Information Geometry and Its Applications

Description

Preface; Contents; Part I Geometry of Divergence Functions: Dually Flat Riemannian Structure; 1 Manifold, Divergence and Dually Flat Structure; 1.1 Manifolds; 1.1.1 Manifold and Coordinate Systems; 1.1.2 Examples of Manifolds; 1.2 Divergence Between Two Points; 1.2.1 Divergence; 1.2.2 Examples of Divergence; 1.3 Convex Function and Bregman Divergence; 1.3.1 Convex Function; 1.3.2 Bregman Divergence; 1.4 Legendre Transformation; 1.5 Dually Flat Riemannian Structure Derived from Convex Function; 1.5.1 Affine and Dual Affine Coordinate Systems.;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. Preface Contents Part I Geometry of Divergence Functions: Dually Flat Riemannian Structure 1 Manifold, Divergence and Dually Flat Structure 1.1 Manifolds 1.1.1 Manifold and Coordinate Systems 1.1.2 Examples of Manifolds 1.2 Divergence Between Two Points 1.2.1 Divergence 1.2.2 Examples of Divergence 1.3 Convex Function and Bregman Divergence 1.3.1 Convex Function 1.3.2 Bregman Divergence 1.4 Legendre Transformation 1.5 Dually Flat Riemannian Structure Derived from Convex Function 1.5.1 Affine and Dual Affine Coordinate Systems. 1.5.2 Tangent Space, Basis Vectors and Riemannian Metric1.5.3 Parallel Transport of Vector 1.6 Generalized Pythagorean Theorem and Projection Theorem 1.6.1 Generalized Pythagorean Theorem 1.6.2 Projection Theorem 1.6.3 Divergence Between Submanifolds: Alternating Minimization Algorithm 2 Exponential Families and Mixture Families of Probability Distributions 2.1 Exponential Family of Probability Distributions 2.2 Examples of Exponential Family: Gaussian and Discrete Distributions 2.2.1 Gaussian Distribution 2.2.2 Discrete Distribution 2.3 Mixture Family of Probability Distributions. 2.4 Flat Structure: e-flat and m-flat2.5 On Infinite-Dimensional Manifold of Probability Distributions 2.6 Kernel Exponential Family 2.7 Bregman Divergence and Exponential Family 2.8 Applications of Pythagorean Theorem 2.8.1 Maximum Entropy Principle 2.8.2 Mutual Information 2.8.3 Repeated Observations and Maximum Likelihood Estimator 3 Invariant Geometry of Manifold of Probability Distributions 3.1 Invariance Criterion 3.2 Information Monotonicity Under Coarse Graining 3.2.1 Coarse Graining and Sufficient Statistics in Sn 3.2.2 Invariant Divergence.