Calibrating Verbal Uncertainty as a Linear Feature to Reduce Hallucinations
Paper · 2025 · Ziwei Ji, Lei Yu, Yeskendir Koishekenov et al.
Summary
Large language models often state wrong answers in a confident tone, which leads users to trust them too much. The authors argue that these overconfident hallucinations come from a gap between semantic uncertainty (SU) and verbal uncertainty (VU). SU is how unsure the model actually is about an answer's meaning, measured as Semantic Entropy. VU is how much doubt the wording conveys, scored by an LLM-as-a-Judge. They find that the hidden states of questions answered with hedged versus assertive phrasing are linearly separable. Taking the difference of means between these two groups gives a single Verbal Uncertainty Feature (VUF). The authors use the VUF in two ways. First, a hallucination detector that combines SU and VU outperforms detectors based on SU alone. Second, Mechanistic Uncertainty Calibration (MUC) shifts activations along the VUF at inference time, so the answer's wording reflects the model's real uncertainty. MUC requires no fine-tuning and no extra prompt engineering. In short-form QA, it substantially cuts confident hallucinations. This makes model outputs more honest about knowledge gaps.
Key points
- The VUF is found in three models: Llama-3.1-8B-Instruct, Mistral-7B-Instruct-v0.3 and Qwen2.5-7B-Instruct. It is tested on TriviaQA, NQ-Open and PopQA, and VU labels come from Llama3.1-70B-Instruct acting as judge.
- MUC lowers confident hallucinations by 29.6% on average (relative) and improves VU–SU alignment by 28.4%.
- A VUF extracted from TriviaQA still reduces hallucinations on NQ-Open and PopQA. Random steering vectors of similar magnitude help far less.
- On TriviaQA with Llama3.1, many hallucinated answers combine high semantic entropy with low VU, which is the miscalibration the method targets.
- Limitations: the work covers only short-form QA, and MUC makes wrong answers sound appropriately hedged rather than correcting them. Code is available at github.com/facebookresearch/verbal_uncertainty_feature_calibration.
Institutions
Fields, topics and concepts
📚 Shelves 4The field of science this work belongs to (OpenAlex fields).Computer Science · 18,614 worksArtificial Intelligence · 17,965 worksArts and Humanities · 17,450 worksLanguage and Linguistics · 17,296 works
Cite
Cite as Ziwei Ji, Lei Yu, Yeskendir Koishekenov, Yejin Bang, Anthony Hartshorn, Alan Schelten, Cheng Zhang, Pascale Fung and Nicola Cancedda. 2025. Calibrating Verbal Uncertainty as a Linear Feature to Reduce Hallucinations. Proceedings of the 2025 Conference on Empirical Methods in Natural Language Processing.
@inproceedings{ji2025calibrating,
title = {{Calibrating Verbal Uncertainty as a Linear Feature to Reduce Hallucinations}},
author = {Ziwei Ji, Lei Yu, Yeskendir Koishekenov, Yejin Bang, Anthony Hartshorn, Alan Schelten, Cheng Zhang, Pascale Fung and Nicola Cancedda},
booktitle = {Proceedings of the 2025 Conference on Empirical Methods in Natural Language Processing},
year = {2025},
doi = {10.18653/v1/2025.emnlp-main.187},
url = {https://doi.org/10.18653/v1/2025.emnlp-main.187}
}References
- Andy Arditi, Oscar Obeso, Aaquib Syed, Daniel Paleka, Nina Panickssery, Wes Gurnee, and Neel Nanda. 2024. Refusal in language models is mediated by a single direction. arXiv preprint arXiv:2406.11717.
- Amos Azaria and Tom Mitchell. 2023. The internal state of an llm knows when it’s lying. In Findings of the Association for Computational Linguistics: EMNLP 2023, pages 967–976.
- Neil Band, Xuechen Li, Tengyu Ma, and Tatsunori Hashimoto. 2024. Linguistic calibration of longform generations. In Forty-first International Conference on Machine Learning. find
- Yejin Bang, Samuel Cahyawijaya, Nayeon Lee, Wenliang Dai, Dan Su, Bryan Wilie, Holy Lovenia, Ziwei Ji, Tiezheng Yu, Willy Chung, Quyet V. Do, Yan Xu, and Pascale Fung. 2023. A multitask, multilingual, multimodal evaluation of chatgpt on reasoning, hallucination, and interactivity. AACL.
- Yejin Bang, Ziwei Ji, Alan Schelten, Anthony Hartshorn, Tara Fowler, Cheng Zhang, Nicola Cancedda, and Pascale Fung. 2025. HalluLens: LLM hallucination benchmark. In Proceedings of the 63rd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 24128– 24156, V…
- Nora Belrose. 2023. [link]. find
- Tolga Bolukbasi, Kai-Wei Chang, James Y Zou, Venkatesh Saligrama, and Adam T Kalai. 2016. Man is to computer programmer as woman is to homemaker? debiasing word embeddings. Advances in neural information processing systems, 29.
- Trenton Bricken, Adly Templeton, Joshua Batson, Brian Chen, Adam Jermyn, Tom Conerly, Nick Turner, Cem Anil, Carson Denison, Amanda Askell, Robert Lasenby, Yifan Wu, Shauna Kravec, Nicholas Schiefer, Tim Maxwell, Nicholas Joseph, Zac Hatfield-Dodds, Alex Tamkin, Karina Nguyen, Brayden McLean, Josiah… find
- Collin Burns, Haotian Ye, Dan Klein, and Jacob Steinhardt. 2023. Discovering latent knowledge in language models without supervision. In The Eleventh International Conference on Learning Representations. find
- Chao Chen, Kai Liu, Ze Chen, Yi Gu, Yue Wu, Mingyuan Tao, Zhihang Fu, and Jieping Ye. 2024. INSIDE: LLMs’ internal states retain the power of hallucination detection. In The Twelfth International Conference on Learning Representations.
- Qinyuan Cheng, Tianxiang Sun, Xiangyang Liu, Wenwei Zhang, Zhangyue Yin, Shimin Li, Linyang Li, Zhengfu He, Kai Chen, and Xipeng Qiu. 2024. Can ai assistants know what they don’t know? arXiv preprint arXiv:2401.13275. — [arXiv:2401.13275](https://arxiv.org/abs/2401.13275) find
- Yung-Sung Chuang, Yujia Xie, Hongyin Luo, Yoon Kim, James Glass, and Pengcheng He. 2023. Dola: Decoding by contrasting layers improves factuality in large language models. arXiv preprint arXiv:2309.03883.
- Nico Daheim, Nouha Dziri, Mrinmaya Sachan, Iryna Gurevych, and Edoardo M Ponti. 2023. Elastic weight removal for faithful and abstractive dialogue generation. arXiv preprint arXiv:2303.17574. find
- Shehzaad Dhuliawala, Mojtaba Komeili, Jing Xu, Roberta Raileanu, Xian Li, Asli Celikyilmaz, and Jason Weston. 2023. Chain-of-verification reduces hallucination in large language models. arXiv preprint arXiv:2309.11495.
- Jinhao Duan, Hao Cheng, Shiqi Wang, Alex Zavalny, Chenan Wang, Renjing Xu, Bhavya Kailkhura, and Kaidi Xu. 2024. Shifting attention to relevance: Towards the predictive uncertainty quantification of free-form large language models. In Proceedings of the 62nd Annual Meeting of the Association for Com… find
- Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al-Dahle, Aiesha Letman, Akhil Mathur, Alan Schelten, Amy Yang, Angela Fan, et al. 2024. The llama 3 herd of models. arXiv preprint arXiv:2407.21783.
- Nelson Elhage, Tristan Hume, Catherine Olsson, Nicholas Schiefer, Tom Henighan, Shauna Kravec, Zac Hatfield-Dodds, Robert Lasenby, Dawn Drain, Carol Chen, et al. 2022. Toy models of superposition. arXiv preprint arXiv:2209.10652. find
- Wade Fagen-Ulmschneider. 2023. Perception of probability words. Ms., UIUC. find
- Sebastian Farquhar, Jannik Kossen, Lorenz Kuhn, and Yarin Gal. 2024. Detecting hallucinations in large language models using semantic entropy. Nature, 630(8017):625–630. find
- Shangbin Feng, Weijia Shi, Yike Wang, Wenxuan Ding, Vidhisha Balachandran, and Yulia Tsvetkov. 2024. Don‘t hallucinate, abstain: Identifying LLM knowledge gaps via multi-LLM collaboration. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Pape… find
- Javier Ferrando, Oscar Obeso, Senthooran Rajamanoharan, and Neel Nanda. 2024. Do i know this entity? knowledge awareness and hallucinations in language models. arXiv preprint arXiv:2411.14257. — [arXiv:2411.14257](https://arxiv.org/abs/2411.14257) find
- Xiang Gao, Jiaxin Zhang, Lalla Mouatadid, and Kamalika Das. 2024. SPUQ: Perturbation-based uncertainty quantification for large language models. In Proceedings of the 18th Conference of the European Chapter of the Association for Computational Linguistics (Volume 1: Long Papers), pages 2336–2346, St… find
- Yuzhe Gu, Ziwei Ji, Wenwei Zhang, Chengqi Lyu, Dahua Lin, and Kai Chen. 2024. Anah-v2: Scaling analytical hallucination annotation of large language models. NeurIPS, abs/2407.04693. — [arXiv:2407.04693](https://arxiv.org/abs/2407.04693) find
- Hsiu-Yuan Huang, Yutong Yang, Zhaoxi Zhang, Sanwoo Lee, and Yunfang Wu. 2024. A survey of uncertainty estimation in llms: Theory meets practice. arXiv preprint arXiv:2410.15326. — [arXiv:2410.15326](https://arxiv.org/abs/2410.15326) find
- Ziwei Ji, Delong Chen, Etsuko Ishii, Samuel Cahyawijaya, Yejin Bang, Bryan Wilie, and Pascale Fung. Llm internal states reveal hallucination risk faced with a query. In The 7th BlackboxNLP Workshop.
- Ziwei Ji, Yuzhe Gu, Wenwei Zhang, Chengqi Lyu, Dahua Lin, and Kai Chen. 2024. Anah: Analytical annotation of hallucinations in large language models. arXiv preprint arXiv:2405.20315. find
- Ziwei Ji, Nayeon Lee, Rita Frieske, Tiezheng Yu, Dan Su, Yan Xu, Etsuko Ishii, Yejin Bang, Andrea Madotto, and Pascale Fung. 2022. Survey of hallucination in natural language generation. ACM Computing Surveys.
- Ziwei Ji, Zihan Liu, Nayeon Lee, Tiezheng Yu, Bryan Wilie, Min Zeng, and Pascale Fung. 2023a. RHO: Reducing hallucination in open-domain dialogues with knowledge grounding. In Findings of the Association for Computational Linguistics: ACL 2023, pages 4504–4522, Toronto, Canada. Association for Compu… find
- Ziwei Ji, Tiezheng Yu, Yan Xu, Nayeon Lee, Etsuko Ishii, and Pascale Fung. 2023b. Towards mitigating hallucination in large language models via self-reflection. EMNLP Findings.
- Albert Q Jiang, Alexandre Sablayrolles, Arthur Mensch, Chris Bamford, Devendra Singh Chaplot, Diego de las Casas, Florian Bressand, Gianna Lengyel, Guillaume Lample, Lucile Saulnier, et al. 2023. Mistral 7b. arXiv preprint arXiv:2310.06825. find
- Mandar Joshi, Eunsol Choi, Daniel S Weld, and Luke Zettlemoyer. 2017. Triviaqa: A large scale distantly supervised challenge dataset for reading comprehension. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 1601–1611.
- Saurav Kadavath, Tom Conerly, Amanda Askell, Tom Henighan, Dawn Drain, Ethan Perez, Nicholas Schiefer, Zac Hatfield-Dodds, Nova DasSarma, Eli Tran-Johnson, et al. 2022. Language models (mostly) know what they know. arXiv preprint arXiv:2207.05221. find
- Sunnie S. Y. Kim, Q. Vera Liao, Mihaela Vorvoreanu, Stephanie Ballard, and Jennifer Wortman Vaughan. 2024. "i’m not sure, but...": Examining the impact of large language models’ uncertainty expression on user reliance and trust. In Proceedings of the 2024 ACM Conference on Fairness, Accountability,… find
- Jannik Kossen, Jiatong Han, Muhammed Razzak, Lisa Schut, Shreshth Malik, and Yarin Gal. 2024. Semantic entropy probes: Robust and cheap hallucination detection in llms. arXiv preprint arXiv:2406.15927. — [arXiv:2406.15927](https://arxiv.org/abs/2406.15927) find
- Lorenz Kuhn, Yarin Gal, and Sebastian Farquhar. 2023. Semantic uncertainty: Linguistic invariances for uncertainty estimation in natural language generation. arXiv preprint arXiv:2302.09664.
- Tom Kwiatkowski, Jennimaria Palomaki, Olivia Redfield, Michael Collins, Ankur Parikh, Chris Alberti, Danielle Epstein, Illia Polosukhin, Jacob Devlin, Kenton Lee, et al. 2019. Natural questions: a benchmark for question answering research. Transactions of the Association for Computational Linguistic…
- Abhika Mishra, Akari Asai, Vidhisha Balachandran, Yizhong Wang, Graham Neubig, Yulia Tsvetkov, and Hannaneh Hajishirzi. 2024. Fine-grained hallucination detection and editing for language models. arXiv preprint arXiv:2401.06855.
- Chankyu Lee, Rajarshi Roy, Mengyao Xu, Jonathan Raiman, Mohammad Shoeybi, Bryan Catanzaro, and Wei Ping. 2024. Nv-embed: Improved techniques for training llms as generalist embedding models. arXiv preprint arXiv:2405.17428.
- Dor Muhlgay, Ori Ram, Inbal Magar, Yoav Levine, Nir Ratner, Yonatan Belinkov, Omri Abend, Kevin Leyton-Brown, Amnon Shashua, and Yoav Shoham. 2024. Generating benchmarks for factuality evaluation of language models. In Proceedings of the 18th Conference of the European Chapter of the Association for…
- Deren Lei, Yaxi Li, Mingyu Wang, Vincent Yun, Emily Ching, Eslam Kamal, et al. 2023. Chain of natural language inference for reducing large language model ungrounded hallucinations. arXiv preprint arXiv:2310.03951. — [arXiv:2310.03951](https://arxiv.org/abs/2310.03951) find
- Junyi Li, Xiaoxue Cheng, Wayne Xin Zhao, Jian-Yun Nie, and Ji-Rong Wen. 2023a. Halueval: A large-scale hallucination evaluation benchmark for large language models. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, pages 6449–6464.
- Kenneth Li, Oam Patel, Fernanda Viégas, Hanspeter Pfister, and Martin Wattenberg. 2023b. Inference-time intervention: Eliciting truthful answers from a language model. arXiv preprint arXiv:2306.03341.
- Kenneth Li, Oam Patel, Fernanda Viégas, Hanspeter Pfister, and Martin Wattenberg. 2024. Inference-time intervention: Eliciting truthful answers from a language model. Advances in Neural Information Processing Systems, 36.
- Stephanie Lin, Jacob Hilton, and Owain Evans. 2022. Teaching models to express their uncertainty in words. arXiv preprint arXiv:2205.14334. find
- Andrey Malinin and Mark Gales. 2020. Uncertainty estimation in autoregressive structured prediction. arXiv preprint arXiv:2002.07650. find
- Alex Mallen, Akari Asai, Victor Zhong, Rajarshi Das, Hannaneh Hajishirzi, and Daniel Khashabi. 2022. When not to trust language models: Investigating effectiveness and limitations of parametric and non-parametric memories. arXiv preprint. find
- Samuel Marks and Max Tegmark. 2023. The geometry of truth: Emergent linear structure in large language model representations of true/false datasets. arXiv preprint arXiv:2310.06824.
- Sabrina J. Mielke, Arthur Szlam, Emily Dinan, and Y-Lan Boureau. 2022. Reducing conversational agents’ overconfidence through linguistic calibration. Transactions of the Association for Computational Linguistics, 10:857–872. find
- Tomáš Mikolov, Wen-tau Yih, and Geoffrey Zweig. 2013. Linguistic regularities in continuous space word representations. In Proceedings of the 2013 conference of the north american chapter of the association for computational linguistics: Human language technologies, pages 746–751.
- Hadas Orgad, Michael Toker, Zorik Gekhman, Roi Reichart, Idan Szpektor, Hadas Kotek, and Yonatan Belinkov. 2024. Llms know more than they show: On the intrinsic representation of llm hallucinations. arXiv preprint arXiv:2410.02707.
- Nina Panickssery, Nick Gabrieli, Julian Schulz, Meg Tong, Evan Hubinger, and Alexander Matt Turner. 2023. Steering llama 2 via contrastive activation addition. arXiv preprint arXiv:2312.06681. find
- Kiho Park, Yo Joong Choe, and Victor Veitch. 2023. The linear representation hypothesis and the geometry of large language models. arXiv preprint arXiv:2311.03658.
- Clément Rebuffel, Marco Roberti, Laure Soulier, Geoffrey Scoutheeten, Rossella Cancelliere, and Patrick Gallinari. 2022. Controlling hallucinations at word level in data-to-text generation. Data Mining and Knowledge Discovery, 36(1):318–354. find
- Weijia Shi, Xiaochuang Han, Mike Lewis, Yulia Tsvetkov, Luke Zettlemoyer, and Scott Wen-tau Yih. 2023. Trusting your evidence: Hallucinate less with context-aware decoding. arXiv preprint arXiv:2305.14739.
- Ben Snyder, Marius Moisescu, and Muhammad Bilal Zafar. 2023. On early detection of hallucinations in factual question answering. arXiv preprint arXiv:2312.14183. — [arXiv:2312.14183](https://arxiv.org/abs/2312.14183) find
- Elias Stengel-Eskin, Peter Hase, and Mohit Bansal. 2024. Lacie: Listener-aware finetuning for confidence calibration in large language models. arXiv preprint arXiv:2405.21028. find
- Curt Tigges, Oskar John Hollinsworth, Atticus Geiger, and Neel Nanda. 2023. Linear representations of sentiment in large language models. arXiv preprint arXiv:2310.15154.
- Christian Tomani, Kamalika Chaudhuri, Ivan Evtimov, Daniel Cremers, and Mark Ibrahim. 2024. Uncertainty-based abstention in llms improves safety and reduces hallucinations. arXiv preprint arXiv:2404.10960. — [arXiv:2404.10960](https://arxiv.org/abs/2404.10960) find
- Neeraj Varshney, Wenlin Yao, Hongming Zhang, Jianshu Chen, and Dong Yu. 2023. A stitch in time saves nine: Detecting and mitigating hallucinations of llms by validating low-confidence generation. arXiv preprint arXiv:2307.03987.
- Xuezhi Wang, Jason Wei, Dale Schuurmans, Quoc V Le, Ed H Chi, Sharan Narang, Aakanksha Chowdhery, and Denny Zhou. Self-consistency improves chain of thought reasoning in language models. In The Eleventh International Conference on Learning Representations.
- Zhenhailong Wang, Shaoguang Mao, Wenshan Wu, Tao Ge, Furu Wei, and Heng Ji. 2023. Unleashing cognitive synergy in large language models: A task-solving agent through multi-persona selfcollaboration. arXiv preprint arXiv:2307.05300, 1(2):3.
- Yotam Wolf, Noam Wies, Dorin Shteyman, Binyamin Rothberg, Yoav Levine, and Amnon Shashua. 2024. Tradeoffs between alignment and helpfulness in language models. arXiv preprint arXiv:2401.16332. — [arXiv:2401.16332](https://arxiv.org/abs/2401.16332) find
- Zhengxuan Wu, Aryaman Arora, Zheng Wang, Atticus Geiger, Dan Jurafsky, Christopher D Manning, and Christopher Potts. 2024. Reft: Representation finetuning for language models. arXiv preprint arXiv:2404.03592.
- Yijun Xiao and William Yang Wang. 2021. On hallucination and predictive uncertainty in conditional language generation. In Proceedings of the 16th Conference of the European Chapter of the Association for Computational Linguistics: Main Volume, pages 2734–2744.
- Miao Xiong, Zhiyuan Hu, Xinyang Lu, YIFEI LI, Jie Fu, Junxian He, and Bryan Hooi. 2023. Can llms express their uncertainty? an empirical evaluation of confidence elicitation in llms. In The Twelfth International Conference on Learning Representations. find
- An Yang, Baosong Yang, Beichen Zhang, Binyuan Hui, Bo Zheng, Bowen Yu, Chengyuan Li, Dayiheng Liu, Fei Huang, Haoran Wei, et al. 2024. Qwen2. 5 technical report. arXiv preprint arXiv:2412.15115. find
- Shiping Yang, Renliang Sun, and Xiaojun Wan. 2023. A new benchmark and reverse validation method for passage-level hallucination detection. arXiv preprint arXiv:2310.06498. find
- Gal Yona, Roee Aharoni, and Mor Geva. 2024. Can large language models faithfully express their intrinsic uncertainty in words? In Proceedings of the 2024 Conference on Empirical Methods in Natural Language Processing, pages 7752–7764, Miami, Florida, USA. Association for Computational Linguistics.
- Lei Yu, Meng Cao, Jackie Chi Kit Cheung, and Yue Dong. 2024a. Mechanistic understanding and mitigation of language model non-factual hallucinations. In Findings of the Association for Computational Linguistics: EMNLP 2024, pages 7943–7956. find
- Lei Yu, Virginie Do, Karen Hambardzumyan, and Nicola Cancedda. 2024b. Robust llm safeguarding via refusal feature adversarial training. arXiv preprint arXiv:2409.20089. — [arXiv:2409.20089](https://arxiv.org/abs/2409.20089) find
- Caiqi Zhang, Fangyu Liu, Marco Basaldella, and Nigel Collier. 2024a. LUQ: Long-text uncertainty quantification for LLMs. In Proceedings of the 2024 Conference on Empirical Methods in Natural Language Processing, pages 5244–5262, Miami, Florida, USA. Association for Computational Linguistics.
- Hanning Zhang, Shizhe Diao, Yong Lin, Yi Fung, Qing Lian, Xingyao Wang, Yangyi Chen, Heng Ji, and Tong Zhang. 2024b. R-tuning: Instructing large language models to say ‘I don‘t know’. In Proceedings of the 2024 Conference of the North American Chapter of the Association for Computational Linguistics…
- Tianhang Zhang, Lin Qiu, Qipeng Guo, Cheng Deng, Yue Zhang, Zheng Zhang, Chenghu Zhou, Xinbing Wang, and Luoyi Fu. 2023. Enhancing uncertainty-based hallucination detection with stronger focus. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, pages 915–932,… find
- Lianmin Zheng, Wei-Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi Lin, Zhuohan Li, Dacheng Li, Eric Xing, et al. 2023. Judging llm-as-a-judge with mt-bench and chatbot arena. Advances in Neural Information Processing Systems, 36:46595–46623.
- Kaitlyn Zhou, Jena Hwang, Xiang Ren, and Maarten Sap. 2024a. Relying on the unreliable: The impact of language models’ reluctance to express uncertainty. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 3623–3643, Bangkok, Thai… find
- Kaitlyn Zhou, Jena D Hwang, Xiang Ren, and Maarten Sap. 2024b. Relying on the unreliable: The impact of language models’ reluctance to express uncertainty. arXiv preprint arXiv:2401.06730. find
- Kaitlyn Zhou, Dan Jurafsky, and Tatsunori Hashimoto. 2023. Navigating the grey area: How expressions of uncertainty and overconfidence affect language models. In Proceedings of the 2023 Conference on Empirical Methods in Natural Language Processing, pages 5506–5524, Singapore. Association for Comput… find
- Andy Zou, Long Phan, Sarah Chen, James Campbell, Phillip Guo, Richard Ren, Alexander Pan, Xuwang Yin, Mantas Mazeika, Ann-Kathrin Dombrowski, et al. 2023. Representation engineering: A top-down approach to ai transparency. arXiv preprint arXiv:2310.01405.
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