论文标题
风险措施和上层概率：连贯和分层
Risk Measures and Upper Probabilities: Coherence and Stratification
论文作者
论文摘要
机器学习通常以经典概率理论为前提，这意味着聚集是基于期望的。现在，有多种原因可以激励人们将经典概率理论作为机器学习的数学基础。我们系统地检查了一类强大而丰富的替代聚合功能，即各种被称为光谱风险度量，Choquet积分或Lorentz规范。我们提出了一系列的表征结果，并演示了使这个光谱家族如此特别的原因。在这样做的过程中，我们从它们通过利用重新安排不变的Banach空间理论来诱导的上层概率来实现所有连贯的风险度量的自然分层。我们从经验上证明，这种新的不确定性方法如何有助于解决实用的机器学习问题。
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.
