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Table of Contents
Intro
Preface: Sumio Watanabe-Spreading the Wonder of Bayesian Theory
One-Point Advice for Those Who Struggle with Math
Features of This Series
Contents
1 Overview of Watanabe's Bayes
1.1 Frequentist Statistics
1.2 Bayesian Statistics
1.3 Asymptotic Normality of the Posterior Distribution
1.4 Model Selection
1.5 Why are WAIC and WBIC Bayesian Statistics?
1.6 What is ``Regularity''
1.7 Why is Algebraic Geometry Necessary for Understanding WAIC and WBIC?
1.8 Hironaka's Desingularization, Nothing to Fear
3.3.5 Mixture of Normal Distributions
4 Mathematical Preparation
4.1 Elementary Mathematics
4.1.1 Matrices and Eigenvalues
4.1.2 Open Sets, Closed Sets, and Compact Sets
4.1.3 Mean Value Theorem and Taylor Expansion
4.2 Analytic Functions
4.3 Law of Large Numbers and Central Limit Theorem
4.3.1 Random Variables
4.3.2 Order Notation
4.3.3 Law of Large Numbers
4.3.4 Central Limit Theorem
4.4 Fisher Information Matrix
5 Regular Statistical Models
5.1 Empirical Process
5.2 Asymptotic Normality of the Posterior Distribution
5.3 Generalization Loss and Empirical Loss
6 Information Criteria
6.1 Model Selection Based on Information Criteria
6.2 AIC and TIC
6.3 WAIC
6.4 Free Energy, BIC, and WBIC
Exercises 54-66
7 Algebraic Geometry
7.1 Algebraic Sets and Analytical Sets
7.2 Manifold
7.3 Singular Points and Their Resolution
7.4 Hironaka's Theorem
7.5 Local Coordinates in Watanabe Bayesian Theory
8 The Essence of WAIC
8.1 Formula of State Density
8.2 Generalization of the Posterior Distribution
8.3 Properties of WAIC
8.4 Equivalence with Cross-Validation-like Methods
9 WBIC and Its Application to Machine Learning
9.1 Properties of WBIC
9.2 Calculation of the Learning Coefficient
9.3 Application to Deep Learning
9.4 Application to Gaussian Mixture Models
9.5 Non-informative Prior Distribution
Exercises 87-100
Appendix Bibliography
Index
Preface: Sumio Watanabe-Spreading the Wonder of Bayesian Theory
One-Point Advice for Those Who Struggle with Math
Features of This Series
Contents
1 Overview of Watanabe's Bayes
1.1 Frequentist Statistics
1.2 Bayesian Statistics
1.3 Asymptotic Normality of the Posterior Distribution
1.4 Model Selection
1.5 Why are WAIC and WBIC Bayesian Statistics?
1.6 What is ``Regularity''
1.7 Why is Algebraic Geometry Necessary for Understanding WAIC and WBIC?
1.8 Hironaka's Desingularization, Nothing to Fear
3.3.5 Mixture of Normal Distributions
4 Mathematical Preparation
4.1 Elementary Mathematics
4.1.1 Matrices and Eigenvalues
4.1.2 Open Sets, Closed Sets, and Compact Sets
4.1.3 Mean Value Theorem and Taylor Expansion
4.2 Analytic Functions
4.3 Law of Large Numbers and Central Limit Theorem
4.3.1 Random Variables
4.3.2 Order Notation
4.3.3 Law of Large Numbers
4.3.4 Central Limit Theorem
4.4 Fisher Information Matrix
5 Regular Statistical Models
5.1 Empirical Process
5.2 Asymptotic Normality of the Posterior Distribution
5.3 Generalization Loss and Empirical Loss
6 Information Criteria
6.1 Model Selection Based on Information Criteria
6.2 AIC and TIC
6.3 WAIC
6.4 Free Energy, BIC, and WBIC
Exercises 54-66
7 Algebraic Geometry
7.1 Algebraic Sets and Analytical Sets
7.2 Manifold
7.3 Singular Points and Their Resolution
7.4 Hironaka's Theorem
7.5 Local Coordinates in Watanabe Bayesian Theory
8 The Essence of WAIC
8.1 Formula of State Density
8.2 Generalization of the Posterior Distribution
8.3 Properties of WAIC
8.4 Equivalence with Cross-Validation-like Methods
9 WBIC and Its Application to Machine Learning
9.1 Properties of WBIC
9.2 Calculation of the Learning Coefficient
9.3 Application to Deep Learning
9.4 Application to Gaussian Mixture Models
9.5 Non-informative Prior Distribution
Exercises 87-100
Appendix Bibliography
Index