**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: $83.00

Individual Member Price: $66.40

# Methods of Information Geometry

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*Shun-ichi Amari; Hiroshi Nagaoka*

*Translated by: Daishi Harada*

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 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.

#### Readership

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