Divergence Based Quadrangle and Applications
This paper introduces a novel framework for assessing risk and decision-making in the presence of uncertainty, the \emph{$φ$-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 $φ$-Divergence Quadrangle incorporates the $φ$-divergence as a measure of the difference between probability distributions, thereby providing a more nuanced understanding of risk. Importantly, the $φ$-Divergence Quadrangle is closely connected with the distributionally robust optimization based on the $φ$-divergence approach through the duality theory of convex functionals. To illustrate its practicality and versatility, several examples of the $φ$-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 $φ$-Divergence Quadrangle presents a refined methodology for understanding and managing risk, contributing to the ongoing development of risk assessment and management strategies.