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Bayesian Data Analysis, Second Edition (Chapman & Hall/CRC Texts in Statistical Science) (Hardcover)
Andrew Gelman (Author), John B. Carlin (Author), Hal S. Stern (Author), Donald B. Rubin (Author)Product Details
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Bayesian Data Analysis~2004.pdf
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index
Contents
List of models xv
List of examples xvii
Preface xix
Part I: Fundamentals of Bayesian Inference 1
1 Background 3
1.1 Overview 3
1.2 General notation for statistical inference 4
1.3 Bayesian inference 6
1.4 Example: inference about a genetic probability 9
1.5 Probability as a measure of uncertainty 11
1.6 Example of probability assignment: football point spreads 14
1.7 Example of probability assignment: estimating the accuracy of record linkage 17
1.8 Some useful results from probability theory 22
1.9 Summarizing inferences by simulation 25
1.10 Computation and software 27
1.11 Bibliographic note 27
1.12 Exercises 29
2 Single-parameter models 33
2.1 Estimating a probability from binomial data 33
2.2 Posterior distribution as compromise between data and prior information 36
2.3 Summarizing posterior inference 37
2.4 Informative prior distributions 39
2.5 Example: estimating the probability of a female birth given placenta previa 43
2.6 Estimating the mean of a normal distribution with known variance 46
2.7 Other standard single-parameter models 49
2.8 Example: informative prior distribution and multilevel structure for estimating cancer rates 55
2.9 Noninformative prior distributions 61
2.10 Bibliographic note 65
2.11 Exercises 67
3 Introduction to multiparameter models 73
3.1 Averaging over ‘nuisance parameters’ 73
3.2 Normal data with a noninformative prior distribution 74
3.3 Normal data with a conjugate prior distribution 78
3.4 Normal data with a semi-conjugate prior distribution 80
3.5 The multinomial model 83
3.6 The multivariate normal model 85
3.7 Example: analysis of a bioassay experiment 88
3.8 Summary of elementary modeling and computation 93
3.9 Bibliographic note 94
3.10 Exercises 95
4 Large-sample inference and frequency properties of Bayesian inference 101
4.1 Normal approximations to the posterior distribution 101
4.2 Large-sample theory 106
4.3 Counterexamples to the theorems 108
4.4 Frequency evaluations of Bayesian inferences 111
4.5 Bibliographic note 113
4.6 Exercises 113
Part II: Fundamentals of Bayesian Data Analysis 115
5 Hierarchical models 117
5.1 Constructing a parameterized prior distribution 118
5.2 Exchangeability and setting up hierarchical models 121
5.3 Computation with hierarchical models 125
5.4 Estimating an exchangeable set of parameters from a normal model 131
5.5 Example: combining information from educational testing experiments in eight schools 138
5.6 Hierarchical modeling applied to a meta-analysis 145
5.7 Bibliographic note 150
5.8 Exercises 152
6 Model checking and improvement 157
6.1 The place of model checking in applied Bayesian statistics 157
6.2 Do the inferences from the model make sense? 158
6.3 Is the model consistent with data? Posterior predictive checking 159
6.4 Graphical posterior predictive checks 1656.5 Numerical posterior predictive checks 172
6.6 Model expansion 177
6.7 Model comparison 179
6.8 Model checking for the educational testing example 186
6.9 Bibliographic note 190
6.10 Exercises 192
7 Modeling accounting for data collection 197
7.1 Introduction 197
7.2 Formal models for data collection 200
7.3 Ignorability 203
7.4 Sample surveys 207
7.5 Designed experiments 218
7.6 Sensitivity and the role of randomization 223
7.7 Observational studies 226
7.8 Censoring and truncation 231
7.9 Discussion 236
7.10 Bibliographic note 237
7.11 Exercises 239
8 Connections and challenges 247
8.1 Bayesian interpretations of other statistical methods 247
8.2 Challenges in Bayesian data analysis 252
8.3 Bibliographic note 255
8.4 Exercises 255
9 General advice 259
9.1 Setting up probability models 259
9.2 Posterior inference 264
9.3 Model evaluation 265
9.4 Summary 271
9.5 Bibliographic note 271
Part III: Advanced Computation 273
10 Overview of computation 275
10.1 Crude estimation by ignoring some information 276
10.2 Use of posterior simulations in Bayesian data analysis 276
10.3 Practical issues 278
10.4 Exercises 282
11 Posterior simulation 283
11.1 Direct simulation 283
11.2 Markov chain simulation 285
11.3 The Gibbs sampler 28711.4 The Metropolis and Metropolis-Hastings algorithms 289
11.5 Building Markov chain algorithms using the Gibbs sampler and Metropolis algorithm 292
11.6 Inference and assessing convergence 294
11.7 Example: the hierarchical normal model 299
11.8 Efficient Gibbs samplers 302
11.9 Efficient Metropolis jumping rules 305
11.10 Recommended strategy for posterior simulation 307
11.11 Bibliographic note 308
11.12 Exercises 310
12 Approximations based on posterior modes 311
12.1 Finding posterior modes 312
12.2 The normal and related mixture approximations 314
12.3 Finding marginal posterior modes using EM and related algorithms 317
12.4 Approximating conditional and marginal posterior densities 324
12.5 Example: the hierarchical normal model (continued) 325
12.6 Bibliographic note 331
12.7 Exercises 332
13 Special topics in computation 335
13.1 Advanced techniques for Markov chain simulation 335
13.2 Numerical integration 340
13.3 Importance sampling 342
13.4 Computing normalizing factors 345
13.5 Bibliographic note 348
13.6 Exercises 349
Part IV: Regression Models 351
14 Introduction to regression models 353
14.1 Introduction and notation 353
14.2 Bayesian analysis of the classical regression model 355
14.3 Example: estimating the advantage of incumbency in U.S. Congressional elections 359
14.4 Goals of regression analysis 367
14.5 Assembling the matrix of explanatory variables 369
14.6 Unequal variances and correlations 372
14.7 Models for unequal variances 375
14.8 Including prior information 382
14.9 Bibliographic note 385
14.10 Exercises 38515 Hierarchical linear models 389
15.1 Regression coefficients exchangeable in batches 390
15.2 Example: forecasting U.S. Presidential elections 392
15.3 General notation for hierarchical linear models 399
15.4 Computation 400
15.5 Hierarchical modeling as an alternative to selecting predictors 405
15.6 Analysis of variance 406
15.7 Bibliographic note 411
15.8 Exercises 412
16 Generalized linear models 415
16.1 Introduction 415
16.2 Standard generalized linear model likelihoods 416
16.3 Setting up and interpreting generalized linear models 418
16.4 Computation 421
16.5 Example: hierarchical Poisson regression for police stops 425
16.6 Example: hierarchical logistic regression for political opinions 428
16.7 Models for multinomial responses 430
16.8 Loglinear models for multivariate discrete data 433
16.9 Bibliographic note 439
16.10 Exercises 440
17 Models for robust inference 443
17.1 Introduction 443
17.2 Overdispersed versions of standard probability models 445
17.3 Posterior inference and computation 448
17.4 Robust inference and sensitivity analysis for the educational testing example 451
17.5 Robust regression using Student-t errors 455
17.6 Bibliographic note 457
17.7 Exercises 458
Part V: Specific Models and Problems 461
18 Mixture models 463
18.1 Introduction 463
18.2 Setting up mixture models 463
18.3 Computation 467
18.4 Example: reaction times and schizophrenia 468
18.5 Bibliographic note 479
19 Multivariate models 481
19.1 Linear regression with multiple outcomes 481
19.2 Prior distributions for covariance matrices 483
19.3 Hierarchical multivariate models 48619.4 Multivariate models for nonnormal data 488
19.5 Time series and spatial models 491
19.6 Bibliographic note 493
19.7 Exercises 494
20 Nonlinear models 497
20.1 Introduction 497
20.2 Example: serial dilution assay 498
20.3 Example: population toxicokinetics 504
20.4 Bibliographic note 514
20.5 Exercises 515
21 Models for missing data 517
21.1 Notation 517
21.2 Multiple imputation 519
21.3 Missing data in the multivariate normal and t models 523
21.4 Example: multiple imputation for a series of polls 526
21.5 Missing values with counted data 533
21.6 Example: an opinion poll in Slovenia 534
21.7 Bibliographic note 539
21.8 Exercises 540
22 Decision analysis 541
22.1 Bayesian decision theory in different contexts 542
22.2 Using regression predictions: incentives for telephone surveys 544
22.3 Multistage decision making: medical screening 552
22.4 Decision analysis using a hierarchical model: home radon measurement and remediation 555
22.5 Personal vs. institutional decision analysis 567
22.6 Bibliographic note 568
22.7 Exercises 569
Appendixes 571
A Standard probability distributions 573
A.1 Introduction 573
A.2 Continuous distributions 573
A.3 Discrete distributions 582
A.4 Bibliographic note 584
B Outline of proofs of asymptotic theorems 585
B.1 Bibliographic note 589
C Example of computation in R and Bugs 591
C.1 Getting started with R and Bugs 591C.2 Fitting a hierarchical model in Bugs 592
C.3 Options in the Bugs implementation 596
C.4 Fitting a hierarchical model in R 600
C.5 Further comments on computation 607
C.6 Bibliographic note 608
References 611
Author index 647
Subject index 655
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