Mathematical Analysis of Hallucination Dynamics in Large Language Models: Uncertainty Quantification, Advanced Decoding, and Principled Mitigation

TL;DR

This paper introduces a mathematical framework to understand, quantify, and reduce hallucinations in large language models by leveraging probabilistic, information-theoretic, and signal analysis techniques, leading to improved safety and reliability.

Contribution

It provides a novel, mathematically grounded approach to analyze hallucination dynamics, proposes advanced uncertainty metrics, and develops principled mitigation strategies for large language models.

Findings

Refined uncertainty metrics including semantic and phase-aware variants.

Effective mitigation strategies such as contrastive decoding and retrieval-augmented grounding.

Unified framework connecting calibration, retrieval, and alignment techniques.

Abstract

Large Language Models (LLMs) are powerful linguistic engines but remain susceptible to hallucinations: plausible-sounding outputs that are factually incorrect or unsupported. In this work, we present a mathematically grounded framework to understand, measure, and mitigate these hallucinations. Drawing on probabilistic modeling, information theory, trigonometric signal analysis, and Bayesian uncertainty estimation, we analyze how errors compound autoregressively, propose refined uncertainty metrics, including semantic and phase-aware variants, and develop principled mitigation strategies such as contrastive decoding, retrieval-augmented grounding, factual alignment, and abstention. This unified lens connects recent advances in calibration, retrieval, and alignment to support safer and more reliable LLMs.