Divergence Based Quadrangle and Applications

Abstract

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

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Reproductions