Methods of Information Geometry

Front Cover
American Mathematical Soc., 2007 - Mathematics - 206 pages
0 Reviews
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.
  

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Elementary differential geometry
1
The geometric structure of statistical models
25
Dual connections
51
Statistical inference and differential geometry
81
The geometry of time series and linear systems
115
Multiterminal information theory and statistical inference
133
Information geometry for quantum systems
145
Miscellaneous topics
167
Guide to the Bibliography
181
Bibliography
187
Index
203
Copyright

Common terms and phrases

References to this book

All Book Search results »

Bibliographic information