Softcover ISBN:  9780821843024 
Product Code:  MMONO/191.S 
List Price:  $165.00 
MAA Member Price:  $148.50 
AMS Member Price:  $132.00 
eBook ISBN:  9781470446055 
Product Code:  MMONO/191.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 
Softcover ISBN:  9780821843024 
eBook: ISBN:  9781470446055 
Product Code:  MMONO/191.S.B 
List Price:  $320.00 $242.50 
MAA Member Price:  $288.00 $218.25 
AMS Member Price:  $256.00 $194.00 
Softcover ISBN:  9780821843024 
Product Code:  MMONO/191.S 
List Price:  $165.00 
MAA Member Price:  $148.50 
AMS Member Price:  $132.00 
eBook ISBN:  9781470446055 
Product Code:  MMONO/191.E 
List Price:  $155.00 
MAA Member Price:  $139.50 
AMS Member Price:  $124.00 
Softcover ISBN:  9780821843024 
eBook ISBN:  9781470446055 
Product Code:  MMONO/191.S.B 
List Price:  $320.00 $242.50 
MAA Member Price:  $288.00 $218.25 
AMS Member Price:  $256.00 $194.00 

Book DetailsTranslations of Mathematical MonographsVolume: 191; 2000; 206 ppMSC: Primary 00; 53; 62; 93; 81; 94
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 oneparameter 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 wide areas of applications, such as statistics, linear systems, information theory, quantum mechanics, convex analysis, neural networks, and affine differential geometry. The book will serve as a suitable text for a topics course for advanced undergraduates and graduate students.
This volume is copublished by the AMS and Oxford University Press. The AMS has exclusive distribution rights in North America. AMS members in Europe may purchase the book from the AMS. Both the AMS and OUP have worldwide distribution rights.
ReadershipAdvanced undergraduates, graduate students, and research mathematicians interested in differential geometry, statistics, probability theory, information theory, and physics; applied mathematicians.

Table of Contents

Chapters

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


Additional Material

Reviews

Presents for the first time the entirety of the emerging field of information geometry ... this book will play a key role for the further progress of information geometry.
Zentralblatt MATH 
Until now there have been no expository textbooks covering the methods and applications of information geometry. The present monograph will be extremely useful in dissemination of knowledge in this area and promoting muchneeded research ...
This monograph is a welcome and muchneeded addition to the literature on the use of differential geometry methods in statistics, information theory and control theory. Research workers in statistics and information theory will greatly benefit by studying the concepts and methods discussed in the monograph.
Mathematical Reviews


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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 oneparameter 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 wide areas of applications, such as statistics, linear systems, information theory, quantum mechanics, convex analysis, neural networks, and affine differential geometry. The book will serve as a suitable text for a topics course for advanced undergraduates and graduate students.
This volume is copublished by the AMS and Oxford University Press. The AMS has exclusive distribution rights in North America. AMS members in Europe may purchase the book from the AMS. Both the AMS and OUP have worldwide distribution rights.
Advanced undergraduates, graduate students, and research mathematicians interested in differential geometry, statistics, probability theory, information theory, and physics; applied mathematicians.

Chapters

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

Presents for the first time the entirety of the emerging field of information geometry ... this book will play a key role for the further progress of information geometry.
Zentralblatt MATH 
Until now there have been no expository textbooks covering the methods and applications of information geometry. The present monograph will be extremely useful in dissemination of knowledge in this area and promoting muchneeded research ...
This monograph is a welcome and muchneeded addition to the literature on the use of differential geometry methods in statistics, information theory and control theory. Research workers in statistics and information theory will greatly benefit by studying the concepts and methods discussed in the monograph.
Mathematical Reviews