# CoE: Collaborative Entropy for Uncertainty Quantification in Agentic Multi-LLM Systems

arXiv: https://arxiv.org/abs/2603.28360 · HTML: https://mlsanity.com/paper/2603.28360/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in General LLM agent frameworks and benchmarks, one of 16 directions in LLM agents and tool use.
- Builds on 32 of its 42 references that are in the map.
- First posted on arXiv 30 Mar 2026.

## Placed in

- LLM agents and tool use / [General LLM agent frameworks and benchmarks](https://mlsanity.com/agents/general-llm-agent-frameworks-and-benchmarks/)

## Builds on (most cited first, 30 of 32)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [SQuAD: 100,000+ Questions for Machine Comprehension of Text](https://mlsanity.com/paper/1606.05250/) (2016)
- [Self-Refine: Iterative Refinement with Self-Feedback](https://mlsanity.com/paper/2303.17651/) (2023)
- [TriviaQA: A Large Scale Distantly Supervised Challenge Dataset for Reading Comprehension](https://mlsanity.com/paper/1705.03551/) (2017)
- [The Curious Case of Neural Text Degeneration](https://mlsanity.com/paper/1904.09751/) (2019)
- [Large Language Models Encode Clinical Knowledge](https://mlsanity.com/paper/2212.13138/) (2022)
- [Mixtral of Experts](https://mlsanity.com/paper/2401.04088/) (2024)
- [Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation](https://mlsanity.com/paper/2302.09664/) (2023)
- [ChatEval: Towards Better LLM-based Evaluators through Multi-Agent Debate](https://mlsanity.com/paper/2308.07201/) (2023)
- [Why Do Multi-Agent LLM Systems Fail?](https://mlsanity.com/paper/2503.13657/) (2025)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [A Survey of Confidence Estimation and Calibration in Large Language Models](https://mlsanity.com/paper/2311.08298/) (2023)
- [A Survey on Mixture of Experts in Large Language Models](https://mlsanity.com/paper/2407.06204/) (2024)
- [Fact-Checking the Output of Large Language Models via Token-Level Uncertainty Quantification](https://mlsanity.com/paper/2403.04696/) (2024)
- [Rethinking the Bounds of LLM Reasoning: Are Multi-Agent Discussions the Key?](https://mlsanity.com/paper/2402.18272/) (2024)
- [Theory of Mind for Multi-Agent Collaboration via Large Language Models](https://mlsanity.com/paper/2310.10701/) (2023)
- [LM-Polygraph: Uncertainty Estimation for Language Models](https://mlsanity.com/paper/2311.07383/) (2023)
- [A Survey on Uncertainty Quantification of Large Language Models: Taxonomy, Open Research Challenges, and Future Directions](https://mlsanity.com/paper/2412.05563/) (2024)
- [DEUP: Direct Epistemic Uncertainty Prediction](https://mlsanity.com/paper/2102.08501/) (2021)
- [Why think step by step? Reasoning emerges from the locality of experience](https://mlsanity.com/paper/2304.03843/) (2023)
- [Cooperation, Competition, and Maliciousness: LLM-Stakeholders Interactive Negotiation](https://mlsanity.com/paper/2309.17234/) (2023)
- [A Closer Look at AUROC and AUPRC under Class Imbalance](https://mlsanity.com/paper/2401.06091/) (2024)
- [Uncertainty Quantification for In-Context Learning of Large Language Models](https://mlsanity.com/paper/2402.10189/) (2024)
- [MAQA: Evaluating Uncertainty Quantification in LLMs Regarding Data Uncertainty](https://mlsanity.com/paper/2408.06816/) (2024)
- [Understanding the Uncertainty of LLM Explanations: A Perspective Based on Reasoning Topology](https://mlsanity.com/paper/2502.17026/) (2025)
- [Combining Confidence Elicitation and Sample-based Methods for Uncertainty Quantification in Misinformation Mitigation](https://mlsanity.com/paper/2401.08694/) (2024)
- [Collaborative Inference and Learning between Edge SLMs and Cloud LLMs: A Survey of Algorithms, Execution, and Open Challenges](https://mlsanity.com/paper/2507.16731/) (2025)
- [Toward Edge General Intelligence with Multiple-Large Language Model (Multi-LLM): Architecture, Trust, and Orchestration](https://mlsanity.com/paper/2507.00672/) (2025)
- [EVINCE: Optimizing Multi-LLM Dialogues Using Conditional Statistics and Information Theory](https://mlsanity.com/paper/2408.14575/) (2024)

## Abstract (arXiv)

Uncertainty estimation in multi-LLM systems remains largely single-model-centric: existing methods quantify uncertainty within each model but do not adequately capture semantic disagreement across models. To address this gap, we propose Collaborative Entropy (CoE), a unified information-theoretic metric for semantic uncertainty in multi-LLM collaboration. CoE is defined on a shared semantic cluster space and combines two components: intra-model semantic entropy and inter-model divergence to the ensemble mean. CoE is not a weighted ensemble predictor; it is a system-level uncertainty measure that characterizes collaborative confidence and disagreement. We analyze several core properties of CoE, including non-negativity, zero-value certainty under perfect semantic consensus, and the behavior of CoE when individual models collapse to delta distributions. These results clarify when reducing per-model uncertainty is sufficient and when residual inter-model disagreement remains. We also present a simple CoE-guided, training-free post-hoc coordination heuristic as a practical application of the metric. Experiments on textitTriviaQA and textitSQuAD with LLaMA-3.1-8B-Instruct, Qwen-2.5-7B-Instruct, and Mistral-7B-Instruct show that CoE provides stronger uncertainty estimation than standard entropy- and divergence-based baselines, with gains becoming larger as additional heterogeneous models are introduced. Overall, CoE offers a useful uncertainty-aware perspective on multi-LLM collaboration.