Divergence Measures

Data science, information theory, probability theory, statistical learning and other related disciplines greatly benefit from non-negative measures of dissimilarity between pairs of probability measures. These are known as divergence measures, and exploring their mathematical foundations and diverse applications is of significant interest. The present Special Issue, entitled “Divergence Measures: Mathematical Foundations and Applications in Information-Theoretic and Statistical Problems”, includes eight original contributions, and it is focused on the study of the mathematical properties and applications of classical and generalized divergence measures from an information-theoretic perspective. It mainly deals with two key generalizations of the relative entropy: namely, the R_ényi divergence and the important class of f -divergences. It is our hope that the readers will find interest in this Special Issue, which will stimulate further research in the study of the mathematical foundations and applications of divergence measures

The Statistical Foundations of Entropy

In the last two decades, the understanding of complex dynamical systems underwent important conceptual shifts. The catalyst was the infusion of new ideas from the theory of critical phenomena (scaling laws, renormalization group, etc.), (multi)fractals and trees, random matrix theory, network theory, and non-Shannonian information theory. The usual Boltzmann–Gibbs statistics were proven to be grossly inadequate in this context. While successful in describing stationary systems characterized by ergodicity or metric transitivity, Boltzmann–Gibbs statistics fail to reproduce the complex statistical behavior of many real-world systems in biology, astrophysics, geology, and the economic and social sciences.The aim of this Special Issue was to extend the state of the art by original contributions that could contribute to an ongoing discussion on the statistical foundations of entropy, with a particular emphasis on non-conventional entropies that go significantly beyond Boltzmann, Gibbs, and Shannon paradigms. The accepted contributions addressed various aspects including information theoretic, thermodynamic and quantum aspects of complex systems and found several important applications of generalized entropies in various systems

Information Geometry

This Special Issue of the journal Entropy, titled “Information Geometry I”, contains a collection of 17 papers concerning the foundations and applications of information geometry. Based on a geometrical interpretation of probability, information geometry has become a rich mathematical field employing the methods of differential geometry. It has numerous applications to data science, physics, and neuroscience. Presenting original research, yet written in an accessible, tutorial style, this collection of papers will be useful for scientists who are new to the field, while providing an excellent reference for the more experienced researcher. Several papers are written by authorities in the field, and topics cover the foundations of information geometry, as well as applications to statistics, Bayesian inference, machine learning, complex systems, physics, and neuroscience

Information geometry

This Special Issue of the journal Entropy, titled “Information Geometry I”, contains a collection of 17 papers concerning the foundations and applications of information geometry. Based on a geometrical interpretation of probability, information geometry has become a rich mathematical field employing the methods of differential geometry. It has numerous applications to data science, physics, and neuroscience. Presenting original research, yet written in an accessible, tutorial style, this collection of papers will be useful for scientists who are new to the field, while providing an excellent reference for the more experienced researcher. Several papers are written by authorities in the field, and topics cover the foundations of information geometry, as well as applications to statistics, Bayesian inference, machine learning, complex systems, physics, and neuroscience