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，其中聚合基于期望构建。目前有多个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。我们系统研究了一类强大且丰富的替代聚合泛函，即谱风险度量、Choquet积分或Lorentz范数。我们呈现了一系列刻画结果，并展示了该谱族为何具有特殊性。在此过程中，通过利用重排不变Banach空间理论的结果，我们得到了所有一致风险度量基于其所诱导的上概率的自然分层。实验表明，这种不确定性新方法有助于解决实际机器学习问题。