This paper introduces a novel framework for assessing risk and decision-making in the presence of uncertainty, the \emph{$\varphi$-Divergence Quadrangle}. This approach expands upon the traditional Risk Quadrangle, a model that quantifies uncertainty through four key components: \emph{risk, deviation, regret}, and \emph{error}. The $\varphi$-Divergence Quadrangle incorporates the $\varphi$-divergence as a measure of the difference between probability distributions, thereby providing a more nuanced understanding of risk. Importantly, the $\varphi$-Divergence Quadrangle is closely connected with the distributionally robust optimization based on the $\varphi$-divergence approach through the duality theory of convex functionals. To illustrate its practicality and versatility, several examples of the $\varphi$-Divergence Quadrangle are provided, including the Quantile Quadrangle. The final portion of the paper outlines a case study implementing regression with the Entropic Value-at-Risk Quadrangle. The proposed $\varphi$-Divergence Quadrangle presents a refined methodology for understanding and managing risk, contributing to the ongoing development of risk assessment and management strategies.

翻译：本文提出了一个在不确定性条件下评估风险与决策的新框架——φ-散度四边形。该框架扩展了传统的风险四边形模型，该模型通过风险、偏差、遗憾和误差四个关键要素量化不确定性。φ-散度四边形将φ-散度作为概率分布间差异的度量纳入其中，从而提供了对风险更细致入微的理解。重要的是，通过凸函数对偶理论，φ-散度四边形与基于φ-散度的分布鲁棒优化紧密关联。为说明其实用性与通用性，本文给出了多个φ-散度四边形的实例，包括分位数四边形。论文最后部分概述了采用熵值风险价值四边形的回归案例研究。所提出的φ-散度四边形提供了一种理解和管理风险的优化方法，有助于推动风险评估与管理策略的持续发展。