Copula熵的英文专著《Copula Entropy: Theory and Applications》已在arXiv的数学和统计学分区发布，归在统计学.方法论、数学.概率论、数学.信息论和数学.统计理论四个分类下。

arXiv链接：https://arxiv.org/abs/2512.18168

现分享专著前言Preface，以及Stanford的alphaXiv网站对专著的总结和评论摘录。

alphaXiv链接：https://www.alphaxiv.org/overview/2512.18168

---

Preface

Copula theory is of fundamental importance in probability and become mature after decades of development. As the core of it, Sklar's theorem presents the universal representation of dependence, i.e., copula, which can be used for dependence inference and measurements. Many types of copulas have been proposed and copula based bivariate dependence measures, such as Spearman's ρ and Kendall's τ, have also been given. Copula based multivariate dependence measure remains a problem in the field.

This monograph originated from the author's PhD study at Tsinghua University, during which the author defined the concept of copula entropy (CE), proved its equivalence to mutual information in information theory, proposed a nonparametric CE estimator, and applied CE to structure learning and copula component analysis. Since then, the author continued his PhD research and applied CE to eleven fundamental problems in statistics. Particularly, he proved the CE representation of transfer entropy / conditional mutual information and therefore built a theoretical framework that unified the concepts of correlation/dependence and causality/conditional independence based on CE. Additionally, he proposed five hypothesis testing methodologies based on CE. Meanwhile, researchers have introduced many mathematical generalizations of CE and applied CE to real problems in every branch of science. CE has become a mature theory and an integrated area of copula theory. This monograph aims to present the theoretical framework of CE and its latest developments.

CE theory presents a multivariate dependence measure based on copula and has many theoretical meanings, including

• developed copula theory by introducing a perfect copula based multivariate dependence measure;

• built a bridge between copula theory and information theory with the proof of the equivalence of CE and mutual information and the CE representation of conditional mutual information, and hence made probability and information theory a much more integrated mathematical field;

• built a unified framework of correlation and causality based on CE, proposed the system of statistical methodologies including independence test, conditional independence test, multivariate normality test, copula hypothesis test, two-sample test, change point detection, and symmetry test and made mathematical statistics much more mature.

This monograph is not finished yet, and any comments and suggestions are welcome.

Jian Ma

Haidian, Beijing

December 2025

---

alphaXiv's comments:

"Copula Entropy (CE) represents a fundamental advancement in statistical methodology, bridging the gap between copula theory and information theory to create a unified framework for measuring and analyzing dependence relationships. This monograph by Jian Ma presents over a decade of research establishing CE as both a theoretical cornerstone and practical toolkit for understanding complex data relationships across diverse scientific domains.

The work emerges from a simple yet profound observation: if copulas universally capture dependence structures and mutual information measures dependence content, there must exist a deep mathematical connection between these concepts. This insight led to the development of CE as the entropy of copula density functions, revealing that mutual information is fundamentally the negative of copula entropy - a discovery that unifies two previously distinct mathematical fields."

"This monograph establishes CE as a fundamental advancement in statistical methodology with profound theoretical and practical implications. By unifying copula theory and information theory, it provides both deep mathematical insights and powerful practical tools for analyzing complex data relationships. The comprehensive system of CE-based methodologies offers researchers and practitioners robust, distribution-free alternatives to traditional methods, particularly valuable in our era of increasingly complex and high-dimensional data.

The work demonstrates how fundamental mathematical insights can lead to practical breakthroughs, establishing CE as an essential tool for modern statistical analysis across diverse scientific domains."