Regularized Multi-LLMs Collaboration for Enhanced Score-based Causal Discovery

Lina Yao; Ruoyu Wang; Xiaoxuan Li; Yao Liu

REVIEW 4 major objections 6 minor 21 references

Regularized Multi-LLMs Collaboration for Enhanced Score-based Causal Discovery

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper's central claim: a BIC-weighted penalty from several LLM-proposed causal graphs improves score-based causal discovery on benchmark datasets.

desk verdict A modest multi-LLM penalty idea whose own experiments undercut the 'improvements across all datasets' claim. read the letter →

arxiv 2411.17989 v1 pith:3MAXG7EN submitted 2024-11-27 cs.LG cs.AIstat.ME

Xiaoxuan Li , Yao Liu , Ruoyu Wang , Lina Yao This is my paper · ORCID

classification cs.LGcs.AIstat.ME

keywords causaldiscoverylargelanguagemodelsscore-basedmethodspriorknowledgeGESNOTEARSKCRLBICpenalty

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to establish that score-based causal discovery can be improved by asking several large language models (LLMs) to propose causal graphs and adding a weighted penalty to the score function that pulls the final graph toward those proposals. The motivation is economic: expert priors and randomized trials are expensive, while LLM queries are cheap. The authors argue that no single LLM is reliable enough, so they weight each LLM's graph by the Bayesian Information Criterion (BIC) score of that graph on the observed data and sum the penalties. If the claim holds, any score-based discovery algorithm can accept LLM advice without retraining, replacing costly prior-knowledge pipelines with a prompt and a penalty term.

What carries the argument

The load-bearing mechanism is the penalty term itself. Each LLM is prompted with a three-stage protocol—variable understanding, causal statement, self-revision—to produce an adjacency matrix $\hat{M}_m$. A quality weight $\mu_m$ is computed by evaluating the BIC score of $\hat{M}_m$ on the same dataset and normalizing the scores so they sum to one. The penalty $\lambda\sum_m \mu_m\|\hat{M}-\hat{M}_m\|_p$ is then added to the base score, with $\lambda$ expressing confidence in the LLM advice; decomposability of the $l_1$ form and differentiability of the $l_2$ form are what let existing search and optimization routines run unchanged.

What would settle it

On a dataset with known ground truth, compute each LLM's BIC-derived weight $\mu_m$ and its edge-level accuracy; if the most accurate LLM is not the highest-weighted one, or if re-fitting $\lambda$ by data-only validation removes the reported gains, the penalty is not carrying the causal knowledge.

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Extended reading notes

Core claim

The paper's central claim is that adding a weighted penalty built from several LLM-proposed causal graphs to an existing score function improves score-based causal discovery. The augmented objective is $S(G)=S_{\mathrm{score}}(G)+\lambda\sum_m \mu_m\|\hat{M}-\hat{M}_m\|_p$, where $\hat{M}_m$ is the adjacency matrix proposed by LLM $m$, $\mu_m$ is that proposal's normalized BIC score on the observed data, and $p=1$ or $2$. The paper proves the $l_1$ penalty is node-wise decomposable and the $l_2$ penalty is differentiable, so the penalties can be dropped into GES and NOTEARS respectively, and it reports experiments with GES, NOTEARS, and KCRL on LUCAS, Asia, Earthquake, SACHS, and Child showing improved accuracy metrics, with TPR reaching 1.0 for enhanced GES on Asia and FDR reaching 0.0 for enhanced KCRL on three datasets.

Load-bearing premise

The load-bearing premise is that an LLM graph's BIC score on the observed data measures how much that LLM should be trusted, and that one penalty weight $\lambda$ can be set without tuning on the ground-truth graph.

Editorial extensions

If this is right

Any score-based causal discovery method can accept LLM advice as a plug-in penalty without retraining the underlying search, since the penalty is additive to the score.
The node-wise decomposability proof means the $l_1$ penalty extends beyond GES to every additive score-based search, not just the three algorithms tested.
The differentiable $l_2$ penalty lets continuous optimizers such as NOTEARS carry LLM advice through the same augmented-Lagrangian procedure used without it.
Combining several LLMs can recover correct edges that no single LLM finds, which the paper's case study illustrates by showing distinct correct predictions from GPT-3.5, GPT-4, and Gemini.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension is to replace BIC-derived weights with weights learned from a small labeled validation set; if accuracy gains persist, the score-proxy assumption is not load-bearing.
The disagreement among LLM proposals could be converted into an uncertainty signal, treating edges all models agree on as near-certain priors and disputed edges as open.
The same penalty construction could seed constraint-based discovery, using high-confidence LLM edges as conditional-independence constraints instead of score penalties.
Because the paper's own results table shows some metrics moving against the enhancement, practical deployment would need automatic $\lambda$ selection rather than a fixed confidence weight.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the multi-LLM penalty idea is real but modest, and the paper's own experiments undercut its headline claim. The framework is simple: query several LLMs for adjacency matrices, weight each by its BIC score on the observed data, then add a weighted l1/l2 penalty to any score-based causal discovery objective. That specific combination is new as far as I can tell; prior work used a single LLM or externally supplied expert priors. The proofs that the penalties are decomposable (for GES) and differentiable (for NOTEARS) are correct — but they are also immediate consequences of the definitions, so the theoretical contribution is thin.

What the paper does well: it is clearly written, the framework is genuinely general, and the case study in Section 4.2 is honest about per-LLM performance. That honesty is what exposes the problem. GPT-4 alone recovers all 8 Asia edges with zero false positives, yet the multi-LLM-enhanced GES has SHD 4 and FDR 0.33 — worse than the original GES (SHD 3) and far worse than GPT-4 alone. There is no single-LLM baseline anywhere in Table 1. Without that control, you cannot attribute any gain to the collaboration mechanism rather than to the best LLM or to per-dataset tuning. And the weight computation is circular in a soft way: each LLM's weight comes from the same BIC score on the same dataset used for the final fit, so the penalty is influenced by the objective it is meant to regularize.

The empirical section has other problems: several Table 1 entries regress (GES on Child SHD 23→26, NOTEARS on Asia TPR 0.5→0.375), the abstract claims improvements across all datasets, which the table contradicts, and lambda is a tuned hyperparameter with no values or selection protocol reported. The Discussion admits tuning 'remains a complex task.' No code and no error bars either.

Who this is for: someone working on LLM priors for causal discovery who wants to see one more knob on the penalty approach. The central claim is unsupported as stated, but the idea is plausible and the paper does not hide its limitations. If the authors add single-LLM baselines, fix a lambda protocol, and report proper uncertainty, it could become a solid workshop paper or a minor conference contribution.

My recommendation: send it to peer review, but expect major revision. The novelty is enough to deserve referee time; the current evidence is not enough to accept.

Referee Report

4 major / 6 minor

Summary. The paper proposes a framework that uses multiple large language models (GPT-3.5, GPT-4, Gemini) to produce candidate causal graphs, combines them with weights derived from the score function, and injects the weighted combination into score-based causal discovery as a regularization term. The augmented score is applied to GES and NOTEARS directly, and to KCRL by replacing expert priors with LLM priors. Experiments on LUCAS, Asia, Earthquake, SACHS, and Child report SHD, TPR, and FDR for GES, NOTEARS, and KCRL with and without the LLM enhancement. The central claim is that the multi-LLM penalty improves accuracy across datasets and algorithms.

Significance. If the claimed improvements were fully supported, this would be a useful practical contribution: LLM-derived priors could replace costly expert knowledge in score-based causal discovery, and the multi-LLM weighting scheme would add a simple, algorithm-agnostic way to fuse imperfect priors. The paper also ships correct but elementary observations: the l1 and l2 penalties are decomposable in the GES sense, and the l2 penalty is differentiable for NOTEARS. The idea is timely and the presentation is generally clear. However, the experimental evidence is the load-bearing part, and it is currently mixed: several Table 1 entries regress relative to the baselines, no single-LLM control is reported, and the lambda selection protocol is unspecified. The significance of the contribution therefore cannot be assessed from the manuscript as it stands.

major comments (4)

[Abstract; §4.2, Table 1] The abstract and Section 4.2 state that 'the LLM enhancement leads to improvements across all datasets,' but Table 1 contains clear regressions: for Asia, GES SHD worsens from 3 to 4 and FDR from 0.22 to 0.33; for Child, GES SHD worsens from 23 to 26 and FDR from 0.60 to 0.62; for Asia, NOTEARS TPR drops from 0.5 to 0.375 while SHD rises from 4 to 5. The claim should either be restated as dataset-dependent improvement, or the paper should provide an aggregate, statistically grounded comparison (for example, mean or median SHD over datasets with paired significance tests).
[§3.3, §4.2] The central novelty of the paper is the collaboration of multiple LLMs, yet no experiment compares the weighted multi-LLM penalty against a single-LLM penalty under the same framework. The case study in Section 4.2 shows that GPT-4 alone recovers all 8 Asia edges with no false positives, while the enhanced GES result in Table 1 has SHD 4 and FDR 0.33, so the multi-LLM penalty is worse than following the best single LLM on that dataset. A control using each LLM individually in the same penalty mechanism is needed to attribute any observed gains to the collaboration rather than to the prior information itself.
[§3.3, §4.3] The hyperparameter lambda is load-bearing, but its selection is not specified. Section 4.3 admits that 'tuning this parameter remains a complex task,' and the manuscript reports no lambda values and no ground-truth-free protocol. If lambda was tuned per dataset using the true graph, the reported improvements are not a fair evaluation of the method. Please report the chosen lambda values and provide a selection protocol that does not use the test graph, or a sensitivity analysis showing that the improvements are robust to reasonable lambda choices.
[§3.3] The weights mu_model are computed by scoring each LLM graph with the same score function Sscore on the same dataset that is later used to fit the final graph. This makes the objective function serve as both the fitting criterion and the quality measure for the prior, which can select LLM outputs that happen to fit the data rather than LLM outputs with genuine causal knowledge. This potential circularity should be addressed directly, for example by using a held-out portion of the data for weighting or by comparing against equal weights.

minor comments (6)

[§3.3] The heading 'Muti-LLM Enhanced GES' contains a typo and should read 'Multi-LLM Enhanced GES.'
[§3.3] The penalty definitions use the notation ||M_hat - M_hat_LLM|| and ||M_hat - M_hat_LLM||^2 without specifying the norm; please state explicitly that this is the Frobenius norm and keep the matrix notation consistent throughout.
[§3.2] The composition K = U ∘ C ∘ R is described as a sequential operation, but the order in which the three stages are applied is ambiguous; please clarify which stage is executed first and whether R (revision) is applied last.
[§4.2] Table 1 reports single values without standard deviations or repeated runs; if the optimization procedures are deterministic, this should be stated explicitly.
[§4.2] The KCRL experiments are not connected to the methodological section: Section 3 derives the penalty only for GES and NOTEARS, so the manuscript should state precisely how the LLM prior is inserted into KCRL rather than only saying that LLM prior knowledge was used.
[References] Several references are incomplete: [4] lacks full publication details and [15] lacks a venue; please complete the bibliography.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the multi-LLM penalty is an additive regularizer with heuristic weights, not a fitted prediction.

full rationale

We find no circular step. The augmented objective S(G)=Sscore(G)+lambda * sum_model mu_model ||M - M_model|| is a genuine modification of the base score; the weights mu_model are a heuristic normalized inverse-score weighting, not a parameter fitted to the evaluation metric or to the final graph. The proofs in Section 3.3 are simple algebraic decomposability/differentiability checks and do not assume the conclusion. No load-bearing self-citation or imported uniqueness theorem appears; the cited LLM-prompt work (Ban et al.) is external. The experimental comparison is empirical and falsifiable, though Table 1 contains regressions and missing single-LLM controls; those are correctness and validity concerns, not circularity.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The framework's central premise is that LLM outputs are useful priors. It rests on: (1) a three-stage prompt elicits causal knowledge from variable names, (2) BIC score of an LLM graph measures its quality, (3) weighted adjacency-matrix averaging preserves complementary signal, (4) the greedy/continuous optimizers tolerate the added penalty. No new entities are introduced.

free parameters (1)

lambda (penalty weight) = not reported
Hyperparameter weighting LLM penalty in S(G)=S_score(G)+lambda*P(G); no value or selection procedure given, presumably tuned per dataset.

assumptions (4)

domain assumption LLM outputs elicited from variable names and possible values via a three-stage prompt contain causal information usable as prior knowledge.
Central premise of the framework; Section 3.2 relies on prompt design from Ban et al. [2].
domain assumption The BIC score of an LLM-produced graph on the observed data is a valid proxy for the quality of that LLM's causal knowledge.
Section 3.3 normalizes Sscore(G_LLM) into weights mu_model; this assumes the score reflects graph quality on the same data used for final fitting.
domain assumption Weighted sum of adjacency matrices from multiple LLMs preserves complementary signal and produces a meaningful combined prior.
Section 3.3; averaging adjacency matrices assumes individual LLM errors are not systematically aligned in a way that survives averaging.
standard math Adding a decomposable (for GES) and differentiable (for NOTEARS) penalty to the score does not break the optimization procedure.
Proved via row decomposition in Section 3.3; relies on standard properties of L1/L2 norms and BIC.

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Cite this review

Pith. "Pith review of Regularized Multi-LLMs Collaboration for Enhanced Score-based Causal Discovery." pith.science (2026). https://pith.science/paper/3MAXG7EN

@misc{pith2026241117989,
  author       = {Pith},
  title        = {Pith review of: Regularized Multi-LLMs Collaboration for Enhanced Score-based Causal Discovery},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3MAXG7EN}},
  note         = {Machine review of arXiv:2411.17989}
}

read the original abstract

As the significance of understanding the cause-and-effect relationships among variables increases in the development of modern systems and algorithms, learning causality from observational data has become a preferred and efficient approach over conducting randomized control trials. However, purely observational data could be insufficient to reconstruct the true causal graph. Consequently, many researchers tried to utilise some form of prior knowledge to improve causal discovery process. In this context, the impressive capabilities of large language models (LLMs) have emerged as a promising alternative to the costly acquisition of prior expert knowledge. In this work, we further explore the potential of using LLMs to enhance causal discovery approaches, particularly focusing on score-based methods, and we propose a general framework to utilise the capacity of not only one but multiple LLMs to augment the discovery process.

Figures

Figures reproduced from arXiv: 2411.17989 by the authors.

**Figure 2.** Overview of our framework. 3.1 Problem Definition The causal structure learning problem is defined as the following: Given the observational dataset X ∈ Rn×d , consisting of n data and each data contains d features. The task of causal discovery is to learn a Directed Acyclic Graph (DAG) G(V, E) where each node v ∈ V corresponding to a feature and the presence of an edge e = (i, j) ∈ E indicates that feature xi is a … view at source ↗

**Figure 3.** Information retrieved from GPT3.5, GPT4 and Gemini. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Alonso-Barba, J.I., delaOssa, L., Gámez, J.A., Puerta, J.M.: Scaling up the greedy equivalence search algorithm by constraining the search space of equivalence classes. Int. J. Approx. Reason. (2013) Title Suppressed Due to Excessive Length

work page 2013

[2] [2]

CoRR (2023)

Ban, T., Chen, L., Wang, X., Chen, H.: From query tools to causal architects: Harnessing large language models for advanced causal discovery from data. CoRR (2023)

work page 2023

[3] [3]

Chickering, D.M.: Optimal structure identification with greedy search. J. Mach. Learn. Res. (2002)

work page 2002

[4] [4]

Chickering, D.M., Heckerman, D.: Efficient approximations for the marginal like- lihood of bayesian networks with hidden variables. Mach. Learn. (1997)

work page 1997

[5] [5]

In: Proceedings of the 12th Interna- tional Conference on Pattern Recognition Applications and Methods, ICPRAM

Chowdhury, J., Rashid, R., Terejanu, G.: Evaluation of induced expert knowledge in causal structure learning by NOTEARS. In: Proceedings of the 12th Interna- tional Conference on Pattern Recognition Applications and Methods, ICPRAM

work page

[6] [6]

In: Proceedings of the Machine Learning for Healthcare Conference, MLHC 2022

Hasan, U., Gani, M.O.: KCRL: A prior knowledge based causal discovery frame- work with reinforcement learning. In: Proceedings of the Machine Learning for Healthcare Conference, MLHC 2022. Proceedings of Machine Learning Research, PMLR (2022)

work page 2022

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Hasan, U., Gani, M.O.: KGS: causal discovery using knowledge-guided greedy equivalence search. CoRR (2023)

work page 2023

[8] [8]

Heckerman, D., Geiger, D., Chickering, D.M.: Learning bayesian networks: The combination of knowledge and statistical data. Mach. Learn. (1995)

work page 1995