Bayesian methods: a social and behavioral sciences approach
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
2015
|
| Edition: | 3., revised ed. |
| Series: | Statistics in the social and behavioral sciences series
A Chapman & Hall book |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XLIII, 680 S. graph. Darst. |
| ISBN: | 9781439862483 |
Staff View
MARC
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Record in the Search Index
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| adam_text | Titel: Bayesian methods
Autor: Gill, Jeff
Jahr: 2015
Contents
List of Figures xix
List of Tables xxi
Preface to the Third Edition xxv
Preface to the Second Edition xxix
Preface to the First Edition xxxvii
1 Background and Introduction 1
1.1 Introduction 1
1.2 General Motivation and Justification 4
1.3 Why Are We Uncertain about Uncertainty? 7
1.3.1 Required Probability Principles 8
1.4 Bayes Law 9
1.4.1 Bayes Law for Multiple Events 12
1.5 Conditional Inference with Bayes Law 15
1.5.1 Statistical Models with Bayes Law 17
1.6 Science and Inference 20
1.6.1 The Scientific Process in Our Social Sciences 20
1.6.2 Bayesian Statistics as a Scientific Approach to Social and Behavioral
Data Analysis 22
1.7 Introducing Markov Chain Monte Carlo Techniques 24
1.7.1 Simple Gibbs Sampling 25
1.7.2 Simple Metropolis Sampling 27
1.8 Historical Comments 30
1.9 Exercises 33
2 Specifying Bayesian Models 37
2.1 Purpose 37
2.2 Likelihood Theory and Estimation 37
2.3 The Basic Bayesian Framework 40
2.3.1 Developing the Bayesian Inference Engine 40
2.3.2 Summarizing Posterior Distributions with Intervals 42
ix
X
Contents
2.3.2.1 Bayesian Credible Intervals 43
2.3.2.2 Bayesian Highest Posterior Density Intervals 46
2.3.3 Quantile Posterior Summaries 48
2.3.4 Beta-Binoinial Model 49
2.4 Bayesian Learning 53
2.5 Conmients on Prior Distributions 56
2.6 Bayesian versus Non-Bayesian Approaches 57
2.7 Exercises 61
2.8 Computational Addendum: R for Basic Analysis 66
3 The Normal and Student s-t Models 69
3.1 WhyBe Normal? 69
3.2 The Normal Model with Variance Known 70
3.3 The Normal Model with Mean Known 72
3.4 The Normal Model with Both Mean and Variance Unknown 74
3.5 Multivariate Normal Model, /j, and S Both Unknown 76
3.6 Simulated Effects of Differing Priors 80
3.7 Some Normal Comments 82
3.8 The Student s-f Model 83
3.9 Normal Mixture Models 87
3.10 Exercises 89
3.11 Computational Addendum: Normal Examples 93
3.11.1 Normal Example with Variance Known 93
3.11.2 Bivariate Normal Simulation Example 94
3.11.3 Multivariate Normal Example, Health Data 95
4 The Bayesian Prior 97
4.1 A Prior Discussion of Priors 97
4.2 A Plethora of Priors 98
4.3 Conjugate Prior Forms 100
4.3.1 Example: Conjugacy in Exponential Specifications 100
4.3.2 The Exponential Family Form 101
4.3.3 Limitations of Conjugacy 104
4.4 Uninformative Prior Distributions 104
4.4.1 Uniform Priors 105
4.4.2 Jeffreys Prior 107
4.4.2.1 Bernoulli Trials and Jeffreys Prior 108
4.4.2.2 Other Forms of Jeffreys Priors 109
4.4.2.3 Jeffreys Prior in the Multiparameter Case 110
4.4.3 Reference Priors 111
4.4.4 Improper Priors 113
Contents xi
4.5 Informative Prior Distributions 114
4.5.1 Power Priors 115
4.5.2 Elicited Priors 116
4.5.2.1 The Community of Elicited Priors 118
4.5.2.2 Simple Elicitation Using Linear Regression 119
4.5.2.3 Variance Components Elicitation 121
4.5.2.4 Predictive Modal Elicitation 123
4.5.2.5 Prior Elicitation for the Normal-Linear Model 126
4.5.2.6 Elicitation Using a Beta Distribution 127
4.5.2.7 Eliciting Some Final Thoughts on Elicited Priors 128
4.6 Hybrid Prior Forms 129
4.6.1 Spike and Slab Priors for Linear Models 130
4.6.2 Maximum Entropy Priors 131
4.6.3 Histogram Priors 133
4.7 Nonparametric Priors 133
4.8 Bayesian Shrinkage 136
4.9 Exercises 138
5 The Bayesian Linear Model 145
5.1 The Basic Regression Model 145
5.1.1 Uninformative Priors for the Linear Model 147
5.1.2 Conjugate Priors for the Linear Model 151
5.1.3 Conjugate Caveats for the Cautious and Careful 154
5.2 Posterior Predictive Distribution for the Data 155
5.3 Linear Regression with Heteroscedasticity 161
5.4 Exercises 165
5.5 Computational Addendum 169
5.5.1 Palm Beach County Normal Model 169
5.5.2 Educational Outcomes Model 171
5.5.3 Ancient China Conflict Model 172
6 Assessing Model Quality 175
6.1 Motivation 175
6.1.1 Posterior Data Replication 177
6.1.2 Likelihood Function Robustness 180
6.2 Basic Sensitivity Analysis 181
6.2.1 Global Sensitivity Analysis 181
6.2.1.1 Specific Cases of Global Prior Sensitivity Analysis 182
6.2.1.2 Global Sensitivity in the Normal Model Case 182
6.2.1.3 Example: Prior Sensitivity in the Analysis of the 2000 U.S.
Election in Palm Beach County 183
xii Contents
6.2.1.4 Problems with Global Sensitivity Analysis 183
6.2.2 Local Sensitivity Analysis 184
6.2.2.1 Normal-Normal Model 185
6.2.2.2 Local Sensitivity Analysis Using Hyperparameter Changes 186
6.2.3 Global and Local Sensitivity Analysis with Recidivism Data .... 187
6.3 Robustness Evaluation 189
6.3.1 Global Robustness 190
6.3.2 Local Robustness 192
6.3.2.1 Bayesian Linear Outlier Detection 193
6.3.3 Bayesian Specification Robustness 195
6.4 Comparing Data to the Posterior Predictive Distribution 196
6.5 Simple Bayesian Model Averaging 198
6.6 Concluding Comments on Model Quality 200
6.7 Exercises 202
7 Bayesian Hypothesis Testing and the Bayes Factor 207
7.1 Motivation 207
7.2 Bayesian Inference and Hypothesis Testing 209
7.2.1 Problems with Conventional Hypothesis Testing 209
7.2.1.1 One-Sided Testing 211
7.2.1.2 Two-Sided Testing 214
7.2.2 Attempting a Bayesian Approximation to Frequentist Hypothesis
Testing 215
7.3 The Bayes Factor as Evidence 216
7.3.1 Bayes Factors for a Mean 218
7.3.2 Bayes Factors for Difference of Means Test 219
7.3.3 Bayes Factor for the Linear Regression Model 219
7.3.4 Bayes Factors and Improper Priors 223
7.3.4.1 Local Bayes Factor 224
7.3.4.2 Intrinsic Bayes Factor 225
7.3.4.3 Partial Bayes Factor 226
7.3.4.4 Fractional Bayes Factor 226
7.3.4.5 Redux 227
7.3.5 Two-Sided Hypothesis Tests and Bayes Factors 228
7.3.6 Challenging Aspects of Bayes Factors 229
7.4 The Bayesian Information Criterion (BIC) 231
7.5 The Deviance Information Criterion (DIC) 233
7.5.1 Some Qualifications 236
7.6 Comparing Posterior Distributions with the Kullback-Leibler Distance . . 237
7.7 Laplace Approximation of Bayesian Posterior Densities 239
7.8 Exercises 243
Contents xiii
8 Bayesian Decision Theory 247
8.1 Introducing Decision Theory 247
8.2 Basic Definitions 249
8.2.1 Personal Preference 250
8.2.2 Rules, Rules, Rules 250
8.2.3 LotsofLoss 251
8.2.4 Risky Business 253
8.2.4.1 Notes on Bayes Rules 256
8.2.5 Minimax Decision Rules 258
8.3 Regression-Style Models with Decision Theory 259
8.3.1 Prediction from the Linear Model 261
8.4 James-Stein Estimation 262
8.5 Empirical Bayes 267
8.6 Exercises 271
9 Monte Carlo and Related Iterative Methods 275
9.1 Background 275
9.2 Basic Monte Carlo Integration 277
9.3 Rejection Sampling 280
9.3.1 Continuous Form with Bounded Support 281
9.3.2 Continuous Form with Unbounded Support 284
9.4 Classical Numerical Integration 288
9.4.1 Newton-Cotes 289
9.4.1.1 Riemann Integrals 289
9.4.1.2 Trapezoid Rule 289
9.4.1.3 Simpson s Rule 290
9.5 Gaussian Quadrature 292
9.5.1 Redux 295
9.6 Importance Sampling and Sampling Importance Resampling 296
9.6.1 Importance Sampling for Producing HPD Regions 301
9.7 Mode Finding and the EM Algorithm 302
9.7.1 Deriving the EM Algorithm 304
9.7.2 Convergence of the EM Algorithm 307
9.7.3 Extensions to the EM Algorithm 313
9.7.4 Additional Comments on EM 315
9.7.5 EM for Exponential Families 316
9.8 Survey of Random Number Generation 320
9.9 Concluding Remarks 322
9.10 Exercises 323
9.11 Computational Addendum: R Code for Importance Sampling 328
xiv Contents
10 Basics of Markov Chain Monte Carlo 333
10.1 Who Is Markov and What Is He Döing with Chams? 333
10.1.1 What Is a Markov Chain? 334
10.1.2 A Markov Chain Illustration 335
10.1.3 The Chapman-Kolrnogorov Equations 338
10.1.4 Marginal Distributions 339
10.2 General Properties of Markov Chains 339
10.2.1 Homogeneity 340
10.2.2 Irreducibility 340
10.2.3 Recurrence 340
10.2.4 Stationarity 341
10.2.5 Ergodicity 342
10.3 The Gibbs Sampler 343
10.3.1 Description of the Algorithm 343
10.3.2 Handling Missing Dichotomous Data with the Gibbs Sampler .... 345
10.3.3 Summary of Properties of the Gibbs Sampler 353
10.4 The Metropolis-Hastings Algorithm 353
10.4.1 Background 354
10.4.2 Description of the Algorithm 354
10.4.3 Metropolis-Hastings Properties 356
10.4.4 Metropolis-Hastings Derivation 356
10.4.5 The Transition Kernel 358
10.4.6 Example: Estimating a Bivariate Normal Density 359
10.5 The Hit-and-Run Algorithm 360
10.6 The Data Augmentation Algorithm 362
10.7 Historical Comments 367
10.7.1 Füll Circle? 368
10.8 Exercises 368
10.9 Computational Addendum: Simple R Graphing Routines for MCMC . . . 375
11 Implementing Bayesian Models with Markov Chain Monte Carlo 377
11.1 Introduction to Bayesian Software Solutions 377
11.2 It s Only a Name: BUGS 378
11.3 Model Specification with BUGS 380
11.3.1 Model Specification 383
11.3.2 Running the Model in WinBUGS 385
11.3.3 Running the Model in JAGS 388
11.4 Differences between WinBUGS and JAGS Code 392
11.5 Technical Background about the Algorithm 401
11.6 Epilogue 408
11.7 Exercises 408
Contents xv
12 Bayesian Hierarchical Models 413
12.1 Introduction to Multilevel Specifications 413
12.2 Basic Multilevel Linear Models 414
12.3 Comparing Variances 416
12.4 Exchangeability 420
12.5 Essential Structure of the Bayesian Hierarchical Model 425
12.5.1 A Poisson-Gamma Hierarchical Specification 427
12.6 The General Role of Priors and Hyperpriors 434
12.7 Bayesian Multilevel Linear Regression Models 436
12.7.1 The Bayesian Hierarchical Linear Model of Lindley and Smith . . . 436
12.8 Bayesian Multilevel Generalized Linear Regression Models 441
12.9 Exercises 446
12.10 Computational Addendum 451
12.10.1 R Function for importing BUGS output 451
13 Some Markov Chain Monte Carlo Theory 453
13.1 Motivation 453
13.2 Measure and Probability Preliminaries 453
13.3 Specific Markov Chain Properties 455
13.3.1 ^-Irreducibility 455
13.3.2 Closed and Absorbing Sets 455
13.3.3 Homogeneity and Periodicity 455
13.3.4 Null and Positive Recurrence 456
13.3.5 Transience 456
13.3.6 Markov Chain Stability 457
13.3.7 Ergodicity 458
13.4 Defining and Reaching Convergence 458
13.5 Rates of Convergence 460
13.6 Implementation Concerns 464
13.6.1 Mixing 466
13.6.2 Partial Convergence for Metropolis-Hastings 467
13.6.3 Partial Convergence for the Gibbs Sampler 469
13.7 Exercises 471
14 Utilitarian Markov Chain Monte Carlo 475
14.1 Objectives 475
14.2 Practical Considerations and Admonitions 476
14.2.1 Starting Points 476
14.2.2 Thinning the Chain 477
14.2.3 The Burn-In Period 478
14.3 Assessing Convergence of Markov Chains 479
xvi Contents
14.3.1 Autocorrelation 485
14.3.2 Graphical Diagnostics 487
14.3.3 Standard Empirical Diagnostics 493
14.3.3.1 The Geweke Time-Series Diagnostic 494
14.3.3.2 Gelman and Rubin s Multiple Sequence Diagnostic .... 496
14.3.3.3 The Heidelberger and Welch Diagnostic 499
14.3.3.4 The Raftery and Lewis Integrated Diagnostic 503
14.3.4 Summary of Diagnostic Similarities and Differences 505
14.3.5 Other Empirical Diagnostics 507
14.3.6 Why Not to Worry Too Much about Stationarity 509
14.4 Mixing and Acceleration 510
14.4.1 Reparameterization 510
14.4.2 Grouping and Collapsing the Gibbs Sampler 512
14.4.3 Adding Auxiliary Variables 513
14.4.4 The Slice Sampler 513
14.5 Chib s Method for Calculating the Marginal Likelihood Integral 515
14.6 Rao-Blackwellizing for Improved Variance Estimation 517
14.7 Exercises 520
14.8 Computational Addendum: Code for Chapter Examples 523
14.8.1 R Code for the Death Penalty Support Model 523
14.8.2 JAGS Code for the Military Personnel Model 524
15 Markov Chain Monte Carlo Extensions 525
15.1 Simulated Annealing 525
15.1.1 General Points on Simulated Annealing 529
15.1.2 Metropolis-Coupling 530
15.1.3 Simulated Tempering 531
15.1.4 Tempored Transitions 532
15.1.5 Comparison of Algorithms 533
15.1.6 Dynamic Tempered Transitions 535
15.2 Reversible Jump Algorithms 536
15.3 Perfect Sampling 538
15.4 Hamiltonian Monte Carlo 542
15.5 Exercises 547
Appendix A Generalized Linear Model Review 553
A.l Terms 553
A.l.l The Linear Regression Model 555
A.2 The Generalized Linear Model 557
A.2.1 Defining the Link Function 558
A.2.2 Deviance Residuais 560
Contents xvii
A.3 Numerical Maximum Likelihood 564
A.3.1 Newton-Raphson and Root Finding 564
A.3.1.1 Newton-Raphson for Statistical Problems 566
A.3.1.2 Weighted Least Squares 566
A.3.1.3 Iterative Weighted Least Squares 567
A.4 Quasi-Likelihood 568
A.5 Exercises 572
Appendix B Common Probability Distributions 579
References 583
Author Index 643
Subject Index 665
|
| any_adam_object | 1 |
| author | Gill, Jeff |
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| ctrlnum | (OCoLC)780110195 (DE-599)BVBBV039870620 |
| dewey-full | 519.5/42 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.5/42 |
| dewey-search | 519.5/42 |
| dewey-sort | 3519.5 242 |
| dewey-tens | 510 - Mathematics |
| discipline | Soziologie Psychologie Mathematik Wirtschaftswissenschaften |
| edition | 3., revised ed. |
| format | Book |
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| id | DE-604.BV039870620 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:13:04Z |
| institution | BVB |
| isbn | 9781439862483 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-024729951 |
| oclc_num | 780110195 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-634 DE-11 DE-20 DE-355 DE-BY-UBR DE-M347 DE-188 DE-83 |
| owner_facet | DE-19 DE-BY-UBM DE-634 DE-11 DE-20 DE-355 DE-BY-UBR DE-M347 DE-188 DE-83 |
| physical | XLIII, 680 S. graph. Darst. |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | CRC Press |
| record_format | marc |
| series2 | Statistics in the social and behavioral sciences series A Chapman & Hall book |
| spelling | Gill, Jeff Verfasser aut Bayesian methods a social and behavioral sciences approach Jeff Gill 3., revised ed. Boca Raton [u.a.] CRC Press 2015 XLIII, 680 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Statistics in the social and behavioral sciences series A Chapman & Hall book Statistik (DE-588)4056995-0 gnd rswk-swf Bayes-Entscheidungstheorie (DE-588)4144220-9 gnd rswk-swf Bayes-Verfahren (DE-588)4204326-8 gnd rswk-swf Sozialwissenschaften (DE-588)4055916-6 gnd rswk-swf Bayes-Entscheidungstheorie (DE-588)4144220-9 s DE-604 Sozialwissenschaften (DE-588)4055916-6 s Statistik (DE-588)4056995-0 s Bayes-Verfahren (DE-588)4204326-8 s DE-188 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024729951&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gill, Jeff Bayesian methods a social and behavioral sciences approach Statistik (DE-588)4056995-0 gnd Bayes-Entscheidungstheorie (DE-588)4144220-9 gnd Bayes-Verfahren (DE-588)4204326-8 gnd Sozialwissenschaften (DE-588)4055916-6 gnd |
| subject_GND | (DE-588)4056995-0 (DE-588)4144220-9 (DE-588)4204326-8 (DE-588)4055916-6 |
| title | Bayesian methods a social and behavioral sciences approach |
| title_auth | Bayesian methods a social and behavioral sciences approach |
| title_exact_search | Bayesian methods a social and behavioral sciences approach |
| title_full | Bayesian methods a social and behavioral sciences approach Jeff Gill |
| title_fullStr | Bayesian methods a social and behavioral sciences approach Jeff Gill |
| title_full_unstemmed | Bayesian methods a social and behavioral sciences approach Jeff Gill |
| title_short | Bayesian methods |
| title_sort | bayesian methods a social and behavioral sciences approach |
| title_sub | a social and behavioral sciences approach |
| topic | Statistik (DE-588)4056995-0 gnd Bayes-Entscheidungstheorie (DE-588)4144220-9 gnd Bayes-Verfahren (DE-588)4204326-8 gnd Sozialwissenschaften (DE-588)4055916-6 gnd |
| topic_facet | Statistik Bayes-Entscheidungstheorie Bayes-Verfahren Sozialwissenschaften |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024729951&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT gilljeff bayesianmethodsasocialandbehavioralsciencesapproach |