mlsanityData to 14 Oct 2026

Properties of the ENCE and other MAD-based calibration metrics

arXiv 2305.11905 ↗First posted 17 May 2023cs.LG physics.chem-ph physics.data-ancited by 7

Not on the map. Why?

Abstract (arXiv)

The Expected Normalized Calibration Error (ENCE) is a popular calibration statistic used in Machine Learning to assess the quality of prediction uncertainties for regression problems. Estimation of the ENCE is based on the binning of calibration data. In this short note, I illustrate an annoying property of the ENCE, i.e. its proportionality to the square root of the number of bins for well calibrated or nearly calibrated datasets. A similar behavior affects the calibration error based on the variance of z-scores (ZVE), and in both cases this property is a consequence of the use of a Mean Absolute Deviation (MAD) statistic to estimate calibration errors. Hence, the question arises of which number of bins to choose for a reliable estimation of calibration error statistics. A solution is proposed to infer ENCE and ZVE values that do not depend on the number of bins for datasets assumed to be calibrated, providing simultaneously a statistical calibration test. It is also shown that the ZVE is less sensitive than the ENCE to outstanding errors or uncertainties.

Key facts. Cited by 7 papers in the map (data through 14 Oct 2026). Builds on 5 of its 12 references that are in the map. First posted on arXiv 17 May 2023.

Builds on

5 of 12 references.

References · 5

  1. 2017On Calibration of Modern Neural Networkscited by 4,960
  2. 2021A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantificationcited by 792
  3. 2019Evaluating and Calibrating Uncertainty Prediction in Regression Taskscited by 87
  4. 2021Calibrated Uncertainty for Molecular Property Prediction using Ensembles of Message Passing Neural Networkscited by 21
  5. 2022Uncertainty quantification for predictions of atomistic neural networkscited by 5

Built on by

Cited by year

Citing papers per year: 2023: 3; 2024: 2; 2025: 2.

7 papers.

  1. 2023Calibration in Machine Learning Uncertainty Quantification: beyond consistency to target adaptivitycited by 13
  2. 2023Stratification of uncertainties recalibrated by isotonic regression and its impact on calibration error statisticscited by 6
  3. 2023Can bin-wise scaling improve consistency and adaptivity of prediction uncertainty for machine learning regression ?cited by 4
  4. 2024Negative impact of heavy-tailed uncertainty and error distributions on the reliability of calibration statistics for machine learning regression taskscited by 1
  5. 2025On Uncertainty Calibration for Equivariant Functionscited by 1
  6. 2025A systematic evaluation of uncertainty quantification techniques in deep learning: a case study in photoplethysmography signal analysiscited by 1
  7. 2024Validation of ML-UQ calibration statistics using simulated reference values: a sensitivity analysiscited by 0