# Probability is Not Enough: Exploring and Counting Divergent Tokens for Reasoning Uncertainty Quantification in LLMs

arXiv: https://arxiv.org/abs/2609.38070 · HTML: https://mlsanity.com/paper/2609.38070/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 10 of its 39 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

## Builds on (most cited first, 10 of 10)

- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Gemma 3 Technical Report](https://mlsanity.com/paper/2503.19786/) (2025)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [DeepSeek-V3.2: Pushing the Frontier of Open Large Language Models](https://mlsanity.com/paper/2512.02556/) (2025)
- [From Calibration to Collaboration: LLM Uncertainty Quantification Should Be More Human-Centered](https://mlsanity.com/paper/2506.07461/) (2025)
- [CrossCheckGPT: Universal Hallucination Ranking for Multimodal Foundation Models](https://mlsanity.com/paper/2405.13684/) (2024)
- [Cross-Model Disagreement as a Label-Free Correctness Signal](https://mlsanity.com/paper/2603.25450/) (2026)
- [Let the Model Distribute Its Doubt: Confidence Estimation through Verbalized Probability Distribution](https://mlsanity.com/paper/2511.14275/) (2025)
- [A Systematic Evaluation of Black-Box Uncertainty Estimation Methods for Large Language Models](https://mlsanity.com/paper/2606.19868/) (2026)

## Abstract (arXiv)

As the chain-of-thought reasoning capabilities of large language models improve, evaluating and calibrating their reasoning confidence is becoming increasingly important for quantifying the uncertainty of their answers. Current methods for estimating the confidence of large language models are generally based on probabilities of selected key tokens, but the underlying mechanism remains unclear. Our pilot study finds that replacing selected token probabilities with coarse substitutes can also improve calibration, motivating us to further explore effective signals of model confidence. We introduce Divergent Token Confidence (DTC), a framework that estimates confidence by counting tokens at which two models strongly disagree during decoding. DTC identifies these divergent tokens using the Jensen-Shannon divergence between next-token distributions evaluated along the same reasoning trajectory. We find that their count is almost negatively associated with answer accuracy, thereby serving as a simple yet effective signal for uncertainty quantification. DTC supports both white-box and black-box evaluation using auxiliary models, without explicit training and affecting the generation process. Experiments across multiple model families and six mathematical benchmarks demonstrate improved calibration over probability-based and verbalized baselines. Under white-box evaluation, the count-only estimator achieves an average expected calibration error of 13.0%, compared with 32.7%-42.4% for standard full-sequence confidence methods. In black-box settings, it also improves calibration over the original verbalized scores. For example, mean expected calibration error falls from 32.1%-40.2% to 13.7%-16.3% on DeepSeek-V3.2. These findings provide new insights for improving reasoning uncertainty quantification in large language models. The code is released at https://github.com/szu-tera/DTC.git.