---
title: "Uncertainty Distillation — paper by Duh; and Andrews (2025)"
description: "The Johns Hopkins authors propose uncertainty distillation. 39 references, cited by 5 papers here."
authors: ["Sophia Hager; David Mueller; Kevin Duh;", "Nicholas Andrews"]
year: 2025
venue: "arXiv:2503.14749"
source: "https://arxiv.org/abs/2503.14749"
doi: "10.48550/arxiv.2503.14749"
arxiv: "2503.14749"
licence: "CC BY-SA 4.0 (arXiv)"
licence_url: "http://creativecommons.org/licenses/by-sa/4.0/"
summary_licence: "Elamur Cosmic License"
canonical: "https://research.elamur.ai/free/en/papers/hager-2025-uncertainty-distillation-teaching-language-models/"
---
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# Uncertainty Distillation: Teaching Language Models to Express Semantic Confidence

**Authors:** Sophia Hager; David Mueller; Kevin Duh; and [Nicholas Andrews](https://md.research.elamur.ai/free/en/authors/nicholas-andrews/) · **Year:** 2025 · **Venue:** arXiv:2503.14749 · **Licence:** [CC BY-SA 4.0 (arXiv)](http://creativecommons.org/licenses/by-sa/4.0/)

**Links:** [Publisher's page](https://arxiv.org/abs/2503.14749) · [PDF](https://arxiv.org/pdf/2503.14749) · [DOI 10.48550/arxiv.2503.14749](https://doi.org/10.48550/arxiv.2503.14749) · [arXiv 2503.14749](https://arxiv.org/abs/2503.14749) · [OpenAlex](https://openalex.org/W4414902044)

**Citations:** 0 citations (OpenAlex, 2026-10-14) · [5](https://md.research.elamur.ai/free/en/papers/hager-2025-uncertainty-distillation-teaching-language-models/cited-by/) in our library, of the 149,439 papers whose bibliographies we read

## Summary
LLMs used for factual question answering often state confidence levels that do not match how often they are actually wrong, and prompting alone does not fix this. Lexical methods based on token probabilities score one particular wording. Users usually care about semantic uncertainty instead: whether the answer itself is right, however it is phrased. The Johns Hopkins authors propose uncertainty distillation. For each training question, they sample many answers, merge semantically equivalent ones with a normalization function, and treat the resulting frequencies as Monte Carlo confidence estimates. Isotonic regression, fitted on a held-out calibration set, converts these frequencies into calibrated probabilities. The probabilities are then rewritten as verbal labels (e.g., "high confidence") and used for supervised fine-tuning. The tuned model outputs an answer and its confidence in a single forward pass. Measured by AUROC, its stated confidences track observed errors as well as strong baselines that can be over 20× slower at inference. The approach also works on black-box models that offer API fine-tuning, making calibrated, phrasing-independent confidence cheap to obtain.

## Key points
- The pipeline has three stages: Monte Carlo sampling with semantic normalization, post-hoc calibration via isotonic regression, and self-annotation followed by fine-tuning on verbalized confidences.
- In the paper's illustration, an answer sampled 650/1000 times (65%) is remapped by the learned calibration function to about 85%.
- Baselines include P(True), semantic clustering, token-level probabilities, P(IK) and prompting. Only uncertainty distillation needs a single model and one pass, works without open weights, is length-agnostic, and aligns with error rates.
- Drawing more samples during training raises AUROC, but the gains diminish as the sample count grows.
- The method was applied to a closed model through API-based fine-tuning and gave uncertainty estimates that were both more effective and more efficient than every baseline.

*Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.*

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## Cite

Cite as: Sophia Hager; David Mueller; Kevin Duh; and Nicholas Andrews. 2025. Uncertainty Distillation: Teaching Language Models to Express Semantic Confidence. arXiv:2503.14749.

The paper itself, not this page. BibTeX from the library's record of the work.

```bibtex
@misc{duh2025uncertainty,
  title = {{Uncertainty Distillation: Teaching Language Models to Express Semantic Confidence}},
  author = {Sophia Hager; David Mueller; Kevin Duh; and Nicholas Andrews},
  year = {2025},
  doi = {10.48550/arxiv.2503.14749},
  eprint = {2503.14749},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.48550/arxiv.2503.14749}
}
```

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## References

39 references, 18 of them with a card of ours. ↗ leads to the work, «find» searches for one we have no id for yet.

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*Updated 30 October 2026*