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

Kangkang Sun, Jun Wu, Jianhua Li, Minyi Guo, Xiuzhen Che, Jianwei Huang

Abstract

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 \textit{TriviaQA} and \textit{SQuAD} 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.

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

Abstract

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 \textit{TriviaQA} and \textit{SQuAD} 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.

Paper Structure

This paper contains 41 sections, 3 theorems, 15 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The measurement gap.
  3. Collaborative Entropy.
  4. Contributions.
  5. Related Work
  6. Methodology
  7. System Overview
  8. Semantic Entropy: Single-Model Uncertainty Baseline
  9. Collaborative Entropy: Decomposing System-Level Uncertainty
  10. Intra-model aleatoric uncertainty $\mathcal{U}_{\mathcal{A}}$.
  11. Inter-model epistemic uncertainty $\mathcal{U}_{\mathcal{E}}$.
  12. Collaborative Entropy (CoE).
  13. CoE Decomposition: The Four Uncertainty Quadrants
  14. Experiments
  15. Experimental Setup
  16. ...and 26 more sections

Key Result

Theorem 1

For any input $x$ and model set $\mathcal{M} = \{\mathcal{M}_1, \dots, \mathcal{M}_K\}$, $\mathcal{U}_{CoE}(\mathcal{K}) \geq 0$.

Figures (7)

  • Figure 1: Overview of the proposed Collaborative Entropy (CoE) framework for agentic multi-LLM uncertainty quantification. Each LLM generates candidate responses that are clustered semantically to form probability distributions, which are then aggregated to minimize system-level collaborative entropy.
  • Figure 2: Visualization of the four CoE quadrants using Dirichlet distributions over three semantic clusters. Each simplex (colour intensity = probability density) and accompanying bar chart illustrates a characteristic combination of $\mathcal{U}_{\mathcal{A}}$ (horizontal axis) and $\mathcal{U}_{\mathcal{E}}$ (vertical axis).
  • Figure 3: Performance comparison of CoE with different divergence metrics on TriviaQA dataset (200 samples, KL divergence).
  • Figure 4: Performance comparison of CoE with different divergence metrics on SQuAD dataset (200 samples, KL divergence).
  • Figure 5: Robustness of the CoE-reduction algorithm to the convergence threshold $\epsilon$ (Llama + Qwen ensemble on TriviaQA, 200 shots).
  • ...and 2 more figures

Theorems & Definitions (12)

  • Definition 1: Semantic Entropy, farquhar2024detecting
  • Definition 2: Collaborative Entropy
  • Theorem 1: Non-Negativity
  • proof : Proof of Theorem \ref{['th: non-negativity']} (Non-Negativity)
  • Theorem 2: Zero Value Certainty
  • proof : Proof of Theorem \ref{['th: zero value certainty']} (Zero Value Certainty)
  • Remark 1
  • Definition 3: Delta Distribution
  • Remark 2
  • Theorem 3: Delta-Distribution Behaviour
  • ...and 2 more