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| Author | : Hiroshi Matsuzoe |
| Publisher | : |
| Total Pages | : 350 |
| Release | : 2015-11-30 |
| Genre | : Mathematics |
| ISBN | : 9789814618762 |
| Author | : Shun'ichi Amari |
| Publisher | : IMS |
| Total Pages | : 254 |
| Release | : 1987 |
| Genre | : Geometry, Differential |
| ISBN | : 9780940600126 |
| Author | : Robert E. Kass |
| Publisher | : Wiley-Interscience |
| Total Pages | : 384 |
| Release | : 1997-07-17 |
| Genre | : Mathematics |
| ISBN | : |
Differential geometry provides an aesthetically appealing and often revealing view of statistical inference. Beginning with an elementary treatment of one-parameter statistical models and ending with an overview of recent developments, this is the first book to provide an introduction to the subject that is largely accessible to readers not already familiar with differential geometry. It also gives a streamlined entry into the field to readers with richer mathematical backgrounds. Much space is devoted to curved exponential families, which are of interest not only because they may be studied geometrically but also because they are analytically convenient, so that results may be derived rigorously. In addition, several appendices provide useful mathematical material on basic concepts in differential geometry. Topics covered include the following: Basic properties of curved exponential families Elements of second-order, asymptotic theory The Fisher-Efron-Amari theory of information loss and recovery Jeffreys-Rao information-metric Riemannian geometry Curvature measures of nonlinearity Geometrically motivated diagnostics for exponential family regression Geometrical theory of divergence functions A classification of and introduction to additional work in the field
| Author | : Shun-ichi Amari |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 302 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461250560 |
From the reviews: "In this Lecture Note volume the author describes his differential-geometric approach to parametrical statistical problems summarizing the results he had published in a series of papers in the last five years. The author provides a geometric framework for a special class of test and estimation procedures for curved exponential families. ... ... The material and ideas presented in this volume are important and it is recommended to everybody interested in the connection between statistics and geometry ..." #Metrika#1 "More than hundred references are given showing the growing interest in differential geometry with respect to statistics. The book can only strongly be recommended to a geodesist since it offers many new insights into statistics on a familiar ground." #Manuscripta Geodaetica#2
| Author | : M.K. Murray |
| Publisher | : Routledge |
| Total Pages | : 292 |
| Release | : 2017-10-19 |
| Genre | : Mathematics |
| ISBN | : 1351455117 |
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.
| Author | : Brayan Cox |
| Publisher | : NY Research Press |
| Total Pages | : 0 |
| Release | : 2023-09-19 |
| Genre | : Mathematics |
| ISBN | : 9781647254636 |
Information geometry refers to an interdisciplinary field that studies statistics and probability theory by applying the techniques of differential geometry. It investigates statistical manifolds, which are Riemannian manifolds, where each point is a probability distribution. Information geometry offers a differential-geometric structure on manifolds that help in designing and studying statistical decision rules. It has been utilized in a variety of applications such as machine learning, quantum systems, mathematical finance, statistical inference, and neural networks. The amount of information integration within various terminals of a causal dynamical system is measured using information geometry. The amount of information lost can be measured using integrated information when a system is divided into parts and information transmission between the parts is stopped. This book provides comprehensive insights into information geometry and its applications. It presents researches and studies performed by experts across the globe. The book aims to equip students and experts with the advanced topics and upcoming concepts in this area of study.
| Author | : Shun'ichi Amari |
| Publisher | : |
| Total Pages | : 206 |
| Release | : 2000 |
| Genre | : Geometry, Differential |
| ISBN | : 9781470446055 |
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 ubiquit.
| Author | : Ovidiu Calin |
| Publisher | : Springer |
| Total Pages | : 389 |
| Release | : 2014-07-17 |
| Genre | : Mathematics |
| ISBN | : 3319077791 |
This book covers topics of Informational Geometry, a field which deals with the differential geometric study of the manifold probability density functions. This is a field that is increasingly attracting the interest of researchers from many different areas of science, including mathematics, statistics, geometry, computer science, signal processing, physics and neuroscience. It is the authors’ hope that the present book will be a valuable reference for researchers and graduate students in one of the aforementioned fields. This textbook is a unified presentation of differential geometry and probability theory, and constitutes a text for a course directed at graduate or advanced undergraduate students interested in applications of differential geometry in probability and statistics. The book contains over 100 proposed exercises meant to help students deepen their understanding, and it is accompanied by software that is able to provide numerical computations of several information geometric objects. The reader will understand a flourishing field of mathematics in which very few books have been written so far.
| Author | : Shun-ichi Amari |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 220 |
| Release | : 2000 |
| Genre | : Computers |
| ISBN | : 9780821843024 |
Information geometry provides the mathematical sciences with a fresh framework of analysis. This book presents a comprehensive introduction to the mathematical foundation of information geometry. It provides an overview of many areas of applications, such as statistics, linear systems, information theory, quantum mechanics, and convex analysis.
| Author | : Ion Mihai |
| Publisher | : MDPI |
| Total Pages | : 166 |
| Release | : 2019-11-21 |
| Genre | : Mathematics |
| ISBN | : 303921800X |
The present book contains 14 papers published in the Special Issue “Differential Geometry” of the journal Mathematics. They represent a selection of the 30 submissions. This book covers a variety of both classical and modern topics in differential geometry. We mention properties of both rectifying and affine curves, the geometry of hypersurfaces, angles in Minkowski planes, Euclidean submanifolds, differential operators and harmonic forms on Riemannian manifolds, complex manifolds, contact manifolds (in particular, Sasakian and trans-Sasakian manifolds), curvature invariants, and statistical manifolds and their submanifolds (in particular, Hessian manifolds). We wish to mention that among the authors, there are both well-known geometers and young researchers. The authors are from countries with a tradition in differential geometry: Belgium, China, Greece, Japan, Korea, Poland, Romania, Spain, Turkey, and United States of America. Many of these papers were already cited by other researchers in their articles. This book is useful for specialists in differential geometry, operator theory, physics, and information geometry as well as graduate students in mathematics.
