Information Geometry and Its Applications

Description

This is the first comprehensive book on information geometry, written by the founder of the field. It begins with an elementary introduction to dualistic geometry and proceeds to a wide range of applications, covering information science, engineering, and neuroscience. It consists of four parts, which on the whole can be read independently. A manifold with a divergence function is first introduced, leading directly to dualistic structure, the heart of information geometry. This part (Part I) can be apprehended without any knowledge of differential geometry. An intuitive explanation of modern differential geometry then follows in Part II, although the book is for the most part understandable without modern differential geometry. Information geometry of statistical inference, including time series analysis and semiparametric estimation (the Neyman–Scott problem), is demonstrated concisely in Part III. Applications addressed in Part IV include hot current topics in machine learning, signal processing, optimization, and neural networks. The book is interdisciplinary, connecting mathematics, information sciences, physics, and neurosciences, inviting readers to a new world of information and geometry. This book is highly recommended to graduate students and researchers who seek new mathematical methods and tools useful in their own fields. Front Matter....Pages i-xiii Front Matter....Pages 1-1 Manifold, Divergence and Dually Flat Structure....Pages 3-30 Exponential Families and Mixture Families of Probability Distributions....Pages 31-49 Invariant Geometry of Manifold of Probability Distributions....Pages 51-69 \(\alpha \) -Geometry, Tsallis q-Entropy and Positive-Definite Matrices....Pages 71-106 Front Matter....Pages 107-107 Elements of Differential Geometry....Pages 109-130 Dual Affine Connections and Dually Flat Manifold....Pages 131-161 Front Matter....Pages 163-163 Asymptotic Theory of Statistical Inference....Pages 165-177 Estimation in the Presence of Hidden Variables....Pages 179-189 Neyman-Scott Problem: Estimating Function and Semiparametric Statistical Model....Pages 191-213 Linear Systems and Time Series....Pages 215-227 Front Matter....Pages 229-229 Machine Learning....Pages 231-278 Natural Gradient Learning and Its Dynamics in Singular Regions....Pages 279-314 Signal Processing and Optimization....Pages 315-358 Back Matter....Pages 359-373