Linear-LLM-SCM: Benchmarking LLMs for Coefficient Elicitation in Linear-Gaussian Causal Models

David Antony Selby; Eyke H\"ullermeier; Kanta Yamaoka; Sebastian Josef Vollmer; Stefan Konigorski; Sumantrak Mukherjee; Thomas G\"artner; Viktor Bengs

REVIEW 5 major objections 5 minor 10 references

Linear-LLM-SCM: Benchmarking LLMs for Coefficient Elicitation in Linear-Gaussian Causal Models

T0 review · 5 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper claims that current LLMs, given only a DAG and variable semantics, cannot reliably estimate linear-Gaussian causal coefficients and quantifies the failure with new scale-invariant metrics.

desk verdict Useful benchmark for a real gap, though the headline claim would be stronger with a baseline and a check that the acceptance gate doesn't reject the ground truth. read the letter →

arxiv 2602.10282 v2 pith:QOBTN2A4 submitted 2026-02-10 cs.LG

Kanta Yamaoka , Sumantrak Mukherjee , Thomas Gärtner , David Antony Selby , Stefan Konigorski , Eyke Hüllermeier , Viktor Bengs , Sebastian Josef Vollmer This is my paper · ORCID

classification cs.LG

keywords largelanguagemodelscausaleffectestimationstructurallinear-GaussianSCMcoefficientelicitationbenchmarkingDAGmisspecificationrobustness

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 is trying to establish whether large language models can perform quantitative causal reasoning: given a directed acyclic graph and plain-language descriptions of its variables, can an LLM produce the regression coefficients of a linear-Gaussian structural causal model without seeing any observational data? It builds a plug-and-play benchmark that asks the model for one equation per node, aggregates the answers, and compares them against ground-truth coefficients from published Gaussian Bayesian networks using four metrics ranging from raw L2 distance to relative-effect ordering. Across three LLMs and seven real-world DAGs, the estimates are highly variable, sensitive to unit changes, and degrade when spurious edges are added, leading the authors to conclude that current LLMs are unreliable as quantitative causal parameterizers. If the paper is right, LLM-based effect-size elicitation is not yet deployable in safety-critical settings, and progress should be tracked with scale-invariant metrics rather than raw coefficient error.

What carries the argument

The framework's load-bearing mechanism is node-level decomposition with an acceptance gate. The DAG is traversed in topological order; for each node, the LLM receives only the node's direct parents, their units, value ranges, and a phenomenon description, and must return a concrete linear equation with numeric coefficients plus an intercept. An iterative feedback loop (Algorithm 2) computes the implied range of the target from the parents' ranges and the proposal, and rejects proposals whose implied range falls outside the node's stated bounds, re-prompting up to five times. Results are scored by four metrics: M1 raw L2 distance, M2 L2 distance after normalizing each node's coefficient vecto

What would settle it

Construct a synthetic linear-Gaussian SCM on a familiar DAG with coefficients that could not appear in any training corpus (e.g., random nonzero values), elicit coefficients from the same LLMs under the same prompting, and check whether M2/M3 and M4 stay poor. If the models approximate the held-out coefficients well, the reported failure is an artifact of memorization or constraint crafting; if they still fail, the limitation is robust.

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Extended reading notes

Core claim

The central claim is that LLMs, when given a pre-specified DAG and variable semantics, cannot reliably parameterize linear-Gaussian SCMs: the node-normalized error metric M2 ranges from roughly 0.45 to 3.25 across model-DAG pairs (far from the perfect score of 0), and the relative-ordering metric M4 falls when spurious edges are added, showing models fail to assign near-zero coefficients to irrelevant causes. The authors additionally report that two of the three tested LLMs remain stochastic even at temperature zero, producing non-deterministic coefficient estimates, while the smallest model is deterministic but less accurate. The claim is specifically about direct, symbolically elicited coe

Load-bearing premise

The benchmark treats the fitted linear-Gaussian coefficients in the source repository as exact ground truth and assumes the hard value ranges supplied (some handcrafted) agree with those fitted effects; if the ranges conflict or if the LLMs have memorized the source papers, the metrics measure agreement with a particular fitted network or retrieval, not causal reasoning.

Editorial extensions

If this is right

Scale-invariant metrics (M2–M4) should replace or supplement raw L2 distance in future LLM causal benchmarks, since M1 is dominated by variable ranges and misleads cross-DAG comparisons.
Predicting full causal effect sizes from prior knowledge alone is currently unsafe for healthcare or public-policy decisions; any LLM-elicited coefficients would need data grounding or strong uncertainty handling.
Adding spurious edges systematically lowers M4 and raises M2/M3, so downstream causal inference should treat DAG structure as uncertain rather than point-estimated.
Even with temperature set to zero, larger models show non-deterministic estimates, implying reproducibility cannot be assumed for LLM-based parameterization in deployment.

Reading between the lines

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

The benchmark never measures training-data leakage directly; if the source papers for the seven DAGs appear in the models' training corpora, the direct RQ1 scores could be inflated by memorization, making actual zero-shot elicitation competence lower than reported.
At least one DAG's 'hard constraints' were crafted by the authors rather than taken from the literature; rerunning the framework with different plausible ranges would show whether the acceptance gate, not the model's causal knowledge, drives the scores.
The unit-tweak experiment found that changing micromolar to nanomolar sometimes improved metrics; a clean test would ask the same model for coefficients in both units and check whether the two answers differ by the correct factor, isolating arithmetic scaling from semantic reasoning.
The framework could be extended to elicit residual variances and confidence intervals, which the authors explicitly left out, turning the benchmark into a full distributional-parameterization test.

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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

Short version: this paper quantifies something most of us suspected — LLMs are not reliable at turning a DAG plus variable semantics into regression coefficients — and it ships an open, reusable framework to keep measuring that. That is a real contribution. The novelty is honest: the closest work uses sampling or discrete CPDs, so direct symbolic coefficient elicitation in continuous linear-Gaussian SCMs is genuinely new. The DAGs come from BnRep, so ground truth is external, and the M2–M4 metrics sensibly move beyond scale-dependent L2 distance. The robustness probes (spurious edges, unit change) are the right kind of stress test.

The soft spots are mostly about calibration, not about the core direction.

First, there is no baseline. M2 values of 0.45–3.25 and M4 counts mean nothing until you compare against, say, random coefficients within the stated ranges or a heuristic like equal weights. The aggregate conclusion 'LLMs are poor' may survive such a baseline, but the paper doesn't show it. Given the claim is about limitation, this is a load-bearing omission.

Second, Algorithm 2's acceptance gate can distort what M1–M4 measure. For the expenditure DAG the value ranges were crafted by the authors. If those ranges are inconsistent with the BnRep fitted coefficients, the gate would reject the ground truth itself and the metrics would reward constraint compliance, not causal accuracy. The paper does not report a compatibility check. This needs to be settled.

Third, some robustness conclusions lean on overlapping CIs. The paper acknowledges the Llama 3.3 70B case, but the conclusion then states degradation as a general result. Soften it or add data.

Fourth, the ground truth is fitted network parameters, not proven causal effects; calling them 'effect ground truth' is a stretch, especially for DAGs that originated as predictive models. Framing it as comparing against a published fitted model would be more accurate.

Finally, data leakage is cited as motivation but never tested. If the BnRep numbers appear in training corpora, direct RQ1 results could be inflated; the perturbations help, but only partially.

None of this breaks the central descriptive finding, and the paper is honest about several limitations. It deserves a serious referee. I'd send it out, asking for a baseline, the compatibility check, and tightened conclusions.

Referee Report

5 major / 5 minor

Summary. The paper introduces Linear-LLM-SCM, a framework for eliciting linear-Gaussian structural equation coefficients from LLMs given a DAG, variable descriptions, units, and hard value-range constraints. The framework decomposes the DAG into local parent-child prompts, optionally refines proposals via an iterative acceptance gate (Algorithm 2), and compares LLM outputs against ground-truth parameters from the BnRep repository using four metrics M1–M4. Experiments on seven DAGs with three LLMs (Gemini 2.5 Flash, Llama 3.1 8B, Llama 3.3 70B) are reported for direct estimation (RQ1), unit-tweak robustness (RQ2-I), and spurious-edge robustness (RQ2-II). The authors conclude that LLMs show high variability, scale-sensitive M1, moderate normalized distances, and degradation under structural misspecification, and they caution against deploying LLMs as quantitative causal parameterizers in safety-critical domains.

Significance. If the findings hold, the paper provides a useful, open-sourced benchmark for a relatively underexplored task—quantitative coefficient elicitation for continuous SCMs—and makes a reasonable case that scale-invariant metrics (M3/M4) are more meaningful than raw L2 distance. The framework is model- and DAG-agnostic, and the authors are honest about several limitations, including the linear assumption, limited unit-tweak coverage, and the possibility of non-determinism. The external anchoring of ground truth in BnRep avoids the circularity of self-generated derivations. However, the strength of the central claim depends on three unresolved issues: causal provenance of the BnRep parameters, whether the Algorithm 2 hard-constraint gate is compatible with the ground truth, and the absence of chance-level or statistical baselines for interpreting M2–M4.

major comments (5)

[Section 4.2, Table 2] Ground-truth validity: The BnRep DAGs are a heterogeneous collection; several entries, e.g., expenditure from Tsagris (2022), are learned predictive networks (FEDHC) rather than established causal graphs. Treating every fitted linear-Gaussian parameter as a causal ground truth conflates conditional association with causal effect size. Since the central claim is about LLMs as 'quantitative causal parameterizers', the benchmark should either restrict to DAGs with explicit causal provenance or add a caveat that it evaluates association-coefficient elicitation. A concrete check would be to compare a subset of coefficients against independent domain-expert values.
[Algorithm 2; Table 2 (VR column)] Compatibility of the acceptance gate with ground truth: For the expenditure DAG, value ranges are 'Crafted' by the authors, not taken from the literature. Algorithm 2 rejects any proposal whose implied range C1 is not contained in the node's hard range C2. If the BnRep ground-truth coefficients themselves imply C1 values outside C2, then the gate would reject the ground-truth parameter vector, and M1–M4 would penalize LLMs for not reproducing an inadmissible parameterization. The paper does not report whether the ground-truth coefficients pass this compatibility check. Without this, a nontrivial portion of the measured 'failure' may reflect the crafted constraints rather than LLM inability.
[Section 5.2.2, Table 5] Statistical support for degradation under spurious edges: The text states that adversarial conditions generally yield lower M4 values, but the 95% confidence intervals overlap substantially across conditions. For example, Gemini M4 is 7.520±0.200 for O and 7.320±0.271 for S4; Llama 3.3 M4 is 6.560±0.199 for O and 6.680±0.271 for S4. Overlapping intervals do not support a claim of performance degradation. Report paired significance tests, bootstrap difference distributions, or a non-overlap criterion before concluding that spurious edges degrade parameterization.
[Section 3.4, Table 3] Missing chance-level baseline: M2–M4 have no null reference. M4 counts the number of nodes with exact effect-size ordering, but the expected value under random ordering depends on the number of parents per node; a score of 1.000 on cachexia1 may be far above chance or simply reflect a single multi-parent node with only two parents. Similarly, M2 and M3 values cannot be judged as 'poor' without knowing the distribution under random coefficients. Add a random-coefficient or permutation baseline for each DAG to calibrate the metrics and support the abstract's claim that LLMs are unreliable.
[Section 4.3, Section 5.1] Data leakage is invoked but not tested: The authors cite Yang et al. (2023) as motivation for adversarial perturbations to mitigate data leakage, yet they never measure whether direct RQ1 scores are inflated by memorization of published BnRep networks. Given that the same networks appear in the literature, LLMs may be recalling coefficients rather than eliciting them. A control condition using permuted variable names/semantics or held-out DAGs would clarify whether the low accuracy reflects a genuine inability to reason about continuous effect sizes.

minor comments (5)

[Abstract] The phrase 'seven real-world DAGs effect ground truth' is ungrammatical; should be 'seven real-world DAGs with effect ground truth'.
[Eq. (4)] Index notation in M4 is inconsistent: the surrounding text uses β_{LLM,j,i} and β_{GT,j,i}, but the equation uses β_{LLM,i,j} and β_{GT,i,j}. Unify the subscript order.
[Algorithm 2] When the loop budget n is reached, the algorithm returns the last proposal even if it failed the validation check. The text does not state whether this is the last accepted proposal or simply the final sampled one; clarify to avoid ambiguity in interpreting rejected samples.
[Table 2] The 'VR' column uses 'N (Crafted)' for expenditure, but the fact that the ranges are author-crafted is not explained until the discussion of Algorithm 2. Move or annotate this earlier so readers do not mistake the ranges as literature-sourced.
[Section 5.1] The interpretation of Llama 3.1 8B's 'determinism' would be stronger if the paper reported repeated runs with different seeds or API call configurations, since temperature=0 is not a guarantee of determinism in all serving stacks.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; benchmark is externally anchored to BnRep ground truth.

full rationale

The paper is an empirical benchmark, not a derivation chain in which a predicted quantity is equivalent to a fitted input. Ground-truth coefficients are taken from the external BnRep repository (Leonelli, 2025), and LLM outputs are compared with those coefficients via the M1–M4 distance metrics; no step defines the target in terms of the inputs or fits a parameter to the data it later 'predicts.' The only author-supplied values are prompt templates and, for the expenditure DAG, hard value ranges marked 'Crafted' in Table 2. These ranges enter Algorithm 2 as a rejection gate, but they are not used to construct the ground truth or to force the conclusion that LLM coefficients deviate from BnRep parameters. Even if the crafted ranges conflict with the BnRep fitted coefficients, that would be a construct-validity threat—the metrics could partly measure constraint compliance rather than causal parameterization quality—but it is not circular, because the comparison target remains external. Similarly, the possibility of LLM memorization of BnRep networks is an unquantified confound, not an equation-level reduction. There are no load-bearing self-citations: the authors do not rely on their own prior results to justify the benchmark's premise. Hence no circular step can be exhibited with the required specificity.

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

The central claim rests on no fitted parameters in the usual sense, but the benchmark embeds several author choices that act as free parameters: value ranges that gate proposal acceptance (crafted for the expenditure DAG), the refinement loop budget, prompt components, and the run configuration. The most consequential hidden input is the coupling between author-crafted ranges and the iterative gate, which can systematically accept or reject coefficient proposals independent of their causal correctness.

free parameters (4)

Per-variable hard value-range constraints (R) = e.g., Share ∈ [0,1], Age ∈ [18,100]; for expenditure DAG: 'Crafted' by authors (Table 2)
These ranges gate the iterative refinement loop (Algorithm 2): a proposal is accepted only if the possible value range C1 implied by parent ranges and the proposed coefficients is contained in the node's range C2. Author-created ranges for the expenditure DAG make this a hand-chosen input that can bias acceptance and all four metrics.
Iterative feedback loop budget (n) = 5
Algorithm 2 retries up to n=5 times per node; performance depends on this budget, and its choice is not studied.
Prompt composition (persona, phenomenon summary, units, unit-effect justification request) = hand-designed; one example shown in Figure 1
The prompt is the measurement instrument and is not varied across the 7 DAGs to assess sensitivity; the full set of prompts is only available in the open-source repo.
Sampling configuration = n=25 runs, temperature=0
All tables use 25 samples and temp 0; observed non-determinism at temp 0 is attributed to external factors. CI widths and all variance claims depend on this configuration.

assumptions (4)

domain assumption Each of the seven phenomena is adequately modeled by the given directed acyclic graph with linear-Gaussian structural equations.
Section 4 states the linear assumption is adopted 'due to verifiability'; real effects may be nonlinear, so the benchmark tests linear elicitation, not true phenomenon reproduction.
domain assumption The BnRep repository's stored parameters are the correct ground-truth effect sizes.
Section 4.2 takes ground truth from BnRep (Leonelli 2025); e.g., the expenditure parameters are from a FEDHC-fitted network (Tsagris 2022). These are fitted estimates with unknown errors, treated as exact, and they are published literature that LLM training corpora may contain.
domain assumption LLM JSON outputs can be reliably parsed into structural equations.
Algorithm 1, line 9; Llama 3.1 8B failed entirely on algal2 (Section 5.1), showing the parsing path is fragile for at least one model-DAG pair.
standard math The range-propagation calculation in Algorithm 2 (C1 from parent ranges and proposed betas) is a correct and complete check of the possible value range of a node.
For linear equations with bounded parents, extrema occur at bound combinations, so the arithmetic is standard; however it ignores covariance/correlation among parents, which is a simplification.

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Cite this review

Pith. "Pith review of Linear-LLM-SCM: Benchmarking LLMs for Coefficient Elicitation in Linear-Gaussian Causal Models." pith.science (2026). https://pith.science/paper/QOBTN2A4

@misc{pith2026260210282,
  author       = {Pith},
  title        = {Pith review of: Linear-LLM-SCM: Benchmarking LLMs for Coefficient Elicitation in Linear-Gaussian Causal Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QOBTN2A4}},
  note         = {Machine review of arXiv:2602.10282}
}

read the original abstract

Large language models (LLMs) have shown potential in identifying qualitative causal relations, but their ability to perform quantitative causal reasoning---estimating effect sizes that parametrize functional relationships---remains underexplored in continuous domains. We introduce Linear-LLM-SCM, a plug-and-play framework for evaluating LLMs on Linear Gaussian structural causal model parametrization when a directed acyclic graph (DAG) is given. The framework decomposes a DAG into local parent-child sets and prompts an LLM to produce a regression-style structural equation per node, which is aggregated and compared against available ground-truth parameters. Our experiments with seven real-world DAGs effect ground truth illustrate limitations of LLMs as quantitative causal parameterizers. Across most models, we observe variability in coefficient estimates and sensitivity to structural perturbations. We open-sourced the framework to further encourage the community to work on studies toward the use of LLM for causal effect elicitation in safety-critical domain, e.g., healthcare.

Figures

Figures reproduced from arXiv: 2602.10282 by the authors.

**Figure 2.** The DAG structure of cachexia1 from BnRep repository. [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** The DAG structure of expenditure from BnRep repository. [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Inclusion and exclusion flowchart for DAG ground-truths from BnRep DAG repository [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]

Discussion (0). Continue with ORCID to comment.

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examine LLMs on causality tasks but focus on causal relationship identification rather than quantitative effect estimation in continuous SCMs. A.2.3 SAMPLES OFDAGSUSED IN THEEXPERIMENTS One may easily find DAGs we used in our experiments from references, but to save time of the readers, we attached the two example DAGs, Figure 2 and Figure 3. GM VB A F GC...