# On the relationship between calibrated predictors and unbiased volume estimation

arXiv: https://arxiv.org/abs/2112.12560 · HTML: https://mlsanity.com/paper/2112.12560/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 31 references that are in the map.
- First posted on arXiv 23 Dec 2021.

## Builds on (most cited first, 13 of 13)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [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)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Identifying the Best Machine Learning Algorithms for Brain Tumor Segmentation, Progression Assessment, and Overall Survival Prediction in the BRATS Challenge](https://mlsanity.com/paper/1811.02629/) (2018)
- [Verified Uncertainty Calibration](https://mlsanity.com/paper/1909.10155/) (2019)
- [A Probabilistic U-Net for Segmentation of Ambiguous Images](https://mlsanity.com/paper/1806.05034/) (2018)
- [PHiSeg: Capturing Uncertainty in Medical Image Segmentation](https://mlsanity.com/paper/1906.04045/) (2019)
- [No New-Net](https://mlsanity.com/paper/1809.10483/) (2018)
- [Non-Parametric Calibration for Classification](https://mlsanity.com/paper/1906.04933/) (2019)
- [Optimization for Medical Image Segmentation: Theory and Practice when evaluating with Dice Score or Jaccard Index](https://mlsanity.com/paper/2010.13499/) (2020)
- [Towards safe deep learning: accurately quantifying biomarker uncertainty in neural network predictions](https://mlsanity.com/paper/1806.08640/) (2018)
- [Post Training Uncertainty Calibration of Deep Networks For Medical Image Segmentation](https://mlsanity.com/paper/2010.14290/) (2020)

## Built on by (most cited first, 11 of 11)

- [Uncertainty Quantification for Machine Learning in Healthcare: A Survey](https://mlsanity.com/paper/2505.02874/) (2025)
- [Jaccard Metric Losses: Optimizing the Jaccard Index with Soft Labels](https://mlsanity.com/paper/2302.05666/) (2023)
- [Beyond Classification: Definition and Density-based Estimation of Calibration in Object Detection](https://mlsanity.com/paper/2312.06645/) (2023)
- [RankSEG: A Consistent Ranking-based Framework for Segmentation](https://mlsanity.com/paper/2206.13086/) (2022)
- [On Image Segmentation With Noisy Labels: Characterization and Volume Properties of the Optimal Solutions to Accuracy and Dice](https://mlsanity.com/paper/2206.06484/) (2022)
- [Dice Semimetric Losses: Optimizing the Dice Score with Soft Labels](https://mlsanity.com/paper/2303.16296/) (2023)
- [Usable Region Estimate for Assessing Practical Usability of Medical Image Segmentation Models](https://mlsanity.com/paper/2207.00156/) (2022)
- [Noisy Image Segmentation With Soft-Dice](https://mlsanity.com/paper/2304.00801/) (2023)
- [Identifying and Combating Bias in Segmentation Networks by leveraging multiple resolutions](https://mlsanity.com/paper/2206.14919/) (2022)
- [Marginal Thresholding in Noisy Image Segmentation](https://mlsanity.com/paper/2304.04116/) (2023)
- [CARE: Confidence-aware Ratio Estimation for Medical Biomarkers](https://mlsanity.com/paper/2505.19585/) (2025)

## Abstract (arXiv)

Machine learning driven medical image segmentation has become standard in medical image analysis. However, deep learning models are prone to overconfident predictions. This has led to a renewed focus on calibrated predictions in the medical imaging and broader machine learning communities. Calibrated predictions are estimates of the probability of a label that correspond to the true expected value of the label conditioned on the confidence. Such calibrated predictions have utility in a range of medical imaging applications, including surgical planning under uncertainty and active learning systems. At the same time it is often an accurate volume measurement that is of real importance for many medical applications. This work investigates the relationship between model calibration and volume estimation. We demonstrate both mathematically and empirically that if the predictor is calibrated per image, we can obtain the correct volume by taking an expectation of the probability scores per pixel/voxel of the image. Furthermore, we show that convex combinations of calibrated classifiers preserve volume estimation, but do not preserve calibration. Therefore, we conclude that having a calibrated predictor is a sufficient, but not necessary condition for obtaining an unbiased estimate of the volume. We validate our theoretical findings empirically on a collection of 18 different (calibrated) training strategies on the tasks of glioma volume estimation on BraTS 2018, and ischemic stroke lesion volume estimation on ISLES 2018 datasets.