# Variable transformations in consistent loss functions

arXiv: https://arxiv.org/abs/2502.16542 · HTML: https://mlsanity.com/paper/2502.16542/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 68 references that are in the map.
- First posted on arXiv 23 Feb 2025.

## Builds on (most cited first, 9 of 9)

- [Regression Diagnostics meets Forecast Evaluation: Conditional Calibration, Reliability Diagrams, and Coefficient of Determination](https://mlsanity.com/paper/2108.03210/) (2021)
- [A review of predictive uncertainty estimation with machine learning](https://mlsanity.com/paper/2209.08307/) (2022)
- [Of Quantiles and Expectiles: Consistent Scoring Functions, Choquet Representations, and Forecast Rankings](https://mlsanity.com/paper/1503.08195/) (2015)
- [Model Comparison and Calibration Assessment: User Guide for Consistent Scoring Functions in Machine Learning and Actuarial Practice](https://mlsanity.com/paper/2202.12780/) (2022)
- [A review of machine learning concepts and methods for addressing challenges in probabilistic hydrological post-processing and forecasting](https://mlsanity.com/paper/2206.08998/) (2022)
- [Proper Scoring Rules for Multivariate Probabilistic Forecasts based on Aggregation and Transformation](https://mlsanity.com/paper/2407.00650/) (2024)
- [Forecast Evaluation of Quantiles, Prediction Intervals, and other Set-Valued Functionals](https://mlsanity.com/paper/1910.07912/) (2019)
- [Sensitivity Measures Based on Scoring Functions](https://mlsanity.com/paper/2203.00460/) (2022)
- [Deep Huber quantile regression networks](https://mlsanity.com/paper/2306.10306/) (2023)

## Abstract (arXiv)

The empirical use of variable transformations within (strictly) consistent loss functions is widespread, yet a theoretical understanding is lacking. To address this gap, we develop a theoretical framework that establishes formal characterizations of (strict) consistency for such transformed loss functions. Our analysis focuses on two interrelated cases: (a) transformations applied solely to the realization variable and (b) bijective transformations applied jointly to both the realization and prediction variables. These cases extend the well-established framework of transformations applied exclusively to the prediction variable, as formalized by Osband's revelation principle. We further develop analogous characterizations for (strict) identification functions. The resulting theoretical framework is broadly applicable to statistical and machine learning methodologies. For instance, we apply the framework to Bregman and expectile loss functions to interpret empirical findings from models trained with transformed loss functions and systematically construct new identifiable and elicitable functionals, which we term respectively g-transformed expectation and g-transformed expectile. Applications of the framework to simulated and real-world data illustrate its practical utility in diverse settings. By unifying theoretical insights with practical applications, this work advances principled methodologies for designing loss functions in complex predictive tasks.