 Will become standard reference of the field
 Introduces new mathematical treatment of parametrised measure models
 Includes general results on sufficient statistics, CramérRao inequality, uniqueness of Fisher metric and AmariChentsov tensor
 Provides new applications of information geometry
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 About this book

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 AmariChentsov 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 CramérRao 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.
 About the authors

Nihat Ay studied mathematics and physics at the Ruhr University Bochum and received his Ph.D. in mathematics from the University of Leipzig in 2001. He was a Postdoc at the Santa Fe Institute and at the Redwood Center for Theoretical Neuroscience, UC Berkeley. Since 2005, he has been group leader at the Max Planck Institute for Mathematics in the Sciences in Leipzig, heading the Information Theory of Cognitive Systems group. His research interests are information geometry, complexity and information theory, mathematical learning theory, embodied cognitive systems, graphical models and causality, and robustness theory. Nihat Ay is Professor at the Santa Fe Institute and honorary Professor of information geometry at the University of Leipzig.
Jürgen Jost studied mathematics, physics, economics and philosophy in Bonn 19751980. He received his Ph.D. in mathematics in 1980, and was a Postdoc at IAS Princeton, UC San Diego, CMA Canberra and Bonn, a Professor at Ruhr University Bochum 19841996, and has been a Director of the Max Planck Institute for Mathematics in the Sciences, Leipzig, since 1996. He is an Honorary Professor of Leipzig University, external member of the Santa Fe Institute. He received the DFG Leibniz Award in 1993, and an ERC advanced grant in 2010. His research interests are Riemannian geometry, geometric analysis, dynamical systems, information theory, network analysis, mathematical biology and neurobiology, mathematical economics and complex systems theory.
Hông Vân Lê studied mathematics at the Moscow State University and received her Ph.D. in mathematics in 1987 and her DrSc in mathematics in 1990. She was a Postdoc at the Moscow State University, the International Center for Theoretical Physics (ICTP) in Trieste, the Max Planck Institute for Mathematics in Bonn, a Heisenberg Fellow at the Max Planck Institute for Mathematics in Bonn, the Henri Poincaré Institute in Paris, the Newton Institute in Cambridge and Leipzig University, a research associate at the Max Planck Institute for Mathematics in the Sciences in Leipzig, foreign Professor at the Abdus Salam School of Mathematics in Lahore, visiting Professor at the Vietnam National University for Sciences in Hanoi. She was awarded the Moscow Mathematical Society prize in 1990, the ICTP prize in 1991, and a DFG Heisenberg fellowship in 1994. Since 2005 she has been a senior researcher at the Czech Academy of Sciences Institute of Mathematics. Her research interests are Riemannian geometry, symplectic topology, representation theory, differential topology and information geometry.
Lorenz Schwachhöfer studied mathematics and computer science in Darmstadt, New Orleans (Tulane) and Philadelphia (UPenn) where he received his Ph.D. in 1992. He was a Postdoc in St.Louis (Washington Univ.), Bonn (Max Planck Institute for Mathematics) and Leipzig University where he completed his habilitation in 1998. He was a Professor (Chargé de cours) at ULB Brussels (20002003), and since 2003 he has been a full Professor at TU Dortmund University. In 2010, he received the Royal Academy of Science (Belgium) E. Catalan prize. His main research interests are in differential geometry and differential topology, representation theory and information geometry.
 Reviews

“The monograph is very timely and welcome. It will become a standard textbook and reference for the field. It is not only accessible and useful for scientists who work in probability or statistics, but also for all graduate students of mathematics.” (Andreas Knauf, Jahresbericht der Deutschen MathematikerVereinigung, Vol. 121, 2019)
“This book also provides a general framework that integrates the differential geometry into the functional analysis, it is an easy read for graduate students in mathematical statistics. I believe that it could be standard reference for information geometry.” (Etsuo Kumagai Hamada, Mathematical Reviews, June, 2018)“The book as well as [Shunichi Amari’s Information Geometry and its Applications] will remain standard textbooks on information geometry in the foreseeable future and will become the classics in the arena afterwards.” (Hirokazu Nishimura, zbMATH 1383.53002)
 Table of contents (6 chapters)


Introduction
Pages 123

Finite Information Geometry
Pages 25119

Parametrized Measure Models
Pages 121184

The Intrinsic Geometry of Statistical Models
Pages 185239

Information Geometry and Statistics
Pages 241293

Table of contents (6 chapters)
 Download Preface 1 PDF (58.4 KB)
 Download Sample pages 1 PDF (722.3 KB)
 Download Sample pages 2 PDF (780.6 KB)
 Download Table of contents PDF (437.1 KB)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Information Geometry
 Authors

 Nihat Ay
 Jürgen Jost
 Hong Van Le
 Lorenz Schwachhöfer
 Series Title
 Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
 Series Volume
 64
 Copyright
 2017
 Publisher
 Springer International Publishing
 Copyright Holder
 Springer International Publishing AG
 eBook ISBN
 9783319564784
 DOI
 10.1007/9783319564784
 Hardcover ISBN
 9783319564777
 Softcover ISBN
 9783319859217
 Series ISSN
 00711136
 Edition Number
 1
 Number of Pages
 XI, 407
 Number of Illustrations
 15 b/w illustrations
 Topics