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E-grāmata: Mathematical Theory of Bayesian Statistics

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  • Formāts: 330 pages
  • Izdošanas datums: 27-Apr-2018
  • Izdevniecība: Chapman & Hall/CRC
  • Valoda: eng
  • ISBN-13: 9781482238082
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Mathematical Theory of Bayesian StatisticsPalielināt
  • Formāts: 330 pages
  • Izdošanas datums: 27-Apr-2018
  • Izdevniecība: Chapman & Hall/CRC
  • Valoda: eng
  • ISBN-13: 9781482238082
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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 Science at Tokyo Institute of Technology. He studies the relationship between algebraic geometry and mathematical statistics.

Recenzijas

"Information criteria are introduced from the two viewpoints, model selection and hyperparameter optimization. In each viewpoint, the properties of the generalization loss and the free energy or the minus log marginal likelihood are investigated. The book is very nicely written with well-defined concepts and contexts. I recommend to all students and researchers." ~Rozsa Horvath-Bokor, Zentralblatt MATH

Preface ix
1 Definition of Bayesian Statistics
1(34)
1.1 Bayesian Statistics
2(2)
1.2 Probability Distribution
4(3)
1.3 True Distribution
7(2)
1.4 Model, Prior, and Posterior
9(2)
1.5 Examples of Posterior Distributions
11(6)
1.6 Estimation and Generalization
17(4)
1.7 Marginal Likelihood or Partition Function
21(4)
1.8 Conditional Independent Cases
25(3)
1.9 Problems
28(7)
2 Statistical Models
35(32)
2.1 Normal Distribution
35(6)
2.2 Multinomial Distribution
41(7)
2.3 Linear Regression
48(5)
2.4 Neural Network
53(3)
2.5 Finite Normal Mixture
56(3)
2.6 Nonparametric Mixture
59(4)
2.7 Problems
63(4)
3 Basic Formula of Bayesian Observables
67(32)
3.1 Formal Relation between True and Model
67(10)
3.2 Normalized Observables
77(3)
3.3 Cumulant Generating Functions
80(5)
3.4 Basic Bayesian Theory
85(9)
3.5 Problems
94(5)
4 Regular Posterior Distribution
99(36)
4.1 Division of Partition Function
99(8)
4.2 Asymptotic Free Energy
107(4)
4.3 Asymptotic Losses
111(7)
4.4 Proof of Asymptotic Expansions
118(5)
4.5 Point Estimators
123(3)
4.6 Problems
126(9)
5 Standard Posterior Distribution
135(42)
5.1 Standard Form
136(10)
5.2 State Density Function
146(6)
5.3 Asymptotic Free Energy
152(2)
5.4 Renormalized Posterior Distribution
154(8)
5.5 Conditionally Independent Case
162(9)
5.6 Problems
171(6)
6 General Posterior Distribution
177(30)
6.1 Bayesian Decomposition
177(4)
6.2 Resolution of Singularities
181(9)
6.3 General Asymptotic Theory
190(6)
6.4 Maximum A Posteriori Method
196(7)
6.5 Problems
203(4)
7 Markov Chain Monte Carlo
207(24)
7.1 Metropolis Method
207(10)
7.1.1 Basic Metropolis Method
209(2)
7.1.2 Hamiltonian Monte Carlo
211(4)
7.1.3 Parallel Tempering
215(2)
7.2 Gibbs Sampler
217(8)
7.2.1 Gibbs Sampler for Normal Mixture
218(3)
7.2.2 Nonparametric Bayesian Sampler
221(4)
7.3 Numerical Approximation of Observables
225(4)
7.3.1 Generalization and Cross Validation Losses
225(1)
7.3.2 Numerical Free Energy
226(3)
7.4 Problems
229(2)
8 Information Criteria
231(36)
8.1 Model Selection
231(20)
8.1.1 Criteria for Generalization Loss
232(8)
8.1.2 Comparison of ISCV with WAIC
240(5)
8.1.3 Criteria for Free Energy
245(5)
8.1.4 Discussion for Model Selection
250(1)
8.2 Hyperparameter Optimization
251(13)
8.2.1 Criteria for Generalization Loss
253(4)
8.2.2 Criterion for Free Energy
257(2)
8.2.3 Discussion for Hyperparameter Optimization
259(5)
8.3 Problems
264(3)
9 Topics in Bayesian Statistics
267(26)
9.1 Formal Optimality
267(3)
9.2 Bayesian Hypothesis Test
270(5)
9.3 Bayesian Model Comparison
275(2)
9.4 Phase Transition
277(5)
9.5 Discovery Process
282(4)
9.6 Hierarchical Bayes
286(5)
9.7 Problems
291(2)
10 Basic Probability Theory
293(16)
10.1 Delta Function
293(1)
10.2 Kullback-Leibler Distance
294(2)
10.3 Probability Space
296(6)
10.4 Empirical Process
302(1)
10.5 Convergence of Expected Values
303(3)
10.6 Mixture by Dirichlet Process
306(3)
References 309(8)
Index 317
Sumio Watanabe is a professor in the Department of Computational Intelligence and Systems Science at Tokyo Institute of Technology, Japan.
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