Information Geometry

Nihat Ay, Jürgen Jost, Hông Vân Lê, Lorenz Schwachhöfer

The book provides a comprehensive introduction and a novel mathematical foundation of the field of information geometry with complete proofs and detailed background material on measure theory, Riemannian geometry and Banach space theory.  Parametrised measure models are defined as fundamental geometric objects, which can be both finite or infinite dimensional. Based on these models, canonical tensor fields are introduced and further studied, including the Fisher metric and the Amari-Chentsov tensor, and embeddings of statistical manifolds are investigated.

This novel foundation then leads to application highlights, such as generalizations and extensions of the classical uniqueness result of Chentsov or the Cramér-Rao inequality.  Additionally, several new application fields of information geometry are highlighted, for instance hierarchical and graphical models, complexity theory, population genetics, or Markov Chain Monte Carlo.

The book will be of interest to mathematicians who are interested in geometry, information theory, or the foundations of statistics, to statisticians as well as to scientists interested in the mathematical foundations of complex systems.

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Information Geometry and Its Applications

On the Occasion of Shun-ichi Amari's 80th Birthday, IGAIA IV Liblice, Czech Republic, June 2016

Editors: Nihat Ay, Paolo Gibilisco, František Matúš

Presents a collection of important papers by leading experts
Contains striking results that will be widely read and cited
Will become a milestone in the field of information geometry

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Springer Journal Information Geometry

Nihat Ay, Editor-in-Chief

This journal, as the first to be dedicated to the interdisciplinary field of information geometry:

• Embraces the challenge of uncovering and synthesizing mathematical foundations of information science;
• Offers a platform for intellectual engagements with overlapping interests and diverse backgrounds in mathematical science;
• Balances both theoretical and computational approaches, with ample attention to applications;
• Covers investigations of core concepts defining and studying invariance principles such as the Fisher–Rao metric, dual connection, divergence functions, exponential and mixture geodesics, information projections, and many more areas.

The journal engages its readership in geometrizing the science of information. It connects diverse branches of mathematical sciences that deal with probability, entropy, measurement, inference, and related concepts. Coverage includes original work and synthesis exploring the foundation and application of information geometry in both mathematical and computational aspects.

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