🤖 machine learning

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

This paper introduces Linear-LLM-SCM, a framework for benchmarking large language models' ability to estimate quantitative causal coefficients in Linear-Gaussian structural causal models, revealing significant limitations in their accuracy and stability across real-world datasets.

Original authors: Kanta Yamaoka, Sumantrak Mukherjee, Thomas Gärtner, David Antony Selby, Stefan Konigorski, Eyke Hüllermeier, Viktor Bengs, Sebastian Josef Vollmer

Published 2026-07-29

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CC BY 4.0

Original authors: Kanta Yamaoka, Sumantrak Mukherjee, Thomas Gärtner, David Antony Selby, Stefan Konigorski, Eyke Hüllermeier, Viktor Bengs, Sebastian Josef Vollmer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a super-smart robot that has read almost every book, article, and website on the internet. You ask it a question, and it answers with the confidence of a seasoned professor. But here's the catch: while this robot is amazing at telling you that two things are connected (like "smoking causes coughs"), it often struggles to tell you how much they are connected (like "smoking increases coughing by exactly 15%"). This is the difference between qualitative reasoning (knowing the story) and quantitative reasoning (knowing the math). In the world of science, we use something called "causal models" to map out these stories. Think of a causal model as a flowchart or a recipe where arrows show how one ingredient changes another. Usually, scientists need to crunch numbers from real experiments to figure out the exact amounts in the recipe. But what if we could just ask our super-smart robot to guess those numbers based on what it has read? That is the big question researchers are asking today: Can an AI that knows everything about the world also do the math to predict the future accurately?

This paper, titled "Linear-LLM-SCM," sets up a playground to test exactly that. The researchers created a game where they gave seven different "Large Language Models" (the fancy name for these super-smart robots) a map of a real-world system—a Directed Acyclic Graph, or DAG for short. You can think of a DAG as a one-way street map of cause and effect, where you can't drive in circles. The map showed which variables (like "Glucose levels" or "Spending money") influenced others. The robots' job was to act like expert statisticians and write down the exact mathematical equations that describe these relationships, specifically guessing the "coefficients" (the numbers that tell you how strong the effect is).

The team tested seven different real-world maps, ranging from how people spend money to how plants grow in algae. They asked the robots to look at the map and the descriptions of the variables, then spit out a regression equation (a math formula) for each part of the map. They then compared the robots' guesses against the "ground truth"—the actual, correct numbers that scientists had already measured in the real world.

The results were a mix of "not bad" and "oops." The researchers found that while the robots could sometimes guess the right direction (positive or negative effect), the actual numbers were all over the place. Even when the researchers told the robots to be as consistent as possible (by setting a "temperature" to zero, which is like telling the robot to stop guessing and just calculate), the answers still varied wildly from one try to the next. For example, on the "Cachexia" map (related to muscle wasting), one robot guessed a coefficient of 1.07, while another guessed 1.04, but the variation was high enough to be worrying. On the "Algal" map, a smaller robot model (Llama 3.1 8B) completely failed to write a readable equation at all.

The paper also put the robots through a stress test to see how tough they were. First, they changed the units of measurement (like switching from micrometers to nanometers). Surprisingly, sometimes this made the robots' answers look better by accident, likely because the numbers became easier to write down, not because the robots understood the physics better. Second, they tricked the robots by adding fake connections to the map (spurious edges). When the map had a fake arrow pointing from one thing to another that didn't actually affect it, the robots' performance dropped. They got confused, and their ability to rank the importance of different factors got worse.

The main takeaway is that while these AI models are great at understanding the story of cause and effect, they are currently unreliable at doing the math to predict the exact size of those effects. The study suggests that using these robots for high-stakes decisions, like in healthcare or clinical settings, is risky right now because their numbers can be inconsistent and sensitive to small changes in how the problem is presented. The authors aren't saying the robots are useless; they are saying we need to be very careful and not trust them with the final numbers until they get much more stable. They have even released their testing framework to the public so other scientists can keep trying to fix this problem, hoping that one day, our digital assistants will be able to not just tell us the story, but also do the math perfectly.

Technical Summary: Linear-LLM-SCM

Problem Statement
While Large Language Models (LLMs) have demonstrated potential in identifying qualitative causal relationships (Type 1 tasks), their ability to perform quantitative causal reasoning—specifically estimating effect sizes that parametrize functional relationships in continuous domains (Type 3 tasks)—remains underexplored. Existing research suggests LLMs struggle with discovering causal structures from data and quantitatively estimating consequences. Furthermore, current methods for LLM-based causal effect estimation often rely on discrete domains, require numerical observational data, or use specialized architectures to sample conditional distributions rather than directly eliciting structural parameters. This paper addresses the gap in evaluating whether LLMs can directly estimate continuous linear causal effect parameters for pre-specified Structural Causal Models (SCMs) using only causal structure and variable semantics, without observational data.

Methodology: Linear-LLM-SCM Framework
The authors introduce Linear-LLM-SCM, a plug-and-play framework designed to benchmark LLMs on Linear-Gaussian SCM parametrization. The framework operates under the assumption that LLMs have learned quantitative or qualitative information from their training corpora, including scientific literature formalized as SCMs or linear regressions.

Input Requirements: The system requires a Directed Acyclic Graph (DAG) structure ( $G$ ), variable descriptions ( $D$ ) including units and short text, and variable constraints ( $R$ ) (lower/upper bounds). Ground-truth effect parameters are provided for evaluation.
Decomposition and Prompting: Instead of feeding the entire DAG to the model, the framework decomposes the graph into local parent-child sets. It traverses the DAG in topological order. For each target node, the LLM is prompted with:
- A domain expert persona and a summary of the phenomenon.
- Descriptions, units, and ranges of the target variable and its direct parents.
- Instructions to output a linear structural equation (intercept and beta coefficients) in a specific string format.
Response Parsing: The system extracts beta coefficients from the LLM's textual output, parsing the linear equation to compare against ground truth.
Iterative Feedback: To enforce global consistency within the DAG, the framework includes an iterative feedback mechanism. It performs analytical sanity checks against pre-specified hard constraints (e.g., variable ranges) and retries elicitation if constraints are violated.
Evaluation Metrics: The framework compares elicited parameters against ground truth using four metrics:
- M1–M3: L2 norm distances between parameter vectors (with different aggregation methods). M2 and M3 are scale-invariant or normalized to address variable range issues.
- M4: A metric capturing the relative effect-size ordering per node.

Experimental Setup

Models: The study evaluated four general-purpose pre-trained LLMs: Gemini 2.5 Flash (sparse Mixture-of-Experts), Llama 3.1 8B (dense), Llama 3.3 70B (dense), and GPT-5.4.
Datasets: Seven real-world DAGs were selected from the BnRep repository, filtered for continuous Linear-Gaussian settings and a maximum of 15 nodes.
Conditions:
- Direct Estimation: Standard parameterization against ground truth.
- Adversarial Conditions (RQ2):
  - Unit Tweaking: Changing variable units (e.g., $\mu$ M to nM) to test robustness.
  - DAG Misspecification: Introducing spurious edges into the DAG structure to simulate modeling errors or observational limitations.
Protocol: Each condition was run 25 times with temperature set to zero to minimize stochasticity, reporting mean values with 95% confidence intervals.

Key Results

Direct Estimation (RQ1):
- Performance varied significantly across DAGs and models. No single model dominated all metrics across all DAGs.
- GPT-5.4 and Gemini 2.5 Flash generally performed best on M3 (distance) and M4 (ordering).
- Llama 3.3 70B generally outperformed the smaller Llama 3.1 8B, suggesting a benefit from larger parameter counts for this task.
- The small model (Llama 3.1 8B) failed to generate parsable equations for the algal2 DAG.
Stochasticity: Even at temperature zero, larger models (Gemini 2.5 Flash, Llama 3.3 70B, GPT-5.4) exhibited substantial variability across runs, with non-Gaussian distributions. This non-determinism is identified as a critical failure mode for safety-critical applications.
Robustness (RQ2):
- Unit Sensitivity: Changing units sometimes counter-intuitively improved metrics, potentially due to increased numerical precision in textual representation (e.g., $1 \to 1000$ ).
- Structural Perturbation: Adding spurious edges (DAG misspecification) generally degraded performance. Most models showed increased distance (M3) and reduced ordering accuracy (M4) under adversarial structural conditions, indicating limited robustness to structural uncertainty.

Key Contributions

Framework Introduction: The paper introduces Linear-LLM-SCM, a novel framework for evaluating LLMs on quantitative causal parameterization in continuous domains without observational data.
Open-Source Pipeline: The authors open-sourced the evaluation pipeline, allowing researchers to plug in their own DAGs, variable metadata, and LLMs.
Empirical Benchmarking: The study provides a systematic benchmark across seven real-world DAGs and four diverse LLM architectures, highlighting the current limitations of LLMs as quantitative causal parameterizers.
Failure Mode Analysis: The work identifies specific failure modes, including high variability in coefficient estimates, sensitivity to structural perturbations, and the inability of smaller models to consistently parse complex equations.

Significance and Claims
The paper positions its work as a necessary step toward understanding the capabilities and limitations of LLMs in safety-critical domains, such as healthcare, where quantitative causal expert knowledge is required. The authors explicitly state that LLMs should not currently be used for quantitative causal effect estimation in real healthcare or clinical decision support due to the observed variability and lack of robustness.

The significance lies in:

Highlighting the Gap: Demonstrating that while LLMs may encode causal knowledge qualitatively, their quantitative reasoning in continuous linear domains is currently unreliable.
Safety Implications: The observed stochasticity and sensitivity to structural noise pose risks for safety-critical applications, necessitating further research into safety testing and mitigation strategies before deployment.
Community Resource: By providing a modular framework, the authors aim to encourage further studies on LLM-based causal effect elicitation, specifically focusing on improving robustness and reducing variance.

The authors conclude that future work should extend the framework to non-linear functional forms and investigate methods to mitigate structural noise, as well as examine whether LLMs' textual uncertainty outputs are calibrated.

Technical Summary: Linear-LLM-SCM

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