Risk Measures and Upper Probabilities: Coherence and Stratification
This work addresses foundational issues in machine learning for researchers and practitioners by offering a richer mathematical framework to model uncertainty, though it is incremental in building on existing theories like Choquet integrals.
The paper tackles the problem of using classical probability theory as the foundation for machine learning by systematically examining spectral risk measures as alternative aggregation functionals, resulting in a natural stratification of coherent risk measures based on upper probabilities and empirical demonstrations of improved handling of uncertainty in practical applications.
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.