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Differential Geometry and Statistics : Monographs on Statistics and Applied Probability - Michael K. Murray

Differential Geometry and Statistics

Monographs on Statistics and Applied Probability

Hardcover Published: 1st April 1993
ISBN: 9780412398605
Number Of Pages: 288

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Several years ago our statistical friends and relations introduced us to the work of Amari and Barndorff-Nielsen on applications of differential geometry to statistics. This book has arisen because we believe that there is a deep relationship between statistics and differential geometry and moreoever that this relationship uses parts of differential geometry, particularly its 'higher-order' aspects not readily accessible to a statistical audience from the existing literature. It is, in part, a long reply to the frequent requests we have had for references on differential geometry! While we have not gone beyond the path-breaking work of Amari and Barndorff­ Nielsen in the realm of applications, our book gives some new explanations of their ideas from a first principles point of view as far as geometry is concerned. In particular it seeks to explain why geometry should enter into parametric statistics, and how the theory of asymptotic expansions involves a form of higher-order differential geometry. The first chapter of the book explores exponential families as flat geometries. Indeed the whole notion of using log-likelihoods amounts to exploiting a particular form of flat space known as an affine geometry, in which straight lines and planes make sense, but lengths and angles are absent. We use these geometric ideas to introduce the notion of the second fundamental form of a family whose vanishing characterises precisely the exponential families.

Industry Reviews

"This book does an excellent job of explaining... geometrical ideas from a statistical point of view, using statistical examples as opposed to, for example, dynamical examples to illustrate the geometric concepts. It should prove most helpful in enlarging the typical statistician's working geometric vocabulary." -Short Book Reviews of the ISI "It is particularly good on the... problem of when to use coordinates and indices, and when to use the more geometric coordinate-free methods. Examples of this are its treatments of the affine connection and of the string theory of Barndorff-Nielsen, both of which are excellent. The first of these concepts is probably the most sophisticated that has been used in statistical applications. It can be viewed in many different ways, not all of which are necessarily useful to the statistician; the approach taken in the book is both relevant and clear." -Bulletin of London Mathematical Society

Preface
The geometry of exponential familiesp. 1
Calculus on manifoldsp. 25
Statistical manifoldsp. 63
Connectionsp. 97
Curvaturep. 132
Information metrics and statistical divergencesp. 157
Asymptoticsp. 194
Bundles and tensorsp. 223
Higher order geometryp. 243
Referencesp. 264
Notation indexp. 267
Subject indexp. 270
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780412398605
ISBN-10: 0412398605
Series: Monographs on Statistics and Applied Probability
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 288
Published: 1st April 1993
Publisher: Taylor & Francis Ltd
Country of Publication: US
Dimensions (cm): 22.86 x 13.97  x 1.91
Weight (kg): 0.41
Edition Number: 1

Earn 414 Qantas Points
on this Book