标题： 泛函和理想点预测的可测性 显示英文标题

标题： Measurability of functionals and of ideal point forecasts

摘要： 对于基于信息集$\mathcal{F}$的随机变量$Y$的理想概率预测，是给定$\mathcal{F}$的$Y$的条件分布。在旨在指定一个函数性$T$（如均值、分位数或风险度量）的点预测背景下，理想的点预测是将该函数性应用于条件分布。本文提供了理论依据，说明为什么这个理想预测实际上是一个预测，即一个$\mathcal{F}$-可测的随机变量。为此，明确了$T$的可测性概念，并为一大类实际相关的函数性（包括可诱发的函数性）建立了这种可测性。 更一般地，$T$的可测性意味着通过将$T$应用于概率预测而产生的任何点预测的可测性。对于适当的评分规则，建立了类似的可测性结果，这是评估概率预测预测准确性的主要工具。 显示英文摘要

摘要： 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.