Many forecasts consist not of point predictions but concern the evolution of quantities. For example, a central bank might predict the interest rates during the next quarter, an epidemiologist might predict trajectories of infection rates, a clinician might predict the behaviour of medical markers over the next day, etc. The situation is further complicated since these forecasts sometimes only concern the approximate "shape of the future evolution" or "order of events". Formally, such forecasts can be seen as probability measures on spaces of equivalence classes of paths modulo time-parametrization. We leverage the statistical framework of proper scoring rules with classical mathematical results to derive a principled approach to decision making with such forecasts. In particular, we introduce notions of gradients, entropy, and divergence that are tailor-made to respect the underlying non-Euclidean structure.

翻译：许多预测并不包括点预测,而是涉及数量的变化。例如,央行可能会预测下季度的利率,流行病学家可能会预测感染率的轨迹,临床医生可能会预测次日的医疗标志行为等等。 情况更加复杂,因为这些预测有时只涉及“未来演变的大致形状”或“事件顺序 ” 。 形式上,这些预测可被视为对模式时间平衡路径等同类空间的概率度量。 我们利用适当评分规则的统计框架和经典数学结果来得出一种有原则的预测决策方法。 特别是,我们引入了梯度、诱变和差异的概念,这些概念是针对基本的非欧洲结构而量身定制的。