论文标题
功能和理想点预测的可测量性
Measurability of functionals and of ideal point forecasts
论文作者
论文摘要
基于信息集的随机变量$ y $的理想概率预测$ \ Mathcal {f} $是给定$ y $的条件分布，给定$ y $。在旨在指定功能性$ t $（例如均值，分位数或风险度量）等点预测的背景下，理想点的预测是应用于条件分布的各自的功能。本文提供了一个理论上的理由，为什么这种理想的预测实际上是一个预测，即$ \ Mathcal {f} $ - 可测量的随机变量。为此，阐明了$ t $的适当可衡量性概念，并为包括可靠的功能（包括可靠的功能）建立了可测量性。更一般而言，$ t $的可衡量性意味着通过将$ t $应用于概率预测而产生的任何点预测的可测量性。为适当的评分规则建立了类似的可测量性结果，这是评估概率预测的预测准确性的主要工具。
The ideal probabilistic forecast for a random variable $Y$ based on an information set $\mathcal{F}$ is the conditional distribution of $Y$ given $\mathcal{F}$. In the context of point forecasts aiming to specify a functional $T$ such as the mean, a quantile or a risk measure, the ideal point forecast is the respective functional applied to the conditional distribution. This paper provides a theoretical justification why this ideal forecast is actually a forecast, that is, an $\mathcal{F}$-measurable random variable. To that end, the appropriate notion of measurability of $T$ is clarified and this measurability is established for a large class of practically relevant functionals, including elicitable ones. More generally, the measurability of $T$ implies the measurability of any point forecast which arises by applying $T$ to a probabilistic forecast. Similar measurability results are established for proper scoring rules, the main tool to evaluate the predictive accuracy of probabilistic forecasts.
