# From token probabilities to calibrated confidence: An empirical study of mathematical question answering

arXiv: https://arxiv.org/abs/2608.07827 · HTML: https://mlsanity.com/paper/2608.07827/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 37 of its 67 references that are in the map.
- First posted on arXiv 8 Aug 2026.

## Builds on (most cited first, 30 of 37)

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- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
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- [Language Models (Mostly) Know What They Know](https://mlsanity.com/paper/2207.05221/) (2022)
- [Google's Neural Machine Translation System: Bridging the Gap between Human and Machine Translation](https://mlsanity.com/paper/1609.08144/) (2016)
- [Can LLMs Express Their Uncertainty? An Empirical Evaluation of Confidence Elicitation in LLMs](https://mlsanity.com/paper/2306.13063/) (2023)
- [How Can We Know What Language Models Know?](https://mlsanity.com/paper/1911.12543/) (2019)
- [BloombergGPT: A Large Language Model for Finance](https://mlsanity.com/paper/2303.17564/) (2023)
- [Are NLP Models really able to Solve Simple Math Word Problems?](https://mlsanity.com/paper/2103.07191/) (2021)
- [Teaching Models to Express Their Uncertainty in Words](https://mlsanity.com/paper/2205.14334/) (2022)
- [DailyDialog: A Manually Labelled Multi-turn Dialogue Dataset](https://mlsanity.com/paper/1710.03957/) (2017)
- [A Survey of Confidence Estimation and Calibration in Large Language Models](https://mlsanity.com/paper/2311.08298/) (2023)
- [LLMs Know More Than They Show: On the Intrinsic Representation of LLM Hallucinations](https://mlsanity.com/paper/2410.02707/) (2024)
- [Reducing conversational agents' overconfidence through linguistic calibration](https://mlsanity.com/paper/2012.14983/) (2020)
- [LongWriter: Unleashing 10,000+ Word Generation from Long Context LLMs](https://mlsanity.com/paper/2408.07055/) (2024)
- [Language Model Cascades: Token-level uncertainty and beyond](https://mlsanity.com/paper/2404.10136/) (2024)
- [Calibrating Large Language Models with Sample Consistency](https://mlsanity.com/paper/2402.13904/) (2024)
- [Calibrating Large Language Models Using Their Generations Only](https://mlsanity.com/paper/2403.05973/) (2024)
- [Factual Confidence of LLMs: on Reliability and Robustness of Current Estimators](https://mlsanity.com/paper/2406.13415/) (2024)
- [Confidence Regulation Neurons in Language Models](https://mlsanity.com/paper/2406.16254/) (2024)
- [Mind the Confidence Gap: Overconfidence, Calibration, and Distractor Effects in Large Language Models](https://mlsanity.com/paper/2502.11028/) (2025)

## Abstract (arXiv)

Confidence estimation for large language models (LLMs) aims to estimate the probability that a generated answer is correct, while calibration aligns these estimates with empirical accuracy. Prior work has shown that token probabilities are often overconfident, we investigate whether these readily available signals can nevertheless provide well-calibrated confidence estimation for mathematical question answering. We compare single-pass estimators, which reuse token probabilities from the original generation, with multi-pass estimators, which obtain additional confidence signals through verification or stochastic forward passes. While individual token probabilities can be highly saturated, we find that aggregating token probabilities over the full sequence captures small but consistent differences between correct and incorrect generations, yielding more informative confidence estimates. Multi-pass methods can yield calibrated confidence estimates. We study two such approaches: self-verification through re-prompting, including a lower-cost in-situ variant, and Monte Carlo Dropout, which derives confidence from variation across stochastic forward passes. We further evaluate two post-hoc calibration methods, Platt scaling and isotonic regression, both of which substantially reduce in-domain calibration error. However, their data efficiency varies with dataset difficulty, and the calibration mappings often transfer asymmetrically across datasets and models.