Information geometry and its applications

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Information geometry and its applications

種類:: 電子ブック
責任表示:: Shun-ichi Amari
出版情報:: [Japan] : Springer, 2016
著者名:: Amari, Shun'ichi,
シリーズ名:: Applied mathematical sciences, ; volume 194
ISBN:: 9784431559788 [4431559787]
注記:: 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, optimizati
Includes bibliographical references and index
Online resource; title from PDF title page (SpringerLink, viewed February 16, 2016)
ローカル注記:: 学内専用E-BOOKS (local access only)

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https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=1165017

所蔵情報

子書誌情報

タイトルが類似している資料

1. Information Geometry and Its Applications. 1st ed. 2016
by Shun-ichi Amari , Springer Japan : Imprint: Springer , 2016

詳細

主題:

Electronic books

分類・件名:

DC23 : 510.1154
BSH : Electronic books
LCSH : Electronic books
LCSH : Information theory -- Mathematics
LCSH : Information theory in mathematics
LCSH : Mathematical statistics
LCSH : Geometry, Differential
BISACSH : MATHEMATICS / Essays
BISACSH : MATHEMATICS / Pre-Calculus
BISACSH : MATHEMATICS / Reference
FREE : Geometry, Differential
FREE : Information theory in mathematics
FREE : Information theory -- Mathematics
FREE : Mathematical statistics

TTLL:

eng

言語:

英語

形態:

1 online resource (xiii, 373 pages) : illustrations

目次情報:

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
3.3 Examples of f-Divergence in Sn3.3.1 KL-Divergence; 3.3.2 χ2-Divergence; 3.3.3 α-Divergence; 3.4 General Properties of f-Divergence and KL-Divergence; 3.4.1 Properties of f-Divergence; 3.4.2 Properties of KL-Divergence; 3.5 Fisher Information: The Unique Invariant Metric; 3.6 f-Divergence in Manifold of Positive Measures; 4 α-Geometry, Tsallis q-Entropy and Positive-Definite Matrices; 4.1 Invariant and Flat Divergence; 4.1.1 KL-Divergence Is Unique; 4.1.2 α-Divergence Is Unique in Rn+ ; 4.2 α-Geometry in Sn and Rn+; 4.2.1 α-Geodesic and α-Pythagorean Theorem in Rn+
4.2.2 α-Geodesic in Sn4.2.3 α-Pythagorean Theorem and α-Projection Theorem in Sn; 4.2.4 Apportionment Due to α-Divergence; 4.2.5 α-Mean; 4.2.6 α-Families of Probability Distributions; 4.2.7 Optimality of α-Integration; 4.2.8 Application to α-Integration of Experts; 4.3 Geometry of Tsallis q-Entropy; 4.3.1 q-Logarithm and q-Exponential Function; 4.3.2 q-Exponential Family (α-Family) of Probability Distributions; 4.3.3 q-Escort Geometry; 4.3.4 Deformed Exponential Family: χ-Escort Geometry; 4.3.5 Conformal Character of q-Escort Geometry

書誌ID:

KB00017657

XISBN:

9784431559771 [4431559779]

内容紹介

目次情報

類似資料:

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