Statistical Decision Theory and Bayesian Analysis (2nd Edition)

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统计决策论及贝叶斯分析第二版英语版，概率论和统计学的经典读物。
o Ann. jill. and julie reface Statistical decision theory and Bayesian analysis are related at a number of levels. First, they are both needed to solve real decision problems, each embodying a description of one of the key elements of a decision problem At a deeper level, Bayesian analysis and decision theory provide unified outlooks towards statistics; they give a foundational framework for thinking about statistics and for evaluating proposed statistical methods The relationships(both conceptual and mathematical)between Bayesian analysis and statistical decision theory are so strong that it is somewhat unnatural to learn one without the other. Nevertheless, major portions of each have developed separately On the Bayesian side, there is an extensively developed Bayesian theory of statistical inference (both subjective and objective versions). This theory recognizes the importance of viewing statis tical analysis conditionally (i.e, treating observed data as known rather than unknown), even when no loss function is to be incorporated into the analysis. There is also a well-developed(frequentist )decision theory, which avoids formal utilization of prior distributions and seeks to provide a foundation for frequentist statistical theory. Although the central thread of the book will be bayesian decision theory both bayesian inference and non-Bayesian decision theory will be extensively discussed. Indeed, the book is written so as to allow, say, the teaching of a course on either subject separately Bayesian analysis and, especially, decision theory also have split per sonalities with regard to their practical orientation. Both can be discussed at a very practical level, and yet they also contain some of the most difficult and elegant theoretical developments in statistics. The book contains a fair amount of material of each type. There is extensive discussion on how to actually do Bayesian decision theory and Bayesian inference, including how Ⅵi Preface to construct prior distributions and loss functions, as well as how to utilize them. At the other extreme, introductions are given to some of the beautifui theoretical developments in these areas The statistical level of the book is formally rather low, in that previous knowledge of Bayesian analysis decision theory or advanced statistics is unnecessary. The book will probably be rough going, however, for those without previous exposure to a moderately serious statistics course. For instance, previous exposure to such concepts as sufficiency is desirable. It should also be mentioned that parts of the book are philosophically very challenging; the extreme disagreements that exist among statisticianS, con cerning the correct approach to statistics, suggest that these fundamental issues are conceptually difficult. Periodic rereading of such material (e. g Sections 1.6, 4.1, and 4.12), as one proceeds through the book, is recom mended The mathematical level of the book is, for the most part, at an easy advanced calculus level. Some knowledge of probability is required; at least say, a knowledge of expectations and conditional probability. from time to time (especially in later chapters some higher mathematical facts will be employed, but knowledge of advanced mathematics is not required to follow most of the text. Because of the imposed mathematical limitations some of the stated theorems need, say, ad ditional measurability conditions to be completely precise. Also, less important(but nonignorable) technical conditions for some developments are sometimes omitted, but such develop- ments are called“ Results," rather than‘ Theorems.” The book is primarily concerned with discussing basic issues and prin ciples of bayesian analysis and decision theory. No systematic attempt is made to present a survey of actual developed methodology, i. e, to present specific developments of these ideas in particular areas of statistics. The examples that are given tend to be rather haphazard, and, unfortunately, do not cover some of the more difficult areas of statistics, such as nonpara- metrics.Nevertheless, a fair amount of methodology ends up being intro duced, one way or another This second edition of the book has undergone a title change, with the addition of Bayesian Analysis. This refects the major change in the book namely an extensive upgrading of the Bayesian material, to the point where the book can serve as a text on bayesian analysis alone. The motivation for this upgrading was the realization that, although i professed to be a Rabid Bayesian"in the first edition (and still am), the first edition was not well suited for a primarily Bayesian course; in particular, it did not highlight the conditional bayesian perspective properly. In attempting to correct this problem, I fell into the usual revision trap of being unable to resist adding substantial new material on subjects crucial to Bayesian analysis, such as hierarchical Bayes theory, Bayesian calculation, Bayesian communication, and combination of evidence Preface X For those familiar with the old book the greatest changes are in Chapters 3 and 4, which were substantially enlarged and almost completely rewritten Some sections of Chapter l were redone (particularly 1.6), and some small subsections were added to Chapter 2. The only significant change to Chapter 5 was the inclusion of an introduction to the now vast field of minimax multivariate estimation(Stein estimation); this has become by far the largest statistical area of development within minimax theory. Only very minor changes were made to Chapter 6, and Chapter 7 was changed only by the addition of a section discussing the issue of optional stopping. A number of changes were made to Chapter &, in light of recent developments, but no thorough survey was attempted In general, no attempt was made to update references in parts of the book that were not rewritten. This, unfortunately, perpetuated a problem with the first edition, namely the lack of references to the early period of decision theory. Many of the decision-theoretic ideas and concepts seem to have become part of the folklore, and I apologize for not making the effort to trace them back to their origins and provide references In terms of teaching, the book can be used as a text for a variety of courses. The easiest such use is as a text in a two-semester or three-quarter course on Bayesian analysis and statistical decision theory; one can simply proceed through the book.( Chapters 1 through 4 should take the first semester, and Chapters 5 through 8 the second. The follo wing are outlines for various possible single-semester courses. The first outline is for a master's course, quite different arrangements could also be used successfully p level course, and has a more applied orientation while the other outlines also include theoretical material perhaps best suited for ph D students Bayesian Analysis and decision Theory(Applied) 1( except1.4,1.7,1,8);2;3( except3.4,3.5.5,3.56,3.57);4( except444, 4.74 through4.,7.1,4.8,4.l1);7( except742 through7.4.l0,7.5,7.6) valuable other material to cover if there is time. includes 4.7. 4. 4.7.5. 4.7.9 4.7.10.4.7.11,and4.11. Bayesian Analysis and Decision Theory(more Theoretical 1;2( except2.3.2.4.3,2.44,2.4.5);3( except3.4,3.5.5,3.5.6,3.5.7);4 except444,4.53,4.6.3,46.4,47.4,4.7.6,4.7.7,47.9,4.7.10,4.8.3,4.9, 4.10,4.11) (i)With Minimax Option: 5(except 5.2.3); parts of 8 (ii)With Invariance Option: 6; parts of 8 (iii)With Sequential Option: 7(except 7.4.7 through 7.4.10, 7.5.5, 7.6) parts of 8 A Mainly Bayesian Course(More Theoretical) A(except 1.4, 1.8); 2(except 2.3); 3(except 3.5.5 and 3.5.6); 4(except 4.7.6, 4.7.7): 7(except 7.4.2 through 7.4.10, 7.5, 7.6); more sequential Bayes could be covered if some of the earlier sections were eliminated. A Mainly Decision Theory Course(Very Theoretical) ( eXcept1.6);2( except2.3); Sections3.3,4.1,4.2,4.4,4.8;5( except52.3): 6;7( except7.2,74,77);8. I am very grateful to a number of people who contributed in one way or another. to the book. Useful comments and discussion were received from many sources; particularly helpful were Eric Balder, Mark berliner, Don Berry, Sudip bose, Lawrence Brown, Arthur Cohen, Persi Diaconis Roger Farrell, Leon Gleser, Bruce Hill, Tzou Wu Jien Joe, T C. Kao, Jack Kiefer. Sudhakar Kunte. Erich Lehmann. Carl Morris, Herman Rubin, S Sivaganesan Bill studden, Don Wallace, robert Wolpert, and arnold Zellner. I am especially grateful to Herman Rubin: he provided most of the material in Subsections 7 4.8 and 7.4.9, and was my"foolishness filter on much of the rest of the book The first edition of the book was typed by Lou Anne Scott, Norma Lucas Kathy Woods, and Carolyn Knutsen, to all of whom I am very grateful The highly trying job of typing this revision was undertaken by Norma ucas, and her skill and cheer throughout the process were deeply appreci ated. Finally, I would like to express my appreciation to the John Simon Guggenheim Memorial Foundation, the Alfred P. Sloan Foundation, and the National Science Foundation for support during the writing of the book. West lafayette. Indiana JAMES BERGER March 1985 Contents CHAPTER 1 Basic Concepts 1.1 Introduction 1.2 Basic Elements 3 1.3 Expected Loss, Decision Rules, and risk 8 1.3.1 Bayesian Expected Loss 8 3.2 Frequentist Risk 1.4 Randomized decision rules 12 1.5 Decision Principles 1.5.1 The Conditional Bayes Decision Principle 16 1.5.2 Frequentist Decision Principles 1.6 Foundations 20 1.6.1 Misuse of Classical Inference procedures 20 1.6.2 The Frequentist Perspective 22 1.6.3 The Conditional Perspective 24 1. 6.4 The likelihood Principle 1.6.5 Choosing a Paradigm or Decision Principle 33 1. Suficient statistics 35 1.8 Convexity 38 Exercises 41 CHAPTER 2 Utility and Loss 46 2.1 Introduction 2.2 Utility Theory 47 2.3 The Utility of Mone 53 2.4 The Loss Function 57 2.4.1 Development from Utility Theory 57 X11 ontents 2.4.2 Certain standard Loss functions 60 2.4.3 For Inference Problems 64 2.4.4 For Predictive problems 6 2.4.5 vector valued Loss Functions 68 25 Criticisms 69 Exercises 70 CHAPTER 3 Prior Information and subjective Probability 74 3.1 Subjective probability 74 3.2 Subjective Determination of the Prior Density 77 3. 3 Noninformative priors 82 3.3.1 Introduction 82 3.3.3 Noninformative Priors in General Setting Q 3.3.2 Noninformative Priors for location and Scale problems 83 87 3.3.4 Discussion 89 3.4 Maximum Entropy Priors 3.5 Using the Marginal Distribution to Determine the prior 94 3.5. 1 The Marginal Distribution 94 3.5.2 Information About m 95 3.5.3 Restricted Classes of priors 97 3.5. 4 The ML-II Approach to Prior Selection 99 3.5.s The Moment Approach to Prior Selection 101 3.5.6 The Distance Approach to Prior Selection 103 3.5.7 Marginal exchangeability 104 3.6 Hierarchical Priors 106 3、7 Criticisms 109 3. 8 The Statistician's role Exercises 113 CHAPTER 4 Bayesian analysis 118 4.1 Introduction 118 4.2 The posterior Distribution 126 4.2.1 Definition and Determination 126 4.2.2 Conjugate Families 130 4.2.3 Improper Priors 132 4.3 Bayesian Inference 132 4.3.1 Estimation 133 4.3.2 Credible sets 140 4.3.3 Hypothesis Testing 145 4.3.4 Predictive inference 157 4. 4 Bayesian Decision Theory 158 4.4. 1 Posterior Decision Analysis 158 4. 4.2 Estimation 161 4.4.3 Finite Action Problems and Hypothesis Testing 163 4.4.4 With inference losses 166 Contents XIlI 4.5 Empirical Bayes Analysi 167 4.5.1 Introduction 167 4.5.2 PEB For Normal Means-The Exchangeable Case 169 4.5.3 PEB For Normal Means-The General Case l73 4.5.4 Nonparametric Empirical Bayes Analysis 178 4.6 Hierarchical Bayes analysis 180 4.6.1 Introduction 180 4.6.2 For Normal Means-The Exchangeable Case 183 4.6.3 For normal mea The general o 4.6.4 Comparison with empirical bayes Analysis 193 4.7 Bayesian Robustness 195 4.7.1 Introduction 195 4.7.2 The Role of the Marginal Distribution 199 4.7.3 Posterior Robustness: Basic Concepts 203 4.7.4 Posterior robustness: e-Contamination Class 206 4.7.5 Bayes Risk Robustness and use of Frequentist measures 213 4.7.6 Gamma-Minimax Approach 215 4.7.7 Uses of the risk function 218 4.7.8 Some Robust and Nonrobust Situations 223 4.79 Robust priors 228 4.7.10 Robust Priors for Normal Means 236 4.7.11 Other issues in robustness 247 4.8 Admissibility of Bayes Rules and Long run Evaluations 253 4.8.1 Admissibility of Bayes rules 253 4.8.2 Admissibility of Generalized bayes rules 254 4.8.3 Inadmissibility and Long Run Evaluations 257 4.9 Bayesian Calculation 4.9.1 Numerical Integration 262 4.9.2 Monte Carlo Integration 263 4.9.3 Analytic Approximations 26 4.10 Bayesian Communication 267 4.10.1 Introduction 267 4.10.2 An Illustration: Testing a point Null Hypothesis 268 4.11 Combining Evidence and group decisions 271 4.11.1 Combining Probabilistic Evidence 272 4.11.2 Combining Decision-Theoretic Evidence 277 4.11.3 Group Decision Making 278 4.12 Criticisms 281 4.12.1 Non-Bayesian Criticisms 281 4122 Foundational criticisms 283 EX 286 CHAPTER 5 Minimax Analysis 308 5.1 Introduction 308 5.2 Game Theor 310 5.2.1 Basic elements 310 5.2.2 General Techniques for Solving Games 319 【实例截图】
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