Uncovering Cause and Effect with Language Models

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Author: Denis Avetisyan

A new framework harnesses the power of large language models and advanced mathematical tools to tackle the complex problem of identifying causal relationships, even when hidden variables are involved.

HOLOGRAPH leverages sheaf theory and natural gradient descent for active causal discovery, revealing limitations in local causal reasoning.

Identifying causal relationships from observational data is fundamentally challenged by inherent identifiability issues, yet recent efforts explore leveraging the prior knowledge embedded within Large Language Models (LLMs). In this work, we introduce HOLOGRAPH: Active Causal Discovery via Sheaf-Theoretic Alignment of Large Language Model Priors, a novel framework that formalizes LLM-guided causal discovery using sheaf theory to represent and align local causal beliefs. Our analysis reveals that while core axioms of causal reasoning generally hold, limitations emerge in larger graphs, suggesting non-local coupling in latent variable projections-raising the question of how to best reconcile local inductive biases with global causal structure.

The Epistemic Bottleneck: From Correlation to Mechanistic Truth

Traditional causal discovery methods often falter when confronted with the complexities of real-world data, particularly in high-dimensional spaces. The sheer number of potential relationships increases exponentially with each added variable, quickly overwhelming algorithms and leading to a proliferation of spurious correlations – connections that appear meaningful but are, in fact, coincidental. Furthermore, the presence of hidden confounders – unmeasured variables influencing multiple observed variables – can systematically distort these relationships, creating false causal links. This poses a significant challenge because distinguishing genuine causal effects from these deceptive patterns requires more than just statistical analysis; it demands a robust approach capable of navigating complexity and accounting for unobserved influences, a feat that proves difficult with conventional techniques.

Many current causal discovery techniques rely on stringent assumptions regarding the underlying data distribution – often demanding linearity, Gaussian noise, or faithfulness – which severely restricts their effectiveness when applied to the complexities of real-world phenomena. These methods frequently falter when faced with non-linear relationships, non-Gaussian errors, or situations where the causal relationships are not fully represented in the observed data. Consequently, findings derived from these approaches can be unreliable or even misleading if the assumed data distribution doesn’t accurately reflect the true generative process. This limitation underscores the need for more flexible and robust methods capable of operating with minimal assumptions, allowing for broader applicability across diverse scientific domains and enabling the identification of causal links in inherently complex systems.

Current causal discovery research confronts a significant hurdle: translating statistical correlations into genuine mechanistic understanding, particularly within complex, high-dimensional datasets. A promising pathway forward involves intelligently incorporating existing domain expertise – what is already known about the system – and augmenting it with the reasoning capabilities of Large Language Models. These LLMs, trained on vast amounts of text, can act as ‘prior knowledge engines’, evaluating potential causal relationships not just on statistical grounds, but also on their plausibility given established scientific principles. By framing causal inference as a knowledge-aware process, researchers aim to move beyond simply identifying associations and instead construct more robust, interpretable, and ultimately useful causal models, even in the presence of unobserved variables and limited data.

Holograph: A Sheaf-Theoretic Formalism for Causal Reasoning

Holograph employs sheaf theory, a mathematical framework dealing with locally defined data and their global relationships, to represent and manage causal beliefs. Specifically, local causal knowledge is formalized as sections on a sheaf defined over a context space. This allows for the representation of beliefs that may vary depending on the specific observational context. The sheaf structure guarantees that these local beliefs are consistent when considered globally; transitions between contexts are governed by restriction maps inherent to the sheaf, ensuring that if a causal claim holds locally, any implications of that claim are also valid in overlapping contexts. This approach allows Holograph to reason about causality across heterogeneous data sources and varying levels of abstraction, maintaining a coherent and logically sound causal model.

Algebraic Latent Projection within Holograph facilitates the transfer of causal state representations between contexts of differing granularity. This projection operates by mapping a high-dimensional causal state \mathbf{x} \in \mathbb{R}^n defined in a larger context to a lower-dimensional representation \mathbf{y} \in \mathbb{R}^m relevant to a smaller context. Critically, this projection incorporates an Absorption Matrix \mathbf{A} \in \mathbb{R}^{n \times m} which models the influence of hidden confounders. The matrix \mathbf{A} effectively filters out components of \mathbf{x} that are not directly relevant to the smaller context, preventing spurious correlations and ensuring that projected causal beliefs are grounded in the available evidence within that specific context. The resulting projection \mathbf{y} = \mathbf{A}^T \mathbf{x} thus represents a context-aware causal state, accounting for potential confounding variables that may not be directly observable.

Holograph utilizes Large Language Models (LLMs) through the SGLang framework to enhance causal reasoning by incorporating external knowledge and establishing informed prior beliefs. SGLang facilitates the translation of natural language queries into structured causal queries, allowing the LLM to generate initial probabilistic assumptions about variable relationships. These LLM-generated priors are then integrated into the sheaf-theoretic framework as starting points for causal discovery, effectively guiding the search for valid causal structures and reducing the computational burden of exploring all possible configurations. The LLM’s capacity for knowledge retrieval and reasoning allows Holograph to leverage a broader context than solely the observed data, improving the robustness and accuracy of causal inferences, particularly in situations with limited data or complex relationships.

Optimization and Validation: Ensuring the Rigor of Causal Inference

The Holograph framework employs Natural Gradient Descent (NGD) for optimizing the belief parameters within its causal inference process. Unlike Stochastic Gradient Descent (SGD), which updates parameters based on first-order gradients, NGD utilizes the Fisher information matrix to pre-condition the gradient, effectively accounting for the curvature of the loss landscape. This adaptation results in faster convergence during training, as demonstrated by ablations comparing Holograph’s performance with equivalent models using SGD. Specifically, the use of NGD allows for larger, more stable parameter updates, accelerating the learning process and improving the efficiency of causal graph discovery. The Fisher information matrix is computed based on the expected curvature of the loss function with respect to the parameters, providing a more informed direction for optimization.

Spectral Regularization is implemented within the Holograph framework to address potential numerical instability arising during the projection step required to construct the Absorption Matrix. This technique adds a penalty term to the optimization objective proportional to the spectral norm of the matrix, effectively constraining its singular values and preventing them from becoming excessively large or small. By limiting the condition number of the Absorption Matrix, Spectral Regularization ensures that the inversion process remains stable and accurate, mitigating the risk of introducing significant errors into the learned causal graph. This is particularly important when dealing with high-dimensional data or complex causal structures where ill-conditioned matrices are more likely to occur.

The imposition of an Acyclicity Constraint is critical to the validity of the causal graph generated by the Holograph framework. This constraint operates by penalizing the presence of directed cycles within the graph’s adjacency matrix during the optimization process. A directed cycle indicates a feedback loop, implying that a variable can directly or indirectly influence itself, which violates the fundamental assumption of causal inference – that causes precede effects. By preventing cycles, the constraint ensures the resulting graph represents a Directed Acyclic Graph (DAG), a necessary condition for a coherent and interpretable causal model. This is achieved through modifications to the loss function, effectively discouraging configurations that would introduce cyclical dependencies between variables.

Performance validation of the Holograph framework was conducted through comparative analysis with existing causal discovery algorithms, specifically NOTEARS and Democritus. Results on the Sachs dataset indicate a 27% reduction in Structural Hamming Distance (SHD), a metric quantifying graph difference, when using Holograph. Furthermore, evaluation on Erdős-Rényi (ER) graphs demonstrated an 82% increase in the number of identifiable queries – representing accurately recovered causal relationships – compared to the NOTEARS algorithm. These results suggest Holograph achieves improved accuracy in causal graph estimation across benchmark datasets.

Beyond Locality: Unexpected Departures from Causal Norms

Recent experimentation utilizing the Holograph framework has revealed a significant departure from the principle of causal locality. Specifically, analyses demonstrate a systematic failure of the Locality axiom, a foundational tenet in causal inference stating that a variable is only directly influenced by its immediate neighbors. This breakdown suggests information isn’t simply propagating through direct connections, but is instead exhibiting non-local effects, influencing variables beyond the scope of immediate adjacency. The observed failures aren’t random errors; they represent a consistent pattern, indicating that standard causal models may be inadequate for representing systems where distant variables can exert influence. This discovery necessitates a re-evaluation of how causal relationships are defined and modeled, potentially unlocking a deeper understanding of complex systems where distant interactions play a crucial role.

The conventional understanding of causality often posits that an effect is directly linked to its immediate causes – a principle known as causal locality. However, recent investigations reveal a more nuanced reality, suggesting that causal influences can propagate beyond these immediate neighbors. This challenges the traditional framework by demonstrating that a variable’s state isn’t solely determined by its direct connections, but can be subtly shaped by more distant elements within a complex system. This extended reach of causal influence implies that identifying true causal relationships requires accounting for these non-local interactions, potentially necessitating a re-evaluation of existing causal inference methods and models to accurately capture the full spectrum of dependencies at play. The implications extend to fields reliant on understanding complex systems, from predicting climate patterns to modeling social networks, where acknowledging these extended causal links could drastically improve predictive accuracy and insight.

The identification of a Locality Failure – a breakdown in the expected constraint that causal influence is limited to immediately neighboring variables – relied heavily on the query generation capabilities of the DeepSeek-V3.2-Exp Large Language Model. This model wasn’t simply used for data analysis; it actively constructed the precise questions needed to probe the Holograph system for deviations from established causal norms. By autonomously formulating complex queries, the LLM bypassed potential human biases and explored a vast search space of causal relationships, ultimately revealing patterns that might have otherwise remained hidden. This demonstrates a significant advancement in utilizing LLMs not just as analytical tools, but as active participants in scientific discovery, capable of uncovering unexpected insights and challenging fundamental assumptions about how systems operate.

Investigations into the observed Locality Failure reveal a systematic relationship between the size of the causal graph and the magnitude of the error, scaling approximately as \mathcal{O}(\sqrt{n}). This means that as the complexity of the system-represented by the number of variables, n-increases, the violations of local causal assumptions grow at a rate proportional to the square root of that complexity. This scaling behavior suggests that traditional models of causal inference, which often rely heavily on locality, may become increasingly unreliable when applied to larger, more interconnected systems. Consequently, the findings underscore the necessity for developing more sophisticated and flexible frameworks for causal reasoning-ones capable of accommodating non-local information propagation and mitigating the errors inherent in assuming strict causal locality.

The pursuit within Holograph, of aligning Large Language Model priors through sheaf-theoretic means, echoes a fundamental tenet of mathematical rigor. The framework’s emphasis on consistency and addressing latent variables isn’t merely about improving causal discovery; it’s about establishing invariants as the complexity-let N approach infinity-increases. Donald Davies aptly stated, “The only thing that really matters is correctness.” Holograph, by grounding its methodology in sheaf theory, aims for precisely that – a provable, consistent system for unraveling causality, rather than one reliant on empirical observation alone. This focus on fundamental principles distinguishes it from approaches that prioritize scalable approximations over mathematical certainty.

What Lies Ahead?

The framework presented exposes, with uncomfortable clarity, the inherent fragility of local causal inference. To assume a consistent causal structure discoverable through iterative probing-even with the expressive power of large language models-proves demonstrably naive. The sheaf-theoretic alignment, while providing a rigorous formalism, does not solve the problem of latent variables; it merely reveals the precise nature of the information lost when attempting to reconstruct a global causal graph from local observations. A proof of impossibility is, naturally, more valuable than a thousand empirically successful approximations.

Future work must confront the limitations exposed. The reliance on LLM priors, while currently pragmatic, introduces an inductive bias that requires careful justification. A truly robust causal discovery system cannot depend on the serendipitous alignment of a pre-trained model’s internal representations with the underlying data generating process. Formalizing a notion of ‘minimal sufficient knowledge’-a foundational axiomatic base independent of data-driven heuristics-remains a critical challenge.

Ultimately, the question is not whether a causal graph can be discovered, but whether it is even well-defined. The framework suggests that many real-world systems may be fundamentally non-Markovian, defying any attempt to represent their causal dependencies with a static graphical model. Perhaps the pursuit of a single, comprehensive causal explanation is a category error, and a more fruitful approach lies in embracing a multiplicity of local, context-dependent causal models.

Original article: https://arxiv.org/pdf/2512.24478.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2026-01-03 02:49