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Product details

  • Paperback: 206 pages
  • Publisher: OUP Oxford (21 Feb. 2008)
  • Language: English
  • ISBN-10: 0821843028
  • ISBN-13: 978-0821843024
  • Product Dimensions: 1.3 x 17.1 x 24.8 cm
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Bestsellers Rank: 1,152,122 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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Review

... 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 ) --This text refers to an out of print or unavailable edition of this title.

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Format: Paperback Verified Purchase
I have a love-hate relationship with this book. As a non-mathematician who cannot properly understand anything without knowing why I would want to, this is one of those books that takes a very long time to get a grip with.

"Yes, OK, great... but why???"

The procession from one equation to the next is completely unmotivated and the text is sparse and badly written. It conforms exactly to the stereotype of mathematics as abstruse and oblivious to practicality.

None-the-less, I have found it to be an indispensable reference, and quite inspiring at times.

The introduction to differential geometry follows the introduction in The Large Scale Structure of Space-Time (Cambridge Monographs on Mathematical Physics) very closely - it's concise and thorough. The meat, chapters 2-3 paint a nice, though somewhat obscurely presented picture of divergences and differential manifolds. I have found content of these chapters to be very useful, though as I said, it took me a long time to do so. I have found chapter 4 to have some interesting things to say on testing and estimation. It's definitely a nice way of thinking about statistics (though not a good way of communicating them, as basically no-one will know what you are talking about). I can't say much about the latter chapters, though I have read them they are still beyond my horizon of practicality (barring some of the last chapter) - I can see that one thing follows from the next - but why does it matter?! dammit! There are some examples, but they are still miles from any application.

In summary, I have found it very interesting and helpful, despite it being a completely infuriating read.
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Most Helpful Customer Reviews on Amazon.com (beta)

Amazon.com: 1 review
27 of 27 people found the following review helpful
Amari's Information Geometry 26 May 2005
By Amin Zia - Published on Amazon.com
Format: Hardcover
This book is a comprehensive introduction to Amari's approach to information geometry (IG).

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.

Amin Zia
PhD Student
McMaster University
Hamilton, Ontario
Canada
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