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Methods of Information Geometry

Shun-ichi Amari, Hiroshi Nagaoka ; [translated from the Japanese by Daishi Harada]
Format
Book
Published
Providence, RI : American Mathematical Society, c2000.
Language
English
Japanese (translated from)
Uniform Title
Jōhō Kika No Hōhō English
Series
Translations of Mathematical Monographs
ISBN
0821805312 (alk. paper)
Contents
  • 1 Elementary differential geometry 1
  • 1.1 Differentiable manifolds 1
  • 1.2 Tangent vectors and tangent spaces 5
  • 1.3 Vector fields and tensor fields 8
  • 1.4 Submanifolds 10
  • 1.5 Riemannian metrics 11
  • 1.6 Affine connections and covariant derivatives 13
  • 1.7 Flatness 17
  • 1.8 Autoparallel submanifolds 19
  • 1.9 Projection of connections and embedding curvature 22
  • 1.10 Riemannian connection 23
  • 2 Geometric structure of statistical models 25
  • 2.1 Statistical models 25
  • 2.2 Fisher metric 28
  • 2.3 [Alpha]-connection 32
  • 2.4 Chentsov's theorem and some historical remarks 37
  • 2.5 Geometry of P (X) 40
  • 2.6 [alpha]-affine manifolds and [alpha]-families 45
  • 3 Dual connections 51
  • 3.1 Duality of connections 51
  • 3.2 Divergences: general contrast functions 53
  • 3.3 Dually flat spaces 58
  • 3.4 Canonical divergence 61
  • 3.5 Dualistic structure of exponential families 65
  • 3.6 Dualistic structure of [alpha]-affine manifolds and [alpha]-families 70
  • 3.7 Mutually dual foliations 75
  • 3.8 A further look at the triangular relation 77
  • 4 Statistical inference and differential geometry 81
  • 4.1 Estimation based on independent observations 81
  • 4.2 Exponential families and observed points 85
  • 4.3 Curved exponential families 87
  • 4.4 Consistency and first-order efficiency 89
  • 4.5 Higher-order asymptotic theory of estimation 94
  • 4.6 Asymptotics of Fisher information 97
  • 4.7 Higher-order asymptotic theory of tests 100
  • 4.8 Theory of estimating functions and fiber bundles 107
  • 4.8.1 Fiber bundle of local exponential families 107
  • 4.8.2 Hilbert bundles and estimating functions 109
  • 5 Geometry of time series and linear systems 115
  • 5.1 Space of systems and time series 115
  • 5.2 Fisher metric and the [alpha]-connection on the system space 118
  • 5.3 Geometry of finite-dimensional models 123
  • 5.4 Stable systems and stable feedback 127
  • 6 Multiterminal information theory and statistical inference 133
  • 6.1 Statistical inference for multiterminal information 133
  • 6.2 0-rate testing 136
  • 6.3 0-rate estimation 140
  • 6.4 Inference for general multiterminal information 142
  • 7 Information geometry for quantum systems 145
  • 7.1 Quantum state space 145
  • 7.2 Geometric structure induced from a quantum divergence 150
  • 7.3 Geometric structure induced from a generalized covariance 154
  • 7.4 Applications to quantum estimation theory 159
  • 8 Miscellaneous topics 167
  • 8.1 Geometry of convex analysis, linear programming and gradient flows 167
  • 8.2 Neuro-manifolds and nonlinear systems 170
  • 8.3 Lie groups and transformation models in information geometry 172
  • 8.4 Mathematical problems posed by information geometry 175.
Description
x, 206 p. : ill. ; cm.
Notes
Includes bibliographical references (p. 187-202) and index.
Series Statement
Translations of mathematical monographs 0065-9282 ; v. 191
Technical Details
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  • Staff View

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    g| 1 t| Elementary differential geometry g| 1 -- g| 1.1 t| Differentiable manifolds g| 1 -- g| 1.2 t| Tangent vectors and tangent spaces g| 5 -- g| 1.3 t| Vector fields and tensor fields g| 8 -- g| 1.4 t| Submanifolds g| 10 -- g| 1.5 t| Riemannian metrics g| 11 -- g| 1.6 t| Affine connections and covariant derivatives g| 13 -- g| 1.7 t| Flatness g| 17 -- g| 1.8 t| Autoparallel submanifolds g| 19 -- g| 1.9 t| Projection of connections and embedding curvature g| 22 -- g| 1.10 t| Riemannian connection g| 23 -- g| 2 t| Geometric structure of statistical models g| 25 -- g| 2.1 t| Statistical models g| 25 -- g| 2.2 t| Fisher metric g| 28 -- g| 2.3 t| [Alpha]-connection g| 32 -- g| 2.4 t| Chentsov's theorem and some historical remarks g| 37 -- g| 2.5 t| Geometry of P (X) g| 40 -- g| 2.6 t| [alpha]-affine manifolds and [alpha]-families g| 45 -- g| 3 t| Dual connections g| 51 -- g| 3.1 t| Duality of connections g| 51 -- g| 3.2 t| Divergences: general contrast functions g| 53 -- g| 3.3 t| Dually flat spaces g| 58 -- g| 3.4 t| Canonical divergence g| 61 -- g| 3.5 t| Dualistic structure of exponential families g| 65 -- g| 3.6 t| Dualistic structure of [alpha]-affine manifolds and [alpha]-families g| 70 -- g| 3.7 t| Mutually dual foliations g| 75 -- g| 3.8 t| A further look at the triangular relation g| 77 -- g| 4 t| Statistical inference and differential geometry g| 81 -- g| 4.1 t| Estimation based on independent observations g| 81 -- g| 4.2 t| Exponential families and observed points g| 85 -- g| 4.3 t| Curved exponential families g| 87 -- g| 4.4 t| Consistency and first-order efficiency g| 89 -- g| 4.5 t| Higher-order asymptotic theory of estimation g| 94 -- g| 4.6 t| Asymptotics of Fisher information g| 97 -- g| 4.7 t| Higher-order asymptotic theory of tests g| 100 -- g| 4.8 t| Theory of estimating functions and fiber bundles g| 107 -- g| 4.8.1 t| Fiber bundle of local exponential families g| 107 -- g| 4.8.2 t| Hilbert bundles and estimating functions g| 109 -- g| 5 t| Geometry of time series and linear systems g| 115 -- g| 5.1 t| Space of systems and time series g| 115 -- g| 5.2 t| Fisher metric and the [alpha]-connection on the system space g| 118 -- g| 5.3 t| Geometry of finite-dimensional models g| 123 -- g| 5.4 t| Stable systems and stable feedback g| 127 -- g| 6 t| Multiterminal information theory and statistical inference g| 133 -- g| 6.1 t| Statistical inference for multiterminal information g| 133 -- g| 6.2 t| 0-rate testing g| 136 -- g| 6.3 t| 0-rate estimation g| 140 -- g| 6.4 t| Inference for general multiterminal information g| 142 -- g| 7 t| Information geometry for quantum systems g| 145 -- g| 7.1 t| Quantum state space g| 145 -- g| 7.2 t| Geometric structure induced from a quantum divergence g| 150 -- g| 7.3 t| Geometric structure induced from a generalized covariance g| 154 -- g| 7.4 t| Applications to quantum estimation theory g| 159 -- g| 8 t| Miscellaneous topics g| 167 -- g| 8.1 t| Geometry of convex analysis, linear programming and gradient flows g| 167 -- g| 8.2 t| Neuro-manifolds and nonlinear systems g| 170 -- g| 8.3 t| Lie groups and transformation models in information geometry g| 172 -- g| 8.4 t| Mathematical problems posed by information geometry g| 175.
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