  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.
