Negative impact of heavy-tailed uncertainty and error distributions on the reliability of calibration statistics for machine learning regression tasks

Average calibration of the (variance-based) prediction uncertainties of machine learning regression tasks can be tested in two ways: one is to estimate the calibration error (CE) as the difference between the mean absolute error (MSE) and the mean variance (MV); the alternative is to compare the mean squared z-scores (ZMS) to 1. The problem is that both approaches might lead to different conclusions, as illustrated in this study for an ensemble of datasets from the recent machine learning uncertainty quantification (ML-UQ) literature. It is shown that the estimation of MV, MSE and their confidence intervals becomes unreliable for heavy-tailed uncertainty and error distributions, which seems to be a frequent feature of ML-UQ datasets. By contrast, the ZMS statistic is less sensitive and offers the most reliable approach in this context, still acknowledging that datasets with heavy-tailed z-scores distributions should be considered with great care. Unfortunately, the same problem is expected to affect also conditional calibrations statistics, such as the popular ENCE, and very likely post-hoc calibration methods based on similar statistics. Several solutions to circumvent the outlined problems are proposed.