Risk Measures and Upper Probabilities: Coherence and Stratification

TL;DR

This paper explores alternative probability frameworks for machine learning, focusing on spectral risk measures and upper probabilities, providing theoretical characterizations and demonstrating practical benefits in uncertainty quantification.

Contribution

It introduces a stratification of coherent risk measures via upper probabilities and empirically shows their advantages in machine learning applications.

Findings

Spectral risk measures can be characterized and stratified using upper probabilities.

The new framework improves uncertainty handling in machine learning tasks.

Empirical results demonstrate practical benefits of the approach.

Abstract

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.