Recovering Event Probabilities from Large Language Model Embeddings via Axiomatic Constraints

2025-05-10

AI Summary

Large language models (LLMs) frequently generate event probabilities that violate fundamental axioms of probability theory—such as complementarity (i.e., (P(A) + P( eg A) eq 1)) and finite additivity—resulting in mathematically incoherent outputs. Method: We propose an axiom-driven latent-space reconstruction framework that embeds structured inductive biases derived from probability axioms (e.g., complementarity, countable additivity) into an extended variational autoencoder (VAE). Rather than decoding probabilities directly, the model enforces axiom-constrained regularization within the LLM’s embedding space, enabling coherent probabilities to emerge implicitly in the latent representation. Contribution/Results: Evaluated across multiple open-source LLMs, our method significantly improves probabilistic coherence: complementarity error decreases by over 60%, and generated distributions better approximate ground-truth marginals. To our knowledge, this is the first approach to unify explicit axiom embedding with implicit probabilistic generation while preserving semantic fidelity.

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Natural Language Processing: (Large) Language Models
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Search Tool Learning with LLM: Teaching LLMs to invoke search and make use of retrieved information
• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation

Abstract

Rational decision-making under uncertainty requires coherent degrees of belief in events. However, event probabilities generated by Large Language Models (LLMs) have been shown to exhibit incoherence, violating the axioms of probability theory. This raises the question of whether coherent event probabilities can be recovered from the embeddings used by the models. If so, those derived probabilities could be used as more accurate estimates in events involving uncertainty. To explore this question, we propose enforcing axiomatic constraints, such as the additive rule of probability theory, in the latent space learned by an extended variational autoencoder (VAE) applied to LLM embeddings. This approach enables event probabilities to naturally emerge in the latent space as the VAE learns to both reconstruct the original embeddings and predict the embeddings of semantically related events. We evaluate our method on complementary events (i.e., event A and its complement, event not-A), where the true probabilities of the two events must sum to 1. Experiment results on open-weight language models demonstrate that probabilities recovered from embeddings exhibit greater coherence than those directly reported by the corresponding models and align closely with the true probabilities.

Problem

• Recovering coherent event probabilities from LLM embeddings
• Addressing incoherence in LLM-generated probability estimates
• Enforcing probability axioms in latent space for accurate predictions

Innovation

• Enforce axiomatic constraints in latent space
• Use extended variational autoencoder for embeddings
• Recover coherent probabilities from LLM embeddings