Differential Geometry and Statistics

Nonfiction, Science & Nature, Mathematics, Statistics
Cover of the book Differential Geometry and Statistics by M.K. Murray, CRC Press
View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart
Author: M.K. Murray ISBN: 9781351455114
Publisher: CRC Press Publication: October 19, 2017
Imprint: Routledge Language: English
Author: M.K. Murray
ISBN: 9781351455114
Publisher: CRC Press
Publication: October 19, 2017
Imprint: Routledge
Language: English

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.

View on Amazon View on AbeBooks View on Kobo View on B.Depository View on eBay View on Walmart

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.

More books from CRC Press

Cover of the book Topographic Laser Ranging and Scanning by M.K. Murray
Cover of the book Nonlinear Analysis of Structures (1997) by M.K. Murray
Cover of the book Introduction to Mathematical Oncology by M.K. Murray
Cover of the book Electric Systems, Dynamics, and Stability with Artificial Intelligence Applications by M.K. Murray
Cover of the book Management of Construction Projects by M.K. Murray
Cover of the book A Primer of Algebraic Geometry by M.K. Murray
Cover of the book Turbomachinery by M.K. Murray
Cover of the book Painting and Decorating by M.K. Murray
Cover of the book Silicon Nanomaterials Sourcebook by M.K. Murray
Cover of the book Environmental Management in Construction by M.K. Murray
Cover of the book Intelligent Systems by M.K. Murray
Cover of the book Medical Leadership by M.K. Murray
Cover of the book The Fearless World of Professional Safety in the 21st Century by M.K. Murray
Cover of the book Two-Stroke Cycle Engine by M.K. Murray
Cover of the book Chinese Dates by M.K. Murray
We use our own "cookies" and third party cookies to improve services and to see statistical information. By using this website, you agree to our Privacy Policy