**Translations of Mathematical Monographs**

2000;
206 pp;
Softcover

MSC: Primary 00; 53; 62; 93; 81; 94;

**Print ISBN: 978-0-8218-4302-4
Product Code: MMONO/191.S**

List Price: $88.00

AMS Member Price: $70.40

MAA Member Price: $79.20

**Electronic ISBN: 978-1-4704-4605-5
Product Code: MMONO/191.E**

List Price: $83.00

AMS Member Price: $66.40

MAA Member Price: $74.70

# Methods of Information Geometry

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*Shun-ichi Amari; Hiroshi Nagaoka*

A co-publication of the AMS and Oxford University Press

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 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.

#### Readership

Advanced undergraduates, graduate students, and research mathematicians interested in differential geometry, statistics, probability theory, information theory, and physics; applied mathematicians.

#### Reviews & Endorsements

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 much-needed research …

This monograph is a welcome and much-needed 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

#### Table of Contents

# Table of Contents

## Methods of Information Geometry

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Preface to the English edition ix10
- Elementary differential geometry 112
- The geometric structure of statistical models 2536
- Dual connections 5162
- Statistical inference and differential geometry 8192
- The geometry of time series and linear systems 115126
- Multiterminal information theory and statistical inference 133144
- Information geometry for quantum systems 145156
- Miscellaneous topics 167178
- Guide to the bibliography 181192
- Bibliography 187198
- Index 203214
- Back Cover Back Cover1218