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The Resource Information geometry, Nihat Ay, Jürgen Jost, Hông Vân Lê, Lorenz Schwachhöfer

Information geometry, Nihat Ay, Jürgen Jost, Hông Vân Lê, Lorenz Schwachhöfer

Label
Information geometry
Title
Information geometry
Statement of responsibility
Nihat Ay, Jürgen Jost, Hông Vân Lê, Lorenz Schwachhöfer
Creator
Contributor
Author
Subject
Language
eng
Summary
"The book provides a comprehensive introduction and a novel mathematical foundation of the field of information geometry with complete proofs and detailed background material on measure theory, Riemannian geometry and Banach space theory. Parametrised measure models are defined as fundamental geometric objects, which can be both finite or infinite dimensional. Based on these models, canonical tensor fields are introduced and further studied, including the Fisher metric and the Amari-Chentsov tensor, and embeddings of statistical manifolds are investigated.This novel foundation then leads to application highlights, such as generalizations and extensions of the classical uniqueness result of Chentsov or the Cramér-Rao inequality. Additionally, several new application fields of information geometry are highlighted, for instance hierarchical and graphical models, complexity theory, population genetics, or Markov Chain Monte Carlo. The book will be of interest to mathematicians who are interested in geometry, information theory, or the foundations of statistics, to statisticians as well as to scientists interested in the mathematical foundations of complex systems."--
Member of
Assigning source
Provided by publisher
Cataloging source
YDX
http://library.link/vocab/creatorDate
1970-
http://library.link/vocab/creatorName
Ay, Nihat
Dewey number
519.5
Index
index present
LC call number
QA276.23
Literary form
non fiction
Nature of contents
  • dictionaries
  • bibliography
http://library.link/vocab/relatedWorkOrContributorDate
1956-.
http://library.link/vocab/relatedWorkOrContributorName
  • Jost, Jürgen
  • Vân Lê, Hông
  • Schwachhöfer, Lorenz
Series statement
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge
Series volume
volume 64
http://library.link/vocab/subjectName
  • Geometrical models in statistics
  • Statistics
  • Statistical Theory and Methods
  • Data Structures
  • Differential Geometry
  • Convex and Discrete Geometry
  • Functional Analysis
  • Complex Systems
Label
Information geometry, Nihat Ay, Jürgen Jost, Hông Vân Lê, Lorenz Schwachhöfer
Instantiates
Publication
Copyright
Bibliography note
Includes bibliographical references and index
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
  • Information Geometry; Preface; Acknowledgements; Contents; Chapter 1: Introduction; 1.1 A Brief Synopsis; 1.2 An Informal Description; 1.2.1 The Fisher Metric and the Amari-Chentsov Structure for Finite Sample Spaces; 1.2.2 In nite Sample Spaces and Functional Analysis; 1.2.3 Parametric Statistics; 1.2.4 Exponential and Mixture Families from the Perspective of Differential Geometry; 1.2.5 Information Geometry and Information Theory; 1.3 Historical Remarks; 1.4 Organization of this Book; Chapter 2: Finite Information Geometry; 2.1 Manifolds of Finite Measures; 2.2 The Fisher Metric
  • 2.3 Gradient Fields2.4 The m- and e-Connections; 2.5 The Amari-Chentsov Tensor and the alpha-Connections; 2.5.1 The Amari-Chentsov Tensor; 2.5.2 The alpha-Connections; 2.6 Congruent Families of Tensors; 2.7 Divergences; 2.7.1 Gradient-Based Approach; 2.7.2 The Relative Entropy; 2.7.3 The alpha-Divergence; 2.7.4 The f-Divergence; 2.7.5 The q-Generalization of the Relative Entropy; 2.8 Exponential Families; 2.8.1 Exponential Families as Af ne Spaces; 2.8.2 Implicit Description of Exponential Families; 2.8.3 Information Projections; 2.9 Hierarchical and Graphical Models; 2.9.1 Interaction Spaces
  • 2.9.2 Hierarchical Models2.9.3 Graphical Models; Chapter 3: Parametrized Measure Models; 3.1 The Space of Probability Measures and the Fisher Metric; 3.2 Parametrized Measure Models; 3.2.1 The Structure of the Space of Measures; 3.2.2 Tangent Fibration of Subsets of Banach Manifolds; 3.2.3 Powers of Measures; 3.2.4 Parametrized Measure Models and k-Integrability; 3.2.5 Canonical n-Tensors of an n-Integrable Model; 3.2.6 Signed Parametrized Measure Models; 3.3 The Pistone-Sempi Structure; 3.3.1 e-Convergence; 3.3.2 Orlicz Spaces; 3.3.3 Exponential Tangent Spaces
  • Chapter 4: The Intrinsic Geometry of Statistical Models4.1 Extrinsic Versus Intrinsic Geometric Structures; 4.2 Connections and the Amari-Chentsov Structure; 4.3 The Duality Between Exponential and Mixture Families; 4.4 Canonical Divergences; 4.4.1 Dual Structures via Divergences; 4.4.2 A General Canonical Divergence; 4.4.3 Recovering the Canonical Divergence of a Dually Flat Structure; 4.4.4 Consistency with the Underlying Dualistic Structure; 4.5 Statistical Manifolds and Statistical Models; 4.5.1 Statistical Manifolds and Isostatistical Immersions
  • 4.5.2 Monotone Invariants of Statistical Manifolds4.5.3 Immersion of Compact Statistical Manifolds into Linear Statistical Manifolds; 4.5.4 Proof of the Existence of Isostatistical Immersions; 4.5.5 Existence of Statistical Embeddings; Chapter 5: Information Geometry and Statistics; 5.1 Congruent Embeddings and Suf cient Statistics; 5.1.1 Statistics and Congruent Embeddings; 5.1.2 Markov Kernels and Congruent Markov Embeddings; 5.1.3 Fisher-Neyman Suf cient Statistics; 5.1.4 Information Loss and Monotonicity; 5.1.5 Chentsov's Theorem and Its Generalization
Dimensions
unknown
Extent
1 online resource.
Form of item
online
Isbn
9783319564777
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Other control number
10.1007/978-3-319-56478-4
Specific material designation
remote
System control number
on1002203855
Label
Information geometry, Nihat Ay, Jürgen Jost, Hông Vân Lê, Lorenz Schwachhöfer
Publication
Copyright
Bibliography note
Includes bibliographical references and index
Carrier category
online resource
Carrier category code
  • cr
Carrier MARC source
rdacarrier
Content category
text
Content type code
  • txt
Content type MARC source
rdacontent
Contents
  • Information Geometry; Preface; Acknowledgements; Contents; Chapter 1: Introduction; 1.1 A Brief Synopsis; 1.2 An Informal Description; 1.2.1 The Fisher Metric and the Amari-Chentsov Structure for Finite Sample Spaces; 1.2.2 In nite Sample Spaces and Functional Analysis; 1.2.3 Parametric Statistics; 1.2.4 Exponential and Mixture Families from the Perspective of Differential Geometry; 1.2.5 Information Geometry and Information Theory; 1.3 Historical Remarks; 1.4 Organization of this Book; Chapter 2: Finite Information Geometry; 2.1 Manifolds of Finite Measures; 2.2 The Fisher Metric
  • 2.3 Gradient Fields2.4 The m- and e-Connections; 2.5 The Amari-Chentsov Tensor and the alpha-Connections; 2.5.1 The Amari-Chentsov Tensor; 2.5.2 The alpha-Connections; 2.6 Congruent Families of Tensors; 2.7 Divergences; 2.7.1 Gradient-Based Approach; 2.7.2 The Relative Entropy; 2.7.3 The alpha-Divergence; 2.7.4 The f-Divergence; 2.7.5 The q-Generalization of the Relative Entropy; 2.8 Exponential Families; 2.8.1 Exponential Families as Af ne Spaces; 2.8.2 Implicit Description of Exponential Families; 2.8.3 Information Projections; 2.9 Hierarchical and Graphical Models; 2.9.1 Interaction Spaces
  • 2.9.2 Hierarchical Models2.9.3 Graphical Models; Chapter 3: Parametrized Measure Models; 3.1 The Space of Probability Measures and the Fisher Metric; 3.2 Parametrized Measure Models; 3.2.1 The Structure of the Space of Measures; 3.2.2 Tangent Fibration of Subsets of Banach Manifolds; 3.2.3 Powers of Measures; 3.2.4 Parametrized Measure Models and k-Integrability; 3.2.5 Canonical n-Tensors of an n-Integrable Model; 3.2.6 Signed Parametrized Measure Models; 3.3 The Pistone-Sempi Structure; 3.3.1 e-Convergence; 3.3.2 Orlicz Spaces; 3.3.3 Exponential Tangent Spaces
  • Chapter 4: The Intrinsic Geometry of Statistical Models4.1 Extrinsic Versus Intrinsic Geometric Structures; 4.2 Connections and the Amari-Chentsov Structure; 4.3 The Duality Between Exponential and Mixture Families; 4.4 Canonical Divergences; 4.4.1 Dual Structures via Divergences; 4.4.2 A General Canonical Divergence; 4.4.3 Recovering the Canonical Divergence of a Dually Flat Structure; 4.4.4 Consistency with the Underlying Dualistic Structure; 4.5 Statistical Manifolds and Statistical Models; 4.5.1 Statistical Manifolds and Isostatistical Immersions
  • 4.5.2 Monotone Invariants of Statistical Manifolds4.5.3 Immersion of Compact Statistical Manifolds into Linear Statistical Manifolds; 4.5.4 Proof of the Existence of Isostatistical Immersions; 4.5.5 Existence of Statistical Embeddings; Chapter 5: Information Geometry and Statistics; 5.1 Congruent Embeddings and Suf cient Statistics; 5.1.1 Statistics and Congruent Embeddings; 5.1.2 Markov Kernels and Congruent Markov Embeddings; 5.1.3 Fisher-Neyman Suf cient Statistics; 5.1.4 Information Loss and Monotonicity; 5.1.5 Chentsov's Theorem and Its Generalization
Dimensions
unknown
Extent
1 online resource.
Form of item
online
Isbn
9783319564777
Media category
computer
Media MARC source
rdamedia
Media type code
  • c
Other control number
10.1007/978-3-319-56478-4
Specific material designation
remote
System control number
on1002203855

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