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.

翻译：机器学习通常预设经典概率论，该理论隐含聚合运算建立在期望之上。目前有多重理由促使我们探索比经典概率论更丰富的替代方案，作为机器学习的数学基础。本文系统研究了一类强大而丰富的替代聚合泛函——即谱风险度量、Choquet积分或Lorentz范数。我们呈现一系列刻画性结果，并阐明该谱族为何具有特殊性。通过利用重排不变Banach空间理论中的结论，我们得以将所有相干风险度量按其诱导的上概率进行自然分层。最终，我们通过实证展示了这种不确定性新方法如何助力解决实际机器学习问题。