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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
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