Watanabe Mathematical Theory of Bayesian Statistics

1. Auflage 2018
ISBN: 978-1-315-35569-6
Verlag: Taylor & Francis
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 330 Seiten

Mathematical Theory of Bayesian Statistics introduces the mathematical foundation of Bayesian inference which is well-known to be more accurate in many real-world problems than the maximum likelihood method. Recent research has uncovered several mathematical laws in Bayesian statistics, by which both the generalization loss and the marginal likelihood are estimated even if the posterior distribution cannot be approximated by any normal distribution.

Features

Explains Bayesian inference not subjectively but objectively.

Provides a mathematical framework for conventional Bayesian theorems.

Introduces and proves new theorems.

Cross validation and information criteria of Bayesian statistics are studied from the mathematical point of view.

Illustrates applications to several statistical problems, for example, model selection, hyperparameter optimization, and hypothesis tests.

This book provides basic introductions for students, researchers, and users of Bayesian statistics, as well as applied mathematicians.

Author
Sumio Watanabe is a professor of Department of Mathematical and Computing Scienceat Tokyo Institute of Technology. He studies the relationship between algebraic geometry and mathematical statistics.

Zielgruppe

This book is intended for students and researchers in statistics and related areas.

Autoren/Hrsg.

Watanabe, Sumio

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik

Inhaltsverzeichnis

Definition of Bayesian Statistics

Bayesian Statistics
Probability distribution
True Distribution
Statistical model, prior, and posterior
Examples of Posterior Distributions
Estimation and Generalization
Marginal Likelihood or Partition Function
Conditional Independent Cases

Statistical Models

Normal Distribution
Multinomial Distribution
Linear regression
Neural Network
Finite Normal Mixture
Nonparametric Mixture

Basic Formula of Bayesian Observables

Formal Relation between True and Model
Normalized Observables
Cumulant Generating Functions
Basic Bayesian Theory

Regular Posterior Distribution

Division of Partition Function
Asymptotic Free Energy
Asymptotic Losses
Proof of Asymptotic Expansions
Point Estimators

Standard Posterior Distribution

Standard Form
State Density Function
Asymptotic Free Energy
Renormalized Posterior Distribution
Conditionally Independent Case

General Posterior Distribution

Bayesian Decomposition
Resolution of Singularities
General Asymptotic Theory
Maximum A Posteriori Method

Markov Chain Monte Carlo

Metropolis Method
Basic Metropolis Method
Hamiltonian Monte Carlo
Parallel Tempering
Gibbs Sampler
Gibbs Sampler for Normal Mixture
Nonparametric Bayesian Sampler
Numerical Approximation of Bayesian Observables
Generalization and Cross Validation Losses
Numerical Free Energy

Information Criteria

Model Selection
Criteria for Generalization Loss
Comparison of ISCV with WAIC
Criteria for Free Energy
Discussion for Model Selection
Hyperparameter Optimization
Criteria for Generalization Loss
Criterion for Free energy
Discussion for Hyperparameter Optimization

Topics in Bayesian Statistics

Formal Optimality
Bayesian Hypothesis Test
Bayesian Model Comparison
Phase Transition
Discovery Process
Hierarchical Bayes

Basic Probability Theory

Delta Function
Kullback-Leibler Distance
Probability Space
Empirical Process
Convergence of Expected Values
Mixture by Dirichlet Process

Über Autor(innen)

Sumio Watanabe is a professor in the Department of Computational Intelligence and Systems Science at Tokyo Institute of Technology, Japan.