Methods of Information Geometry
Seiten
2007
American Mathematical Society (Verlag)
978-0-8218-4302-4 (ISBN)
American Mathematical Society (Verlag)
978-0-8218-4302-4 (ISBN)
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Information geometry provides the mathematical sciences with a fresh framework of analysis. This book presents a comprehensive introduction to the mathematical foundation of information geometry. It provides an overview of many areas of applications, such as statistics, linear systems, information theory, quantum mechanics, and convex analysis.
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.
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.
Shun-ichi Amari, RIKEN Brain Science Institute, Saitama, Japan Hiroshi Nagaoka, University of Electro-Communications, Tokyo, Japan
Preface
Preface to the English edition
Elementary differential geometry
The geometric structure of statistical models
Dual connections
Statistical inference and differential geometry
The geometry of time series and linear systems
Multiterminal information theory and statistical inference
Information geometry for quantum systems
Miscellaneous topics
Guide to the bibliography
Bibliography
Index
| Erscheint lt. Verlag | 30.4.2007 |
|---|---|
| Reihe/Serie | Translations of Mathematical Monographs |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 175 x 246 mm |
| Gewicht | 391 g |
| Themenwelt | Mathematik / Informatik ► Informatik ► Theorie / Studium |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| ISBN-10 | 0-8218-4302-8 / 0821843028 |
| ISBN-13 | 978-0-8218-4302-4 / 9780821843024 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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