Bayesian data analysis by Andrew gelmann.pdf

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Contents
Preface xiii
Part I: Fundamentals of Bayesian Inference 1
1 Probability and inference 3
1.1 The three steps of Bayesian data analysis 3
1.2 General notation for statistical inference 4
1.3 Bayesian inference 6
1.4 Discrete examples: genetics and spell checking 8
1.5 Probability as a measure of uncertainty 11
1.6 Example: probabilities from football point spreads 13
1.7 Example: calibration for record linkage 16
1.8 Some useful results from probability theory 19
1.9 Computation and software 22
1.10 Bayesian inference in applied statistics 24
1.11 Bibliographic note 25
1.12 Exercises 27
2 Single-parameter models 29
2.1 Estimating a probability from binomial data 29
2.2 Posterior as compromise between data and prior information 32
2.3 Summarizing posterior inference 32
2.4 Informative prior distributions 34
2.5 Normal distribution with known variance 39
2.6 Other standard single-parameter models 42
2.7 Example: informative prior distribution for cancer rates 46
2.8 Noninformative prior distributions 51
2.9 Weakly informative prior distributions 55
2.10 Bibliographic note 56
2.11 Exercises 57
3 Introduction to multiparameter models 63
3.1 Averaging over ‘nuisance parameters’ 63
3.2 Normal data with a noninformative prior distribution 64
3.3 Normal data with a conjugate prior distribution 67
3.4 Multinomial model for categorical data 69
3.5 Multivariate normal model with known variance 70
3.6 Multivariate normal with unknown mean and variance 72
3.7 Example: analysis of a bioassay experiment 74
3.8 Summary of elementary modeling and computation 78
3.9 Bibliographic note 78
3.10 Exercises 79
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viii CONTENTS
4 Asymptotics and connections to non-Bayesian approaches 83
4.1 Normal approximations to the posterior distribution 83
4.2 Large-sample theory 87
4.3 Counterexamples to the theorems 89
4.4 Frequency evaluations of Bayesian inferences 91
4.5 Bayesian interpretations of other statistical methods 92
4.6 Bibliographic note 97
4.7 Exercises 98
5 Hierarchical models 101
5.1 Constructing a parameterized prior distribution 102
5.2 Exchangeability and hierarchical models 104
5.3 Bayesian analysis of conjugate hierarchical models 108
5.4 Normal model with exchangeable parameters 113
5.5 Example: parallel experiments in eight schools 119
5.6 Hierarchical modeling applied to a meta-analysis 124
5.7 Weakly informative priors for variance parameters 128
5.8 Bibliographic note 132
5.9 Exercises 134
Part II: Fundamentals of Bayesian Data Analysis 139
6 Model checking 141
6.1 The place of model checking in applied Bayesian statistics 141
6.2 Do the inferences from the model make sense? 142
6.3 Posterior predictive checking 143
6.4 Graphical posterior predictive checks 153
6.5 Model checking for the educational testing example 159
6.6 Bibliographic note 161
6.7 Exercises 163
7 Evaluating, comparing, and expanding models 165
7.1 Measures of predictive accuracy 166
7.2 Information criteria and cross-validation 169
7.3 Model comparison based on predictive performance 178
7.4 Model comparison using Bayes factors 182
7.5 Continuous model expansion 184
7.6 Implicit assumptions and model expansion: an example 187
7.7 Bibliographic note 192
7.8 Exercises 194
8 Modeling accounting for data collection 197
8.1 Bayesian inference requires a model for data collection 197
8.2 Data-collection models and ignorability 199
8.3 Sample surveys 205
8.4 Designed experiments 214
8.5 Sensitivity and the role of randomization 218
8.6 Observational studies 220
8.7 Censoring and truncation 224
8.8 Discussion 229
8.9 Bibliographic note 229
8.10 Exercises 230
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CONTENTS ix
9 Decision analysis 237
9.1 Bayesian decision theory in different contexts 237
9.2 Using regression predictions: survey incentives 239
9.3 Multistage decision making: medical screening 245
9.4 Hierarchical decision analysis for home radon 246
9.5 Personal vs. institutional decision analysis 256
9.6 Bibliographic note 257
9.7 Exercises 257
Part III: Advanced Computation 259
10 Introduction to Bayesian computation 261
10.1 Numerical integration 261
10.2 Distributional approximations 262
10.3 Direct simulation and rejection sampling 263
10.4 Importance sampling 265
10.5 How many simulation draws are needed? 267
10.6 Computing environments 268
10.7 Debugging Bayesian computing 270
10.8 Bibliographic note 271
10.9 Exercises 272
11 Basics of Markov chain simulation 275
11.1 Gibbs sampler 276
11.2 Metropolis and Metropolis-Hastings algorithms 278
11.3 Using Gibbs and Metropolis as building blocks 280
11.4 Inference and assessing convergence 281
11.5 Effective number of simulation draws 286
11.6 Example: hierarchical normal model 288
11.7 Bibliographic note 291
11.8 Exercises 291
12 Computationally efficient Markov chain simulation 293
12.1 Efficient Gibbs samplers 293
12.2 Efficient Metropolis jumping rules 295
12.3 Further extensions to Gibbs and Metropolis 297
12.4 Hamiltonian Monte Carlo 300
12.5 Hamiltonian Monte Carlo for a hierarchical model 305
12.6 Stan: developing a computing environment 307
12.7 Bibliographic note 308
12.8 Exercises 309
13 Modal and distributional approximations 311
13.1 Finding posterior modes 311
13.2 Boundary-avoiding priors for modal summaries 313
13.3 Normal and related mixture approximations 318
13.4 Finding marginal posterior modes using EM 320
13.5 Conditional and marginal posterior approximations 325
13.6 Example: hierarchical normal model (continued) 326
13.7 Variational inference 331
13.8 Expectation propagation 338
13.9 Other approximations 343
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13.10 Unknown normalizing factors 345
13.11 Bibliographic note 348
13.12 Exercises 349
Part IV: Regression Models 351
14 Introduction to regression models 353
14.1 Conditional modeling 353
14.2 Bayesian analysis of classical regression 354
14.3 Regression for causal inference: incumbency and voting 358
14.4 Goals of regression analysis 364
14.5 Assembling the matrix of explanatory variables 365
14.6 Regularization and dimension reduction 367
14.7 Unequal variances and correlations 369
14.8 Including numerical prior information 376
14.9 Bibliographic note 378
14.10 Exercises 378
15 Hierarchical linear models 381
15.1 Regression coefficients exchangeable in batches 382
15.2 Example: forecasting U.S. presidential elections 383
15.3 Interpreting a normal prior distribution as extra data 388
15.4 Varying intercepts and slopes 390
15.5 Computation: batching and transformation 392
15.6 Analysis of variance and the batching of coefficients 395
15.7 Hierarchical models for batches of variance components 398
15.8 Bibliographic note 400
15.9 Exercises 402
16 Generalized linear models 405
16.1 Standard generalized linear model likelihoods 406
16.2 Working with generalized linear models 407
16.3 Weakly informative priors for logistic regression 412
16.4 Overdispersed Poisson regression for police stops 420
16.5 State-level opinons from national polls 422
16.6 Models for multivariate and multinomial responses 423
16.7 Loglinear models for multivariate discrete data 428
16.8 Bibliographic note 431
16.9 Exercises 432
17 Models for robust inference 435
17.1 Aspects of robustness 435
17.2 Overdispersed versions of standard models 437
17.3 Posterior inference and computation 439
17.4 Robust inference for the eight schools 441
17.5 Robust regression using t-distributed errors 444
17.6 Bibliographic note 445
17.7 Exercises 446
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CONTENTS xi
18 Models for missing data 449
18.1 Notation 449
18.2 Multiple imputation 451
18.3 Missing data in the multivariate normal and t models 454
18.4 Example: multiple imputation for a series of polls 456
18.5 Missing values with counted data 462
18.6 Example: an opinion poll in Slovenia 463
18.7 Bibliographic note 466
18.8 Exercises 467
Part V: Nonlinear and Nonparametric Models 469
19 Parametric nonlinear models 471
19.1 Example: serial dilution assay 471
19.2 Example: population toxicokinetics 477
19.3 Bibliographic note 485
19.4 Exercises 486
20 Basis function models 487
20.1 Splines and weighted sums of basis functions 487
20.2 Basis selection and shrinkage of coefficients 490
20.3 Non-normal models and regression surfaces 494
20.4 Bibliographic note 498
20.5 Exercises 498
21 Gaussian process models 501
21.1 Gaussian process regression 501
21.2 Example: birthdays and birthdates 505
21.3 Latent Gaussian process models 510
21.4 Functional data analysis 512
21.5 Density estimation and regression 513
21.6 Bibliographic note 516
21.7 Exercises 516
22 Finite mixture models 519
22.1 Setting up and interpreting mixture models 519
22.2 Example: reaction times and schizophrenia 524
22.3 Label switching and posterior computation 533
22.4 Unspecified number of mixture components 536
22.5 Mixture models for classification and regression 539
22.6 Bibliographic note 542
22.7 Exercises 543
23 Dirichlet process models 545
23.1 Bayesian histograms 545
23.2 Dirichlet process prior distributions 546
23.3 Dirichlet process mixtures 549
23.4 Beyond density estimation 557
23.5 Hierarchical dependence 560
23.6 Density regression 568
23.7 Bibliographic note 571
23.8 Exercises 573
This electronic edition is for non-commercial purposes only.
xii CONTENTS
Appendixes 575
A Standard probability distributions 577
A.1 Continuous distributions 577
A.2 Discrete distributions 585
A.3 Bibliographic note 586
B Outline of proofs of limit theorems 587
B.1 Bibliographic note 590
C Computation in R and Stan 591
C.1 Getting started with R and Stan 591
C.2 Fitting a hierarchical model in Stan 592
C.3 Direct simulation, Gibbs, and Metropolis in R 596
C.4 Programming Hamiltonian Monte Carlo in R 603
C.5 Further comments on computation 607
C.6 Bibliographic note 608
References 609
Author Index 643
Subject Index 654