Fine-Grained Uncertainty Decomposition in Large Language Models: A Spectral Approach

TL;DR

The paper introduces Spectral Uncertainty, a theoretically grounded framework that decomposes total predictive uncertainty in LLMs into aleatoric and epistemic components using kernel-based von Neumann entropy within an RKHS. It establishes a general uncertainty decomposition via functional Bregman information, specializes it to the von Neumann entropy, and provides finite-sum spectral estimators to compute total, aleatoric, and epistemic uncertainty from continuous semantic representations. Through a two-stage sampling workflow that generates input clarifications and model answers, the method achieves state-of-the-art performance on ambiguity detection (aleatoric) and correctness prediction (total uncertainty) across multiple datasets and models, outperforming semantic- and clustering-based baselines. While computationally intensive due to n×m outputs, Spectral Uncertainty offers a principled and practical approach to uncertainty in LLMs with strong implications for reliability and interpretability in real-world AI systems.

Abstract

As Large Language Models (LLMs) are increasingly integrated in diverse applications, obtaining reliable measures of their predictive uncertainty has become critically important. A precise distinction between aleatoric uncertainty, arising from inherent ambiguities within input data, and epistemic uncertainty, originating exclusively from model limitations, is essential to effectively address each uncertainty source. In this paper, we introduce Spectral Uncertainty, a novel approach to quantifying and decomposing uncertainties in LLMs. Leveraging the Von Neumann entropy from quantum information theory, Spectral Uncertainty provides a rigorous theoretical foundation for separating total uncertainty into distinct aleatoric and epistemic components. Unlike existing baseline methods, our approach incorporates a fine-grained representation of semantic similarity, enabling nuanced differentiation among various semantic interpretations in model responses. Empirical evaluations demonstrate that Spectral Uncertainty outperforms state-of-the-art methods in estimating both aleatoric and total uncertainty across diverse models and benchmark datasets.

Figures (14)

• Figure 1: Illustration of our Spectral Uncertainty Decomposition. Given an ambiguous query like “Who’s won the most World Series in baseball?”, an LLM may interpret it in multiple valid ways (e.g., by team or by player), leading to high predictive uncertainty. Unlike existing methods, our spectral decomposition quantifies not just the magnitude but also the source of uncertainty, revealing in this case a dominant aleatoric component rooted in semantic ambiguity.
• Figure 2: Clarification prompt template for AmbigQA.
• Figure 3: Answer generation prompt template for AmbigQA.
• Figure 4: Clarification prompt template for AmbigInst.
• Figure 5: Answer generation prompt template for AmbigInst.

Theorems & Definitions (8)

• Definition 3.1: gruber2023uncertainty
• Theorem 3.2
• Definition 3.3: bach2022information
• Definition 3.4: bach2022information
• Definition 3.5: Kernel-based von Neumann entropy bach2022information
• Corollary 3.6: Spectral uncertainty decomposition
• Proposition 3.7: bach2022information
• proof