# Methods of Information Geometry

American Mathematical Soc., 2007 - Mathematics - 206 pages
Information geometry provides the mathematical sciences with a new framework of analysis. It has emerged from the investigation of the natural differential geometric structure on manifolds of probability distributions, which consists of a Riemannian metric defined by the Fisher information and a one-parameter family of affine connections called the $\alpha$-connections. The duality between the $\alpha$-connection and the $(-\alpha)$-connection together with the metric play an essential role in this geometry. This kind of duality, having emerged from manifolds of probability distributions, is ubiquitous, appearing in a variety of problems which might have no explicit relation to probability theory. Through the duality, it is possible to analyze various fundamental problems in a unified perspective. The first half of this book is devoted to a comprehensive introduction to the mathematical foundation of information geometry, including preliminaries from differential geometry, the geometry of manifolds or probability distributions, and the general theory of dual affine connections. The second half of the text provides an overview of many areas of applications, such as statistics, linear systems, information theory, quantum mechanics, convex analysis, neural networks, and affine differential geometry. The book can serve as a suitable text for a topics course for advanced undergraduates and graduate students.

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### Contents

 Elementary differential geometry 1 The geometric structure of statistical models 25 Dual connections 51 Statistical inference and differential geometry 81 The geometry of time series and linear systems 115 Multiterminal information theory and statistical inference 133
 Information geometry for quantum systems 145 Miscellaneous topics 167 Guide to the Bibliography 181 Bibliography 187 Index 203 Copyright