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Calibrating Expressions of Certainty
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Models certainty phrases (e.g., "Maybe", "Likely") as Beta distributions over the probability simplex, generalizes ECE and reliability diagrams to this representation, and proposes an unbalanced optimal-transport post-hoc calibration map that rewrites certainty phrases to improve calibration of radiologists and LLMs.
Problem
Calibration of humans and language models that express confidence with natural-language certainty phrases cannot be measured with standard scalar-score calibration metrics, since such phrases are mapped to single fixed probabilities that ignore their semantic range and population variation.
Approach
Treat each certainty phrase as a Beta distribution over [0,1], generalize the binning and KDE-style ECE estimators to these distributions, and calibrate by solving an unbalanced discrete optimal transport problem that yields a stochastic map between certainty phrases, so the output remains natural-language phrases.
Key results
- On SciQ, gpt-4o verbalized phrases with the proposed OT calibration reduce ECE from 0.07 (uncalibrated phrases) to 0.08 with accuracy 0.73 unchanged, and on TruthfulQA ECE drops from 0.22 to 0.10 for gpt-4o phrases.
- Radiologist calibration maps suggest concrete wording changes, e.g., using 'May' instead of 'Present' for edema-related overconfidence, which reduce ECE* and Brier Score on the test set.
- Calibrating with the proposed method yields calibrated phrases competitive with Platt scaling and histogram binning on scalar-mapped phrases, without reducing accuracy.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Apply optimal-transport calibration to verbalized certainty phrases | gpt-4o · SciQ | ECE: 0.07 -> 0.08 | Mixed |
| Apply optimal-transport calibration to verbalized certainty phrases | gpt-4o · TruthfulQA | ECE: 0.22 -> 0.1 | Helped |
| Apply optimal-transport calibration to verbalized certainty phrases | gpt-4o · SciQ | accuracy: 0.73 -> 0.73 | No clear effect |
| Use a fixed set of gpt-4o-generated confidence phrases rather than survey-derived phrase distributions (ablation) | gpt-4o · SciQ | ECE: 0.2 -> 0.11 | Helped |
| Use a fixed set of gpt-4o-generated confidence phrases rather than survey-derived phrase distributions (ablation) | gpt-4o · TruthfulQA | ECE: 0.37 -> 0.26 | Helped |
| Use a fixed set of phrases rather than generating confidence distributions on-the-fly (ablation) | gpt-4o · SciQ | ECE: 0.16 -> 0.11 | Helped |
| Use a fixed set of phrases rather than generating confidence distributions on-the-fly (ablation) | gpt-4o · TruthfulQA | ECE: 0.3 -> 0.26 | Helped |
| Use a fixed set of gpt-4o-generated confidence phrases rather than survey-derived phrase distributions (ablation) | claude-3.5-sonnet · SciQ | ECE: 0.19 -> 0.09 | Helped |
| Use a fixed set of gpt-4o-generated confidence phrases rather than survey-derived phrase distributions (ablation) | claude-3.5-sonnet · TruthfulQA | ECE: 0.43 -> 0.25 | Helped |
| Use a fixed set of phrases rather than generating confidence distributions on-the-fly (ablation) | claude-3.5-sonnet · SciQ | ECE: 0.13 -> 0.09 | Helped |
| Use a fixed set of phrases rather than generating confidence distributions on-the-fly (ablation) | claude-3.5-sonnet · TruthfulQA | ECE: 0.36 -> 0.25 | Helped |
| Use a fixed set of gpt-4o-generated confidence phrases rather than survey-derived phrase distributions (ablation) | gemini-1.5-pro · SciQ | ECE: 0.22 -> 0.13 | Helped |
Surprising findings
- Predefined fixed sets of certainty phrases generated by gpt-4o yield better calibration than generating beta distribution parameters on-the-fly for each question.
- Radiologists tend to be underconfident for low-certainty terms and overconfident for high-certainty terms like 'Likely'.
Limitations
- Whether radiologists can effectively adopt the calibration suggestions is untested; a clinical study is needed.
- Calibration gains depend on the model's discriminative performance and the small discrete set of certainty phrases limits precision.
- Radiology labels rely on an LLM-based extraction pipeline with lower accuracy on CT reports (macro 0.879).
Experimental setup
Datasets: Brigham and Women's Hospital paired chest X-ray and CT reports (2,662 pairs) SciQ TruthfulQA social media survey of probability terms radiologist survey of diagnostic certainty terms
Benchmarks: SciQ TruthfulQA Brigham and Women's Hospital paired X-ray/CT reports
Metrics: ECE ECE* Brier Score accuracy
Baselines: Platt scaling histogram binning verbalized probability on-the-fly beta distributions
Builds on: expected calibration error Platt scaling histogram binning unbalanced optimal transport verbalized confidence
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