The empirical use of variable transformations within (strictly) consistent loss functions is widespread, yet a theoretical understanding is lacking. To address this gap, we develop a theoretical framework that establishes formal characterizations of (strict) consistency for such transformed loss functions. Our analysis focuses on two interrelated cases: (a) transformations applied solely to the realization variable and (b) bijective transformations applied jointly to both the realization and prediction variables. These cases extend the well-established framework of transformations applied exclusively to the prediction variable, as formalized by Osband's revelation principle. We further develop analogous characterizations for (strict) identification functions. The resulting theoretical framework is broadly applicable to statistical and machine learning methodologies. For instance, we apply the framework to Bregman and expectile loss functions to interpret empirical findings from models trained with transformed loss functions and systematically construct new identifiable and elicitable functionals, which we term respectively $g$-transformed expectation and $g$-transformed expectile. Applications of the framework to simulated and real-world data illustrate its practical utility in diverse settings. By unifying theoretical insights with practical applications, this work advances principled methodologies for designing loss functions in complex predictive tasks.

翻译：在（严格）一致损失函数中应用变量变换的经验做法十分普遍，但相关理论理解尚不充分。为填补这一空白，我们构建了一个理论框架，为这类变换后损失函数的（严格）一致性建立了形式化特征刻画。我们的分析聚焦于两个相互关联的情形：（a）仅对实现变量施加变换；（b）对实现变量和预测变量联合施加双射变换。这些情形拓展了Osband揭示原理所形式化的、仅对预测变量进行变换的经典框架。我们进一步为（严格）识别函数建立了类似的特征刻画。所得理论框架广泛适用于统计学与机器学习方法。例如，我们将其应用于Bregman损失函数和期望损失函数，以解释使用变换损失函数训练模型获得的实证结果，并系统性地构建了新的可识别与可引出泛函，分别称之为$g$-变换期望与$g$-变换期望。该框架在模拟数据与真实数据中的应用，展示了其在多样化场景中的实用价值。通过将理论洞见与实际应用相统一，本研究为复杂预测任务中损失函数的设计推进了基于原理的方法论。