# On the Pitfalls of Verbalized Confidence Priors for Calibrating Large Reasoning Models

arXiv: https://arxiv.org/abs/2609.32470 · HTML: https://mlsanity.com/paper/2609.32470/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 72 of its 82 references that are in the map.
- First posted on arXiv 26 Sep 2026.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)

## Builds on (most cited first, 30 of 72)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [HotpotQA: A Dataset for Diverse, Explainable Multi-hop Question Answering](https://mlsanity.com/paper/1809.09600/) (2018)
- [TriviaQA: A Large Scale Distantly Supervised Challenge Dataset for Reading Comprehension](https://mlsanity.com/paper/1705.03551/) (2017)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Language Models (Mostly) Know What They Know](https://mlsanity.com/paper/2207.05221/) (2022)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [When Not to Trust Language Models: Investigating Effectiveness of Parametric and Non-Parametric Memories](https://mlsanity.com/paper/2212.10511/) (2022)
- [Can LLMs Express Their Uncertainty? An Empirical Evaluation of Confidence Elicitation in LLMs](https://mlsanity.com/paper/2306.13063/) (2023)
- [MuSiQue: Multihop Questions via Single-hop Question Composition](https://mlsanity.com/paper/2108.00573/) (2021)
- [DROP: A Reading Comprehension Benchmark Requiring Discrete Reasoning Over Paragraphs](https://mlsanity.com/paper/1903.00161/) (2019)
- [Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation](https://mlsanity.com/paper/2302.09664/) (2023)
- [Just Ask for Calibration: Strategies for Eliciting Calibrated Confidence Scores from Language Models Fine-Tuned with Human Feedback](https://mlsanity.com/paper/2305.14975/) (2023)
- [Teaching Models to Express Their Uncertainty in Words](https://mlsanity.com/paper/2205.14334/) (2022)
- [Latent Retrieval for Weakly Supervised Open Domain Question Answering](https://mlsanity.com/paper/1906.00300/) (2019)
- [SFT Memorizes, RL Generalizes: A Comparative Study of Foundation Model Post-training](https://mlsanity.com/paper/2501.17161/) (2025)
- [The Internal State of an LLM Knows When It's Lying](https://mlsanity.com/paper/2304.13734/) (2023)
- [The Entropy Mechanism of Reinforcement Learning for Reasoning Language Models](https://mlsanity.com/paper/2505.22617/) (2025)
- [Learn then Test: Calibrating Predictive Algorithms to Achieve Risk Control](https://mlsanity.com/paper/2110.01052/) (2021)
- [On the Global Convergence Rates of Softmax Policy Gradient Methods](https://mlsanity.com/paper/2005.06392/) (2020)
- [LiveBench: A Challenging, Contamination-Limited LLM Benchmark](https://mlsanity.com/paper/2406.19314/) (2024)
- [Gemma 4 Technical Report](https://mlsanity.com/paper/2607.02770/) (2026)
- [A Survey of Confidence Estimation and Calibration in Large Language Models](https://mlsanity.com/paper/2311.08298/) (2023)
- [Beyond Benchmarks: MathArena as an Evaluation Platform for Mathematics with LLMs](https://mlsanity.com/paper/2605.00674/) (2026)
- [Beyond Binary Rewards: Training LMs to Reason About Their Uncertainty](https://mlsanity.com/paper/2507.16806/) (2025)

## Abstract (arXiv)

Large reasoning models (LRMs) often suffer from overconfidence when expressing their uncertainty. Confidence-aware reinforcement learning (RL) offers a promising way to optimize calibration. However, it relies on on-policy rollouts and is thus constrained by the model's pre-RL confidence distribution, which we term confidence prior. In this work, we reveal that off-the-shelf LRMs exhibit a confidence prior heavily concentrated on a few high values, which persists throughout RL. Theoretically, we prove that this concentration suppresses policy gradient updates for rarely sampled confidence values and inflates the lower bound on expected Brier risk. To overcome this exploration bottleneck, we propose CalibSFT, a plug-and-play supervised fine-tuning stage that shapes a calibrated confidence prior with broad support before RL. For each question, CalibSFT constructs confidence targets combining its success rate with response-level correctness, which provably preserves proper-scoring optimality, and then balances training responses across the confidence spectrum to enable diverse confidence exploration during RL. To learn from incorrect responses without imitating their reasoning, CalibSFT introduces correctness-conditional supervision, guiding confidence across all responses while supervising reasoning only on correct ones. Across 16 mathematical and general reasoning benchmarks, incorporating CalibSFT reduces calibration errors and improves discrimination across five representative RL algorithms while preserving comparable accuracy. Furthermore, CalibSFT delivers practical benefits for downstream selective prediction and model routing. Our code is available at https://github.com/ml-stat-Sustech/verbalized-confidence-training.