Methods of Information Geometry (Tanslations of Mathematical Monographs) (英語) ペーパーバック – 2007/4/13
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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 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.
... a welcome and much needed addition to the literature on the use of differential geometry methods in statistics, information theory and control theory (Mathematical Reviews ) --このテキストは、絶版本またはこのタイトルには設定されていない版型に関連付けられています。
The book starts with an introduction to differential geometry (DG). This chapter of the book is difficult to follow for those who have no background in differential geometry. Besides, the introduction contains detail definitions of advanced concepts that could be skipped in the first run. So, as an engineer, I would suggest my fellows to consider another books, e.g. by "W. M. Boothby" or "M. Do Carmo" for building the necessary background.
After the introduction, the statistical models and the fundamental notion of Fisher information and the most important features of Amari's IG, i.e. dual connections, are introduced in the second and third chapters. Reading these two chapters, needs a certain amount of patience for those engineers like myself who are more "goal-oriented".
Finally, the forth chapter provides the application of previously explained methods to the statistical inference and estimation. This part of the book is very informative, although not so smooth to follow. Despite the fact that I had to go back and forth through out this part of the book to find a smooth and thorough understanding of the concepts, I really enjoyed reading this part. The last part of the chapter focuses on more advances topics like higher-order asymptotic statistics (which might be not that necessary for many of engineers who mainly like to talk about first and second order statistics), and fiber bundles which could be skipped for many of us.
The last part of the book provides some examples for the application of IG, e.g. time-series analysis and identification of the linear systems, multiterminal distribution, IG of quantum information, convex optimization, ... These last four chapters could have been published as another editorial book with more details. This part of the book is useful for those who need some motivation for getting involved with IG. A comprehensive list of references is provided for serious readers who want to dive into the subjects.
All in all, Amari's book on IG is a "must-have" book for those interested in information geometry and differential geometry of statistics. The book is my daily reference for my research, as I learn more of it everyday. The book is useful for a wide spectrum of readers, e.g. for the beginners to find the big picture of IG, for advanced readers to find Amari's taste of IG, and for engineers to learn the fundamentals, see some applications, and use the results in a reasonable way.
It is important to mention that there are other approaches to IG. As an engineer, I have found this field very sophisticated and mature and sometimes confusing. So, before using the results of the book, it is a wise decision to always consider other references and consult the experts in the field.