Mathematical Theory of Bayesian StatisticsSumio Wa

文件名称: Mathematical Theory of Bayesian Statistics

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详细说明：Sumio Watanabe。高清原版PDF，已经裁边，适合阅读。用pdf xchange pro恢复裁剪的页面：依次点：左下角“选项”->“视图”->页面缩略图（快捷键是ctrl+T）。左侧面板中的缩略图，页面右键->裁剪页面（快捷键是ctrl+shift+T）。弹出的菜单中:“设为0”->（页码范围框中）选中“全部”->确定。Taylor Francis Taylor Francis Group http://taylorandfrancis.com Mathematical Theory of Bayesian Statistics Sumio Watanabe ( CRC) CRC Pr ress Taylor Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor Francis Group, an informa business a chapman hall book CRC Press Taylor Francis Group 6000 Broken Sound Parkway nw, Suite 300 Boca raton FL 33487-2742 e 2018 by Taylor Francis Group, LLC CRC Press is an imprint of Taylor Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-frcc papcr Version date: 20180402 International Standard Book Number-13: 978-1-482-23806-8(Hardback This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com(http://www.copyright.com/)orcontactthecOpyrightClearanceCenter,Inc ( CCC), 222 Rosewood Drive, Danvers, MAO1923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the taylor francis Web site at http://www.taylorandfrancis.com and the crc Press web site at http://www.crcpress.com Contents Preface 1 Definition of Bayesian Statistics 1.1 Bayesian Statistics 1.2 Probability Distribution 4 1.3 True distribution 1.4 Model, Prior, and Posterior 9 1.5 Examples of Posterior Distributions 1.6 Estimation and generalization 17 1.7 Marginal Likelihood or Partition Function 21 1.8 Conditional Independent Cases 25 1.9 Problems 28 2 Statistical Models 35 2.1 Normal distribution 35 2.2 Multinomial distribution 41 2.3 Linear regression 48 2.4 Neural Network 2.5 Finite normal mixture 36 2.6 Nonparametric Mixture 59 2.7 Problems 63 3 Basic Formula of Bayesian Observables 67 3.1 Formal relation between True and model 67 3.2 Normalized observables 77 3.3 Cumulant generating Functions 80 3.4 Basic Bayesian Theor 85 3.5 Problems 94 CONTENTS 4 Regular Posterior Distribution 99 4.1 Division of partition Function .99 4.2 Asymptotic Free Energy 107 4.3 Asymptotic Losses .111 4.4 Proof of Asymptotic Expansions 118 4.5 Point Estimators 123 4.6 Problems .126 5 Standard Posterior distribution 135 5.1 Standard Form 136 5.2 State Density Function 146 5.3 Asymptotic Free Energy 152 5.4 Renormalized posterior distribution 154 5.5 Conditionally Independent Case 162 5.6 Problems 171 6 General posterior distribution 177 6.1 Bayesian Decomposition .177 6.2 Resolution of Singularities 181 6.3 General Asymptotic Theory 6.4 Maximum a Posteriori method 196 6.5 Problems 203 7 Markov Chain Monte carlo 207 7.1 Metropolis Method 207 7. 1.1 Basic Metropolis Method 209 7.1.2 Hamiltonian Monte Carlo 211 7. 1.3 Parallel Tempering 7.2 Gibbs Sampler 217 7.2.1 Gibbs Sampler for Normal Mixture 218 7.2.2 Nonparametric Bayesian Sampler 221 7.3 Numerical Approximation of Observables 225 7.3.1 Generalization and Cross Validation Losses 225 7.3.2 Numerical Free Energy ..226 7.4 Problems 229 8 Information Criteria 231 8.1 Model selection 231 8.1.1 Criteria for Generalization Loss ..232 8.1.2 Comparison of ScV with WAIC 240 CONTENTS 8. 1.3 Criteria for Free Energy 245 8.1.4 Discussion for Model selection 250 8.2 Hyperparameter Optimization 251 8.2.1 Criteria for Generalization Loss 253 8.2.2 Criterion for Free Energy 257 8.2.3 Discussion for Hyperparameter Optimization 259 8.3 Problems 264 9 Topics in Bayesian Statistics 267 9.1 Formal Optimality 267 9.2 Bayesian Hypothesis Test 270 9. 3 Bayesian Model Comparison 275 9. 4 Phase Transition 277 9.5 Discovery Process 282 6 Hierarchical bayes .286 9. 7 Problems 91 10 Basic Probability Theory 293 10.1 Delta function 293 10.2 Kullback-Leibler Distance 294 10.3 Probability space 296 10.4 Empirical Process 302 10.5 Convergence of Expected values 303 10.6 Mixture by dirichlet Process 306 References 309 Index 317 Taylor Francis Taylor Francis Group http://taylorandfrancis.com Preface The purpose of this book is to establish a mathematical theory of b: ayesian statistics In practical applications of Bayesian statistical inference, we need to pre pare a statistical model and a prior for a given sample, then estimate the unknown true distribution. One of the most important problems is devising a method how to construct a pair of a statistical model and a prior, although we do not know the true distribution The answer based on mathematical theory to this problem is given by the following procedures (1) Firstly, we construct the universal and mathematical laws between Bayesian observables which hold for an arbitrary triple of a true distribution, a sta tistical model, and a prior (2) Secondly, by using such laws, we can evaluate how appropriate a set of a statistical model and a prior is for the unknown true distribution (3)And lastly, the most suitable pair of the statistial model and the prior employe The conventional approach to such a purpose has been based on the assumption that the posterior distribution can be approximated by some normal distribution. However, the new statistical theory introduced by this book holds for arbitrary posterior distribution, demonstrating that the ap plication field will be extended. The author expects that also new statistical methodology which enables us to manupulate complex and hierarchical sta tistical models such as normal mixtures or hierarhical neural networks will be based on the new mathematical theory Sumio watanabe

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