reliable edge-importance scores that, after thresholding, yield both stability and consistency without introducing selection bias or violating the acyclicity guarantee.

What would settle it

Observing a cycle in the thresholded output graph on any dataset where the weighting and thresholding steps are applied as described would falsify the acyclicity preservation claim.

Figures

Figures reproduced from arXiv: 2605.18633 by Chenglong Ye, Chunlin Li, Yue Wang, Yunan Wu.

Figure 1
Figure 1. Figure 1: Representative DAG structures used in the simulation study. [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The golden standard causal network in Sachs et al. (2005) [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Estimated DAGs obtained from the two pruned specifications [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Estimated PDAGs obtained from the two pruned specifications [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
read the original abstract

Directed Acyclic Graphs (DAGs) are central to uncovering causal structure in complex systems, yet learning a single DAG from data is often challenging: model uncertainty, finite samples, and a combinatorially large search space frequently yield unstable estimates. We propose DAGgr, a model averaging framework that aggregates multiple candidate DAGs into a single stable representation. Candidate graphs are weighted by their out-of-sample predictive likelihood across repeated data splits, and a thresholding rule on the resulting edge-importance scores guarantees that the aggregated graph is itself acyclic. We establish a finite-sample risk bound, prove that the procedure preserves acyclicity, and show that edge selection is consistent under mild conditions on the weights. Simulations across random, hub, and chain structures, together with an analysis of the Sachs et al. (2005) protein-signaling network, show that DAGgr matches or exceeds the best individual candidate while consistently outperforming bootstrap-aggregation baselines across structural recovery metrics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper introduces DAGgr, a model-averaging procedure that aggregates multiple candidate DAGs for causal discovery. Candidates are weighted by out-of-sample predictive likelihood computed over repeated data splits; edge-importance scores are then thresholded to produce a single output graph. The authors claim a finite-sample risk bound, a proof that the thresholding step preserves acyclicity, and consistency of edge selection under mild conditions on the weights. Simulations on random, hub, and chain graphs plus an application to the Sachs et al. (2005) protein-signaling network are reported to show performance that matches or exceeds the best single candidate and outperforms bootstrap aggregation on structural recovery metrics.

Significance. If the finite-sample bound, acyclicity guarantee, and consistency result hold, the work supplies a theoretically grounded route to stable causal structure estimates that directly addresses model uncertainty and finite-sample instability. The combination of out-of-sample weighting, an explicit acyclicity-preserving aggregation rule, and reproducible simulation protocols constitutes a concrete advance over existing bootstrap or model-averaging heuristics in causal discovery.

major comments (1)
  1. [theoretical results / acyclicity proof] Abstract and theoretical results section: the claim that thresholding on weighted edge-importance scores always yields an acyclic graph is load-bearing for the central contribution. The argument must explicitly rule out the case in which two or more high-weight acyclic candidates disagree on orientations that, when both edges survive the threshold, close a directed cycle. Without an additional lemma or explicit condition on the weight distribution or candidate pool, the preservation guarantee is not yet established.
minor comments (2)
  1. [method / notation] Notation for the edge-importance score (presumably defined after the weighting step) should be introduced with a single equation and used consistently in both the theoretical statements and the algorithm box.
  2. [experimental setup] The description of the data-splitting scheme for out-of-sample likelihood would benefit from an explicit statement of the number of splits, the split ratio, and whether the same splits are used for weighting and for final evaluation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comment regarding the acyclicity preservation argument is substantive, and we address it directly below with a commitment to strengthen the theoretical section.

read point-by-point responses

  1. Referee: [theoretical results / acyclicity proof] Abstract and theoretical results section: the claim that thresholding on weighted edge-importance scores always yields an acyclic graph is load-bearing for the central contribution. The argument must explicitly rule out the case in which two or more high-weight acyclic candidates disagree on orientations that, when both edges survive the threshold, close a directed cycle. Without an additional lemma or explicit condition on the weight distribution or candidate pool, the preservation guarantee is not yet established.

    Authors: We appreciate the referee highlighting the need for greater explicitness in ruling out conflicting orientations. The manuscript asserts that the thresholding rule on edge-importance scores preserves acyclicity because all input candidates are DAGs and the aggregation is performed via a threshold chosen to respect topological orderings implicit in the weighted scores. However, we agree that the current argument would benefit from an additional lemma that directly addresses the case of opposing orientations (e.g., A→B in one high-weight candidate and B→A in another). In the revision we will insert a new lemma proving that, under the out-of-sample likelihood weighting and the specific form of the threshold (which discards any edge whose aggregate importance falls below the level that would complete a cycle given the remaining edges), no directed cycle can arise. The proof will proceed by contradiction: suppose a cycle forms after thresholding; then at least one edge in the cycle must have been contributed by a candidate whose weight is inconsistent with the predictive likelihood ordering, violating the construction of the importance scores. We will also add a short remark on the mild condition this imposes on the candidate pool (namely that the pool is generated from a consistent search procedure). This addition clarifies rather than alters the existing result. revision: yes

Circularity Check

0 steps flagged

No significant circularity; weighting and acyclicity proof are independent of final output.

full rationale

The paper derives edge weights from out-of-sample predictive likelihood on repeated data splits, which is statistically independent of the final aggregated graph. The finite-sample risk bound, acyclicity preservation proof, and consistency result are stated under mild conditions on those weights rather than being forced by construction or self-citation. No load-bearing step reduces a claimed prediction or theorem to a fitted parameter or prior self-result; the thresholding rule is explicitly designed to enforce acyclicity and the proof follows from that design without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities. The method implicitly relies on standard causal discovery assumptions such as the existence of an underlying DAG and the validity of predictive likelihood as a model-quality proxy, but these are not detailed enough to enumerate.

pith-pipeline@v0.9.0 · 5691 in / 1219 out tokens · 39168 ms · 2026-05-20T08:23:53.916290+00:00 · methodology

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Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages · 2 internal anchors

  1. [1]

    BMC bioinformatics , volume=

    DAGBagM: learning directed acyclic graphs of mixed variables with an application to identify protein biomarkers for treatment response in ovarian cancer , author=. BMC bioinformatics , volume=. 2022 , publisher=

    work page 2022

  2. [2]

    Conference on Causal Learning and Reasoning , year=

    Bootstrap aggregation and confidence measures to improve time series causal discovery , author=. Conference on Causal Learning and Reasoning , year=

    work page

  3. [3]

    Dynamic Expert-Guided Model Averaging for Causal Discovery

    Dynamic Expert-Guided Model Averaging for Causal Discovery , author=. arXiv preprint arXiv:2601.16715 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  4. [4]

    Ensemble framework for causality learning with heterogeneous Directed Acyclic Graphs through the lens of optimization , journal =

    Babak Aslani and Shima Mohebbi , keywords =. Ensemble framework for causality learning with heterogeneous Directed Acyclic Graphs through the lens of optimization , journal =. 2023 , issn =. doi:https://doi.org/10.1016/j.cor.2023.106148 , url =

    work page doi:10.1016/j.cor.2023.106148 2023

  5. [5]

    Journal of the Royal Statistical Society: Series B , volume=

    Stability selection , author=. Journal of the Royal Statistical Society: Series B , volume=

    work page

  6. [6]

    NeurIPS , year=

    DAGs with No Fears: A Closer Look at Continuous Optimization for Learning Bayesian Networks , author=. NeurIPS , year=

    work page

  7. [7]

    NeurIPS , year=

    Beware of the simulated DAG! Causal discovery benchmarks may be easy to game , author=. NeurIPS , year=

    work page

  8. [8]

    A Bayesian Approach to Structure Discovery in Bayesian Networks , author=

    Being Bayesian about Network Structure. A Bayesian Approach to Structure Discovery in Bayesian Networks , author=. Machine Learning , volume=. 2003 , publisher=

    work page 2003

  9. [9]

    Advances in Neural Information Processing Systems , volume=

    Ordering-based causal discovery for linear and nonlinear relations , author=. Advances in Neural Information Processing Systems , volume=

    work page

  10. [10]

    Advances in Neural Information Processing Systems , volume=

    Hybrid top-down global causal discovery with local search for linear and nonlinear additive noise models , author=. Advances in Neural Information Processing Systems , volume=

    work page

  11. [11]

    Advances in Neural Information Processing Systems , volume=

    A scale-invariant sorting criterion to find a causal order in additive noise models , author=. Advances in Neural Information Processing Systems , volume=

    work page

  12. [12]

    Proceedings of the 24th ACM SIGKDD international conference on knowledge discovery & data mining , pages=

    Generalized score functions for causal discovery , author=. Proceedings of the 24th ACM SIGKDD international conference on knowledge discovery & data mining , pages=

    work page

  13. [13]

    arXiv preprint arXiv:1906.02226 , year=

    Gradient-based neural dag learning , author=. arXiv preprint arXiv:1906.02226 , year=

    work page arXiv 1906

  14. [14]

    2023 , title =

    Liu, Huihang and Zhang, Xinyu , journal =. 2023 , title =. doi:10.1111/biom.13758 , pmid =

    work page doi:10.1111/biom.13758 2023

  15. [15]

    Advances in Neural Information Processing Systems , volume=

    Dagma: Learning dags via m-matrices and a log-determinant acyclicity characterization , author=. Advances in Neural Information Processing Systems , volume=

    work page

  16. [16]

    Journal of the American Statistical Association , year=

    Likelihood ratio tests for a large directed acyclic graph , author=. Journal of the American Statistical Association , year=

    work page

  17. [17]

    Statistical Analysis and Data Mining: The ASA Data Science Journal , volume=

    Maximum likelihood estimation over directed acyclic Gaussian graphs , author=. Statistical Analysis and Data Mining: The ASA Data Science Journal , volume=. 2012 , publisher=

    work page 2012

  18. [18]

    Advances in neural information processing systems , volume=

    Dags with no tears: Continuous optimization for structure learning , author=. Advances in neural information processing systems , volume=

    work page

  19. [19]

    International conference on artificial intelligence and statistics , pages=

    Learning sparse nonparametric dags , author=. International conference on artificial intelligence and statistics , pages=. 2020 , organization=

    work page 2020

  20. [20]

    The Annals of Statistics , number =

    Peter B. The Annals of Statistics , number =

    work page

  21. [21]

    Mooij and Dominik Janzing and Bernhard Sch

    Jonas Peters and Joris M. Mooij and Dominik Janzing and Bernhard Sch. Causal Discovery with Continuous Additive Noise Models , journal =. 2014 , volume =

    work page 2014

  22. [22]

    Proceedings of The 28th International Conference on Artificial Intelligence and Statistics , pages =

    Differentiable Causal Structure Learning with Identifiability by NOTIME , author =. Proceedings of The 28th International Conference on Artificial Intelligence and Statistics , pages =. 2025 , editor =

    work page 2025

  23. [23]

    Journal of Machine Learning Research , year=

    Estimating High-Dimensional Directed Acyclic Graphs with the PC-Algorithm , author=. Journal of Machine Learning Research , year=

    work page

  24. [24]

    and Glymour, C

    Spirtes, P. and Glymour, C. and Scheines, R. , biburl =

    work page

  25. [25]

    Journal of machine learning research , volume=

    Optimal structure identification with greedy search , author=. Journal of machine learning research , volume=

    work page

  26. [26]

    Machine learning , volume=

    The max-min hill-climbing Bayesian network structure learning algorithm , author=. Machine learning , volume=. 2006 , publisher=

    work page 2006

  27. [27]

    Learning directed acyclic graphs via bootstrap aggregating

    Learning directed acyclic graphs via bootstrap aggregating , author=. arXiv preprint arXiv:1406.2098 , year=

    work page internal anchor Pith review Pith/arXiv arXiv 2098

  28. [28]

    Science , volume=

    Causal protein-signaling networks derived from multiparameter single-cell data , author=. Science , volume=. 2005 , publisher=

    work page 2005

  29. [29]

    and Guo, Zhigao and Liu, Yang and Chobtham, Kiattikun , journal =

    Kitson, Neville Kenneth and Constantinou, Anthony C. and Guo, Zhigao and Liu, Yang and Chobtham, Kiattikun , journal =. A survey of

    work page

  30. [30]

    International Journal of Approximate Reasoning , volume =

    Effective and efficient structure learning with pruning and model averaging strategies , author =. International Journal of Approximate Reasoning , volume =. 2022 , note =

    work page 2022

  31. [31]

    Proceedings of the 25th International Conference on Artificial Intelligence and Statistics (AISTATS) , series =

    On the convergence of continuous constrained optimization for structure learning , author =. Proceedings of the 25th International Conference on Artificial Intelligence and Statistics (AISTATS) , series =

    work page

  32. [32]

    arXiv preprint arXiv:2511.14206 , year =

    Causal Discovery on Higher-Order Interactions , author =. arXiv preprint arXiv:2511.14206 , year =

    work page arXiv

  33. [33]

    Proceedings of the 4th Conference on Causal Learning and Reasoning , series =

    The Landscape of Causal Discovery Data: Grounding Causal Discovery in Real-World Applications , author =. Proceedings of the 4th Conference on Causal Learning and Reasoning , series =. 2025 , note =

    work page 2025

  34. [34]

    Fast scalable and accurate discovery of

    Andrews, Bryan and Ramsey, Joseph and Sanchez-Romero, Ruben and Camchong, Jazmin and Kummerfeld, Erich , journal =. Fast scalable and accurate discovery of

    work page

  35. [35]

    2009 , publisher =

    Causality: Models, Reasoning, and Inference , author =. 2009 , publisher =

    work page 2009

  36. [36]

    2009 , publisher=

    Probabilistic Graphical Models: Principles and Techniques , author=. 2009 , publisher=

    work page 2009

  37. [37]

    2020 , publisher=

    Causal Inference: What If , author=. 2020 , publisher=

    work page 2020

  38. [38]

    Friedman, Nir and Linial, Michal and Nachman, Iftach and Pe'er, Dana , journal =. Using

    work page

  39. [39]

    A million variables and more: the

    Ramsey, Joseph and Glymour, Madelyn and Sanchez-Romero, Ruben and Glymour, Clark , journal =. A million variables and more: the

    work page

  40. [40]

    Epidemiology , volume =

    Causal diagrams for epidemiologic research , author =. Epidemiology , volume =

    work page

  41. [41]

    Koivisto, Mikko and Sood, Kismat , journal =. Exact

    work page

  42. [42]

    Journal of Machine Learning Research , volume =

    Order-independent constraint-based causal structure learning , author =. Journal of Machine Learning Research , volume =

    work page

  43. [43]

    and Perlman, Michael D

    Madigan, David and Andersson, Steen A. and Perlman, Michael D. and Volinsky, Chris T. , journal =

    work page

  44. [44]

    Data analysis with

    Friedman, Nir and Goldszmidt, Mois. Data analysis with. Proceedings of the 15th Conference on Uncertainty in Artificial Intelligence (UAI) , pages =. 1999 , publisher =

    work page 1999

  45. [45]

    Estimation of genetic networks and functional structures between genes by using

    Imoto, Seiya and Goto, Tomoyuki and Miyano, Satoru , booktitle =. Estimation of genetic networks and functional structures between genes by using

    work page

  46. [46]

    2014 , note =

    dagbag: Learning directed acyclic graphs (DAGs) through bootstrap aggregating , author =. 2014 , note =

    work page 2014

  47. [47]

    Biometrika , volume =

    Yuan, Yiping and Shen, Xiaotong and Pan, Wei and Wang, Zizhuo , title =. Biometrika , volume =. 2019 , doi =

    work page 2019

  48. [48]

    2012 , isbn =

    Edwards, David , title =. 2012 , isbn =

    work page 2012

  49. [49]

    and Picard, Jean

    Catoni, Olivier. and Picard, Jean. , publisher =. Statistical Learning Theory and Stochastic Optimization , year =

    work page