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

arXiv: https://arxiv.org/abs/2402.10043 · HTML: https://mlsanity.com/paper/2402.10043/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 39 references that are in the map.
- First posted on arXiv 15 Feb 2024.

## Builds on (most cited first, 11 of 11)

- [Accurate Uncertainties for Deep Learning Using Calibrated Regression](https://mlsanity.com/paper/1807.00263/) (2018)
- [Deep Evidential Regression](https://mlsanity.com/paper/1910.02600/) (2019)
- [Evaluating and Calibrating Uncertainty Prediction in Regression Tasks](https://mlsanity.com/paper/1905.11659/) (2019)
- [Calibrated Uncertainty for Molecular Property Prediction using Ensembles of Message Passing Neural Networks](https://mlsanity.com/paper/2107.06068/) (2021)
- [Calibration in Machine Learning Uncertainty Quantification: beyond consistency to target adaptivity](https://mlsanity.com/paper/2309.06240/) (2023)
- [Graph Neural Network Interatomic Potential Ensembles with Calibrated Aleatoric and Epistemic Uncertainty on Energy and Forces](https://mlsanity.com/paper/2305.16325/) (2023)
- [One step closer to unbiased aleatoric uncertainty estimation](https://mlsanity.com/paper/2312.10469/) (2023)
- [Properties of the ENCE and other MAD-based calibration metrics](https://mlsanity.com/paper/2305.11905/) (2023)
- [Stratification of uncertainties recalibrated by isotonic regression and its impact on calibration error statistics](https://mlsanity.com/paper/2306.05180/) (2023)
- [Validation of uncertainty quantification metrics: a primer based on the consistency and adaptivity concepts](https://mlsanity.com/paper/2303.07170/) (2023)
- [Can bin-wise scaling improve consistency and adaptivity of prediction uncertainty for machine learning regression ?](https://mlsanity.com/paper/2310.11978/) (2023)

## Built on by (most cited first, 1 of 1)

- [Validation of ML-UQ calibration statistics using simulated reference values: a sensitivity analysis](https://mlsanity.com/paper/2403.00423/) (2024)

## Abstract (arXiv)

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.