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Table of Contents
Intro; Contents; Introduction; Part I Applications of Information Geometry; Geometry of Information Integration; 1 Introduction; 2 Markovian Dynamical Systems; 3 Stochastic Models of Causal Systems; 3.1 Full Model; 3.2 Fully Split Model; 3.3 Diagonally Split Graphical Model; 3.4 Causally Split Model (Geometric Model); 3.5 Mismatched Decoding Model; 4 Comparison of Various Measures of Integrated Information; 5 Conclusions; References; Information Geometry and Game Theory; 1 Introduction; 2 Preliminaries; 3 Information Geometry; 4 The Nash Equilibrium Theorem; 5 Variations of QREs; 6 An Example
7 Another Example
Not so Nice8 Controlling the Game; 9 Channel Dependence; References; Higher Order Equivalence of Bayes Cross Validation and WAIC; 1 Introduction; 2 Main Results; 2.1 Definitions and Conditions; 2.2 Main Theorem; 3 Example; 4 Basic Lemmas; 5 Proof of Theorem1; 5.1 Proof of Theorem1, Cross Validation; 5.2 Proof of Theorem1, WAIC; 5.3 Mathematical Relations Between Priors; 5.4 Proof of Theorem1, Averages; 5.5 Proof of Theorem1, Random Generalization Loss; 5.6 Proof of Theorem2; 6 Discussions; 6.1 Summary of Results; 6.2 Divergence Phenomenon of CV and WAIC
6.3 Training and Testing Sets7 Conclusion; References; Restricted Boltzmann Machines: Introduction and Review; 1 Introduction; 2 Boltzmann Machines; 3 Restricted Boltzmann Machines; 4 Basics of Training; 5 Dimension; 6 Representational Power; 7 Divergence Bounds; 8 Implicit Description; 9 Open Problems; References; Part II Infinite-Dimensional Information Geometry; Information Geometry of the Gaussian Space; 1 Introduction; 2 Orlicz Spaces with Gaussian Weight; 2.1 Generalities; 2.2 Entropy; 2.3 Orlicz and Lebesgue Spaces; 3 Notable Bounds and Examples; 3.1 Polynomial Bounds
3.2 Densities of Exponential Form3.3 Poincaré-Type Inequalities; 4 Exponential Manifold on the Gaussian Space; 4.1 Maximal Exponential Manifold as an Affine Manifold; 4.2 Translations and Mollifiers; 4.3 Gaussian Statistical Bundle; 5 Weighted Orlicz-Sobolev Model Space; 5.1 Orlicz-Sobolev Spaces with Gaussian Weight; 6 Conclusions; References; Congruent Families and Invariant Tensors; 1 Introduction; 2 Preliminary Results; 2.1 The Space of (Signed) Finite Measures and Their Powers; 2.2 Parametrized Measure Models; 2.3 Congruent Markov Morphisms; 2.4 Tensor Algebras; 2.5 Tensor Fields
7 Another Example
Not so Nice8 Controlling the Game; 9 Channel Dependence; References; Higher Order Equivalence of Bayes Cross Validation and WAIC; 1 Introduction; 2 Main Results; 2.1 Definitions and Conditions; 2.2 Main Theorem; 3 Example; 4 Basic Lemmas; 5 Proof of Theorem1; 5.1 Proof of Theorem1, Cross Validation; 5.2 Proof of Theorem1, WAIC; 5.3 Mathematical Relations Between Priors; 5.4 Proof of Theorem1, Averages; 5.5 Proof of Theorem1, Random Generalization Loss; 5.6 Proof of Theorem2; 6 Discussions; 6.1 Summary of Results; 6.2 Divergence Phenomenon of CV and WAIC
6.3 Training and Testing Sets7 Conclusion; References; Restricted Boltzmann Machines: Introduction and Review; 1 Introduction; 2 Boltzmann Machines; 3 Restricted Boltzmann Machines; 4 Basics of Training; 5 Dimension; 6 Representational Power; 7 Divergence Bounds; 8 Implicit Description; 9 Open Problems; References; Part II Infinite-Dimensional Information Geometry; Information Geometry of the Gaussian Space; 1 Introduction; 2 Orlicz Spaces with Gaussian Weight; 2.1 Generalities; 2.2 Entropy; 2.3 Orlicz and Lebesgue Spaces; 3 Notable Bounds and Examples; 3.1 Polynomial Bounds
3.2 Densities of Exponential Form3.3 Poincaré-Type Inequalities; 4 Exponential Manifold on the Gaussian Space; 4.1 Maximal Exponential Manifold as an Affine Manifold; 4.2 Translations and Mollifiers; 4.3 Gaussian Statistical Bundle; 5 Weighted Orlicz-Sobolev Model Space; 5.1 Orlicz-Sobolev Spaces with Gaussian Weight; 6 Conclusions; References; Congruent Families and Invariant Tensors; 1 Introduction; 2 Preliminary Results; 2.1 The Space of (Signed) Finite Measures and Their Powers; 2.2 Parametrized Measure Models; 2.3 Congruent Markov Morphisms; 2.4 Tensor Algebras; 2.5 Tensor Fields