Properties of the ENCE and other MAD-based calibration metrics

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.