Methods of information geometry
Author(s)
Bibliographic Information
Methods of information geometry
（Translations of mathematical monographs, v. 191）
American Mathematical Society , Oxford University Press, c2000
- : pbk
- Other Title
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情報幾何の方法 : JOHO KIKA NO HOHO
情報幾何の方法
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Note
Includes bibliographical reference(p. 187-202) and index
Description and Table of Contents
- Volume
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ISBN 9780821805312
Description
There has been strong progress in the field of information geometry in recent years and yet there are few textbooks which reflect these developments, except for those which deal with statistics. Information geometry provides a new method applicable to various areas including information sciences and physical sciences. It has emerged from investigating the geometrical structures of the manifold of probability distributions, and has been applied successfully to statistical inference problems. However, it has been proved that information geometry opens a new paradigm useful for elucidation of information systems, intelligent systems, physical systems and mathematical systems. Information geometry has become one of the fundamental methods of analyzing neurocomputing and related areas. Its usefulness has also been recognized in multiterminal information theory and portfolio, in nonlinear systems and nonlinear prediction, in mathematical programming, in statistical inference and information theory of quantum mechanical systems, and so on. Its mathematical foundations have also shown a remarkable progress.
The first three chapters of the book provide a comprehensive introduction to the mathematical foundation of information geometry while the remaining chapters provide an overview of the scope of its application. It is hoped that Methods of Information Geometry will become a key text in information geometry and help to inspire further developments in the field.
Table of Contents
- 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
- BIBILOGRAPHY
- INDEX
- Volume
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: pbk ISBN 9780821843024
Description
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.
Table of Contents
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
by "Nielsen BookData"
