Information Geometry and Its Applications

1. Front Matter

   Pages i-xiii

   PDF

2. Geometry of Divergence Functions: Dually Flat Riemannian Structure

   1. Front Matter

      Pages 1-1

      PDF

   2. Manifold, Divergence and Dually Flat Structure

      • Shun-ichi Amari

      Pages 3-30

   3. Exponential Families and Mixture Families of Probability Distributions

      • Shun-ichi Amari

      Pages 31-49

   4. Invariant Geometry of Manifold of Probability Distributions

      • Shun-ichi Amari

      Pages 51-69

   5. \(\alpha \)-Geometry, Tsallis q-Entropy and Positive-Definite Matrices

      • Shun-ichi Amari

      Pages 71-106

3. Introduction to Dual Differential Geometry

   1. Front Matter

      Pages 107-107

      PDF

   2. Elements of Differential Geometry

      • Shun-ichi Amari

      Pages 109-130

   3. Dual Affine Connections and Dually Flat Manifold

      • Shun-ichi Amari

      Pages 131-161

4. Information Geometry of Statistical Inference

   1. Front Matter

      Pages 163-163

      PDF

   2. Asymptotic Theory of Statistical Inference

      • Shun-ichi Amari

      Pages 165-177

   3. Estimation in the Presence of Hidden Variables

      • Shun-ichi Amari

      Pages 179-189

   4. Neyman-Scott Problem: Estimating Function and Semiparametric Statistical Model

      • Shun-ichi Amari

      Pages 191-213

   5. Linear Systems and Time Series

      • Shun-ichi Amari

      Pages 215-227

5. Applications of Information Geometry

   1. Front Matter

      Pages 229-229

      PDF

   2. Machine Learning

      • Shun-ichi Amari

      Pages 231-278

   3. Natural Gradient Learning and Its Dynamics in Singular Regions

      • Shun-ichi Amari

      Pages 279-314

   4. Signal Processing and Optimization

      • Shun-ichi Amari

      Pages 315-358

6. Correction to: Information Geometry and Its Applications

   • Shun-ichi Amari

   Pages C1-C2

7. Back Matter

   Pages 359-374

   PDF

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.

• Dual differential geometry
• Information geometry
• Machine learning
• Mathematical neuroscience
• Natural gradient learning
• Signal processing

“This book gives a reasonably accessible introduction to the subject and then considers various applications. … The book provides a nice introduction to the subject. … the book provides a nice introduction to a difficult subject that has many important applications.” (Marvin H. J. Gruber, Technometrics, Vol. 58 (4), April, 2016) 

• Brain Science Institute, RIKEN, Wako, Japan

  Shun-ichi Amari

Policies and ethics