Measurability of functionals and of ideal point forecasts

TL;DR

This paper provides a theoretical foundation for the measurability of ideal point forecasts derived from probabilistic forecasts, clarifying conditions under which these forecasts are well-defined and applicable in practice.

Contribution

It establishes the measurability of a broad class of functionals and their associated point forecasts, supporting their validity in probabilistic forecasting frameworks.

Findings

Measurability of ideal point forecasts is theoretically justified.

Results apply to a wide class of relevant functionals, including elicitable ones.

Measurability of scoring rules is also established.

Abstract

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.