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Table of Contents
Intro
Preface
Contents
1 Information Geometry of Smooth Densities on the Gaussian Space: Poincaré Inequalities
1.1 Introduction
1.2 Statistical Bundle Modelled on Orlicz Spaces
1.2.1 Orlicz Spaces
1.2.2 Calculus of the Gaussian Space
1.2.3 Exponential Statistical Bundle
1.3 Bounding the Orlicz Norm with the Orlicz Norm of the Gradient
1.3.1 Ornstein-Uhlenbeck Semi-group
1.3.2 Generator of the Ornstein-Uhlenbeck Semi-group
1.4 Discussion and Conclusions
1.4.1 Sub-exponential Random Variables
1.4.2 Hyvärinen Divergence
1.4.3 Otto's Metric
3 Affine Connections with Torsion in (Para- )complexified Structures
3.1 Introduction
3.2 Torsion of and Integrability of L
3.2.1 L Conjugation of
3.2.2 Nijenhuis Tensor NL and Integrability
3.2.3 MC1 Versus MC2
3.3 Torsion of Under (Para- )complexification
3.3.1 Splitting of TmathcalMotimesmathbbL by L
3.3.2 (Para- )holomorphicity of
3.3.3 (Para- )complexifying NL and T
3.3.4 Torsion-Compatibility (MC1)
3.3.5 Torsion-Coupling (MC2)
3.4 Summary and Discussions
References
4 Contact Hamiltonian Systems for Probability Distribution Functions and Expectation Variables: A Study Based on a Class of Master Equations
4.1 Introduction
4.2 Underlying Geometry
4.2.1 Para-Contact Metric Manifolds
4.2.2 Contact Manifold
4.2.3 Legendre Submanifold as Dually Flat Space
4.2.4 Legendre Submanifold as Attractor
4.3 Distribution Functions from Solvable Master Equations
4.3.1 Denormalization
4.4 Observables with Solvable Master Equations
4.5 Geometric Description of Master Equations
4.5.1 Geometry of Equilibrium States
4.5.2 Geometry of Nonequilibrium States
4.6 Geometric Description of Expectation Variables
4.6.1 Geometry of Equilibrium States
4.6.2 Geometry of Nonequilibrium States
4.7 Beyond the Toy Model
4.7.1 Equilibrium States
4.7.2 Nonequilibrium States
4.8 Conclusions
References
5 Invariant Koszul Form of Homogeneous Bounded Domains and Information Geometry Structures
5.1 Preamble
5.2 Invariant Koszul Form for Homogeneous Bounded Domains
5.3 Contextualization with Last Advanced Works
5.4 Koszul Hessian Geometric Structure of Information Geometry
Preface
Contents
1 Information Geometry of Smooth Densities on the Gaussian Space: Poincaré Inequalities
1.1 Introduction
1.2 Statistical Bundle Modelled on Orlicz Spaces
1.2.1 Orlicz Spaces
1.2.2 Calculus of the Gaussian Space
1.2.3 Exponential Statistical Bundle
1.3 Bounding the Orlicz Norm with the Orlicz Norm of the Gradient
1.3.1 Ornstein-Uhlenbeck Semi-group
1.3.2 Generator of the Ornstein-Uhlenbeck Semi-group
1.4 Discussion and Conclusions
1.4.1 Sub-exponential Random Variables
1.4.2 Hyvärinen Divergence
1.4.3 Otto's Metric
3 Affine Connections with Torsion in (Para- )complexified Structures
3.1 Introduction
3.2 Torsion of and Integrability of L
3.2.1 L Conjugation of
3.2.2 Nijenhuis Tensor NL and Integrability
3.2.3 MC1 Versus MC2
3.3 Torsion of Under (Para- )complexification
3.3.1 Splitting of TmathcalMotimesmathbbL by L
3.3.2 (Para- )holomorphicity of
3.3.3 (Para- )complexifying NL and T
3.3.4 Torsion-Compatibility (MC1)
3.3.5 Torsion-Coupling (MC2)
3.4 Summary and Discussions
References
4 Contact Hamiltonian Systems for Probability Distribution Functions and Expectation Variables: A Study Based on a Class of Master Equations
4.1 Introduction
4.2 Underlying Geometry
4.2.1 Para-Contact Metric Manifolds
4.2.2 Contact Manifold
4.2.3 Legendre Submanifold as Dually Flat Space
4.2.4 Legendre Submanifold as Attractor
4.3 Distribution Functions from Solvable Master Equations
4.3.1 Denormalization
4.4 Observables with Solvable Master Equations
4.5 Geometric Description of Master Equations
4.5.1 Geometry of Equilibrium States
4.5.2 Geometry of Nonequilibrium States
4.6 Geometric Description of Expectation Variables
4.6.1 Geometry of Equilibrium States
4.6.2 Geometry of Nonequilibrium States
4.7 Beyond the Toy Model
4.7.1 Equilibrium States
4.7.2 Nonequilibrium States
4.8 Conclusions
References
5 Invariant Koszul Form of Homogeneous Bounded Domains and Information Geometry Structures
5.1 Preamble
5.2 Invariant Koszul Form for Homogeneous Bounded Domains
5.3 Contextualization with Last Advanced Works
5.4 Koszul Hessian Geometric Structure of Information Geometry