Divergence Based Quadrangle and Applications

• Published in: arXiv.org
• Main Authors: Malandii, Anton, Gupte, Siddhartha, Cheng, Peng, Uryasev, Stan
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 12.07.2023
• Subjects: Decision making; Decision theory; Optimization; Risk assessment; Risk management; Statistical analysis; Uncertainty

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.