# Uncertainty Quantification in Large Language Models Through Convex Hull Analysis

arXiv: https://arxiv.org/abs/2406.19712 · HTML: https://mlsanity.com/paper/2406.19712/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 37 references that are in the map.
- First posted on arXiv 28 Jun 2024.

## Builds on (most cited first, 20 of 20)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Benchmarking LLMs via Uncertainty Quantification](https://mlsanity.com/paper/2401.12794/) (2024)
- [LUQ: Long-text Uncertainty Quantification for LLMs](https://mlsanity.com/paper/2403.20279/) (2024)
- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [A Survey on Large Language Models for Critical Societal Domains: Finance, Healthcare, and Law](https://mlsanity.com/paper/2405.01769/) (2024)
- [(A)I Am Not a Lawyer, But...: Engaging Legal Experts towards Responsible LLM Policies for Legal Advice](https://mlsanity.com/paper/2402.01864/) (2024)
- [Principled Instructions Are All You Need for Questioning LLaMA-1/2, GPT-3.5/4](https://mlsanity.com/paper/2312.16171/) (2023)
- [Uncertainty in Natural Language Processing: Sources, Quantification, and Applications](https://mlsanity.com/paper/2306.04459/) (2023)
- [Uncertainty in Language Models: Assessment through Rank-Calibration](https://mlsanity.com/paper/2404.03163/) (2024)
- [Quantifying Uncertainty in Natural Language Explanations of Large Language Models](https://mlsanity.com/paper/2311.03533/) (2023)
- [Improving the Reliability of Large Language Models by Leveraging Uncertainty-Aware In-Context Learning](https://mlsanity.com/paper/2310.04782/) (2023)
- [Uncertainty Quantification in Machine Learning for Engineering Design and Health Prognostics: A Tutorial](https://mlsanity.com/paper/2305.04933/) (2023)
- [Calibration in Machine Learning Uncertainty Quantification: beyond consistency to target adaptivity](https://mlsanity.com/paper/2309.06240/) (2023)
- [Scaling Up Bayesian Uncertainty Quantification for Inverse Problems using Deep Neural Networks](https://mlsanity.com/paper/2101.03906/) (2021)
- [On the Importance of Uncertainty in Decision-Making with Large Language Models](https://mlsanity.com/paper/2404.02649/) (2024)
- [AI as Decision-Maker: Ethics and Risk Preferences of LLMs](https://mlsanity.com/paper/2406.01168/) (2024)
- [Approaches for Uncertainty Quantification of AI-predicted Material Properties: A Comparison](https://mlsanity.com/paper/2310.13136/) (2023)

## Built on by (most cited first, 13 of 13)

- [Evaluating Uncertainty and Quality of Vision-Language-Action-enabled Robots](https://mlsanity.com/paper/2507.17049/) (2025)
- [Geometric Uncertainty for Detecting and Correcting Hallucinations in LLMs](https://mlsanity.com/paper/2509.13813/) (2025)
- [Every Response Counts: Quantifying Uncertainty of LLM-based Multi-Agent Systems through Tensor Decomposition](https://mlsanity.com/paper/2604.08708/) (2026)
- [Comparing Uncertainty Measurement and Mitigation Methods for Large Language Models: A Systematic Review](https://mlsanity.com/paper/2504.18346/) (2025)
- [Improving Medical Diagnostics with Vision-Language Models: Convex Hull-Based Uncertainty Analysis](https://mlsanity.com/paper/2412.00056/) (2024)
- [A Systematic Evaluation of Black-Box Uncertainty Estimation Methods for Large Language Models](https://mlsanity.com/paper/2606.19868/) (2026)
- [MedBayes-Lite: A Clinical Uncertainty Governance Layer for Risk-Aware Medical Decision Support](https://mlsanity.com/paper/2511.16625/) (2025)
- [Georeferencing complex relative locality descriptions with large language models](https://mlsanity.com/paper/2512.14228/) (2025)
- [A Geometric Taxonomy of Hallucinations in LLMs](https://mlsanity.com/paper/2602.13224/) (2026)
- [Surgical Knowledge Rewrite in Compact LLMs: An 'Unlearn-then-Learn' Strategy with (IA3) for Localized Factual Modulation and Catastrophic Forgetting Mitigation](https://mlsanity.com/paper/2508.07075/) (2025)
- [Semantic Grounding Index: Geometric Bounds on Context Engagement in RAG Systems](https://mlsanity.com/paper/2512.13771/) (2025)
- [Can We Trust LLM's Logic? Quantifying Uncertainty, Coherence, and Robustness via a Graph-Based Framework](https://mlsanity.com/paper/2607.08017/) (2026)
- [Revealing Epistemic Uncertainty in MLLMs via Causal-Invariant Masking](https://mlsanity.com/paper/2610.02887/) (2026)

## Abstract (arXiv)

Uncertainty quantification approaches have been more critical in large language models (LLMs), particularly high-risk applications requiring reliable outputs. However, traditional methods for uncertainty quantification, such as probabilistic models and ensemble techniques, face challenges when applied to the complex and high-dimensional nature of LLM-generated outputs. This study proposes a novel geometric approach to uncertainty quantification using convex hull analysis. The proposed method leverages the spatial properties of response embeddings to measure the dispersion and variability of model outputs. The prompts are categorized into three types, i.e., `easy', `moderate', and `confusing', to generate multiple responses using different LLMs at varying temperature settings. The responses are transformed into high-dimensional embeddings via a BERT model and subsequently projected into a two-dimensional space using Principal Component Analysis (PCA). The Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm is utilized to cluster the embeddings and compute the convex hull for each selected cluster. The experimental results indicate that the uncertainty of the model for LLMs depends on the prompt complexity, the model, and the temperature setting.