# Risk Measures and Upper Probabilities: Coherence and Stratification

arXiv: https://arxiv.org/abs/2206.03183 · HTML: https://mlsanity.com/paper/2206.03183/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2022.

## Built on by (most cited first, 9 of 9)

- [Scoring Rules and Calibration for Imprecise Probabilities](https://mlsanity.com/paper/2410.23001/) (2024)
- [On the Richness of Calibration](https://mlsanity.com/paper/2302.04118/) (2023)
- [Tailoring to the Tails: Risk Measures for Fine-Grained Tail Sensitivity](https://mlsanity.com/paper/2208.03066/) (2022)
- [An Axiomatic Approach to Loss Aggregation and an Adapted Aggregating Algorithm](https://mlsanity.com/paper/2406.02292/) (2024)
- [On Calibration in Multi-Distribution Learning](https://mlsanity.com/paper/2412.14142/) (2024)
- [Bulk-Calibrated Credal Ambiguity Sets: Fast, Tractable Decision Making under Out-of-Sample Contamination](https://mlsanity.com/paper/2601.21324/) (2026)
- [Fast Computation of Superquantile-Constrained Optimization Through Implicit Scenario Reduction](https://mlsanity.com/paper/2405.07965/) (2024)
- [Optimizing Shortfall Risk Metric for Learning Regression Models](https://mlsanity.com/paper/2505.17777/) (2025)
- [Policy Newton methods for Distortion Riskmetrics](https://mlsanity.com/paper/2508.07249/) (2025)

## Abstract (arXiv)

Machine learning typically presupposes classical probability theory which implies that aggregation is built upon expectation. There are now multiple reasons to motivate looking at richer alternatives to classical probability theory as a mathematical foundation for machine learning. We systematically examine a powerful and rich class of alternative aggregation functionals, known variously as spectral risk measures, Choquet integrals or Lorentz norms. We present a range of characterization results, and demonstrate what makes this spectral family so special. In doing so we arrive at a natural stratification of all coherent risk measures in terms of the upper probabilities that they induce by exploiting results from the theory of rearrangement invariant Banach spaces. We empirically demonstrate how this new approach to uncertainty helps tackling practical machine learning problems.