# Robust Statistical Comparison of Random Variables with Locally Varying Scale of Measurement

arXiv: https://arxiv.org/abs/2306.12803 · HTML: https://mlsanity.com/paper/2306.12803/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 80 references that are in the map.
- First posted on arXiv 22 Jun 2023.

## Builds on (most cited first, 13 of 13)

- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Predictive Uncertainty Estimation via Prior Networks](https://mlsanity.com/paper/1802.10501/) (2018)
- [Pitfalls of Epistemic Uncertainty Quantification through Loss Minimisation](https://mlsanity.com/paper/2203.06102/) (2022)
- [Statistical Comparisons of Classifiers by Generalized Stochastic Dominance](https://mlsanity.com/paper/2209.01857/) (2022)
- [Ensemble-based Uncertainty Quantification: Bayesian versus Credal Inference](https://mlsanity.com/paper/2107.10384/) (2021)
- [Credal Self-Supervised Learning](https://mlsanity.com/paper/2106.11853/) (2021)
- [Statistical comparison of classifiers through Bayesian hierarchical modelling](https://mlsanity.com/paper/1609.08905/) (2016)
- [Accounting for Gaussian Process Imprecision in Bayesian Optimization](https://mlsanity.com/paper/2111.08299/) (2021)
- [Information efficient learning of complexly structured preferences: Elicitation procedures and their application to decision making under uncertainty](https://mlsanity.com/paper/2110.12879/) (2021)
- [Attention-based Random Forest and Contamination Model](https://mlsanity.com/paper/2201.02880/) (2022)
- [Multi-Target Decision Making under Conditions of Severe Uncertainty](https://mlsanity.com/paper/2212.06832/) (2022)
- [Learning to Optimize with Stochastic Dominance Constraints](https://mlsanity.com/paper/2211.07767/) (2022)
- [Depth Functions for Partial Orders with a Descriptive Analysis of Machine Learning Algorithms](https://mlsanity.com/paper/2304.09872/) (2023)

## Built on by (most cited first, 13 of 13)

- [Explaining Bayesian Optimization by Shapley Values Facilitates Human-AI Collaboration](https://mlsanity.com/paper/2403.04629/) (2024)
- [Statistical Multicriteria Benchmarking via the GSD-Front](https://mlsanity.com/paper/2406.03924/) (2024)
- [Towards Better Open-Ended Text Generation: A Multicriteria Evaluation Framework](https://mlsanity.com/paper/2410.18653/) (2024)
- [Statistical Multicriteria Evaluation of LLM-Generated Text](https://mlsanity.com/paper/2506.18082/) (2025)
- [Partial Rankings of Optimizers](https://mlsanity.com/paper/2402.16565/) (2024)
- [Semi-Supervised Learning guided by the Generalized Bayes Rule under Soft Revision](https://mlsanity.com/paper/2405.15294/) (2024)
- [A Statistical Case Against Empirical Human-AI Alignment](https://mlsanity.com/paper/2502.14581/) (2025)
- [Generalization Bounds and Stopping Rules for Learning with Self-Selected Data](https://mlsanity.com/paper/2505.07367/) (2025)
- [Pseudo Label Selection is a Decision Problem](https://mlsanity.com/paper/2309.13926/) (2023)
- [Towards Bayesian Data Selection](https://mlsanity.com/paper/2406.12560/) (2024)
- [Reciprocal Learning](https://mlsanity.com/paper/2408.06257/) (2024)
- [Empirical Decision Theory](https://mlsanity.com/paper/2512.05677/) (2025)
- [Self-Supervised Laplace Approximation for Bayesian Uncertainty Quantification](https://mlsanity.com/paper/2605.12208/) (2026)

## Abstract (arXiv)

Spaces with locally varying scale of measurement, like multidimensional structures with differently scaled dimensions, are pretty common in statistics and machine learning. Nevertheless, it is still understood as an open question how to exploit the entire information encoded in them properly. We address this problem by considering an order based on (sets of) expectations of random variables mapping into such non-standard spaces. This order contains stochastic dominance and expectation order as extreme cases when no, or respectively perfect, cardinal structure is given. We derive a (regularized) statistical test for our proposed generalized stochastic dominance (GSD) order, operationalize it by linear optimization, and robustify it by imprecise probability models. Our findings are illustrated with data from multidimensional poverty measurement, finance, and medicine.