Frontiers of Statistical Decision Making and Bayesian Analysis (eBook)

Ming-Hui Chen is Professor of Statistics at the University of Connecticut; Dipak K. Dey is Head and Professor of Statistics at the University of Connecticut; Peter Müller is Professor of Biostatistics at the University of Texas M. D. Anderson Cancer Center; Dongchu Sun is Professor of Statistics at the University of Missouri- Columbia; and Keying Ye is Professor of Statistics at the University of Texas at San Antonio.

Preface 6
Contents 8
List of Contributors 16
1 Introduction 23
1.1 Biography of James O. Berger 23
1.2 The Frontiers of Research at SAMSI 24
1.2.1 Research Topics from Past SAMSI Programs 25
1.2.2 Research Topics from Current SAMSI Programs 44
1.2.3 Research Topics in Future Programs 46
1.3 Overview of the Book 49
2 Objective Bayesian Inference with Applications 53
2.1 Bayesian Reference Analysis of the Hardy-Weinberg Equilibrium 53
José M. Bernardo and Vera Tomazella 53
2.1.1 Problem Statement 54
2.1.2 Objective Precise Bayesian Testing 55
2.1.3 Testing for Hardy-Weinberg Equilibrium 57
2.1.4 Examples 63
2.2 Approximate Reference Priors in the Presence of Latent Structure 66
Brunero Liseo, Andrea Tancredi, and Maria M. Barbieri 66
2.2.1 The Method 67
2.2.2 Examples 69
2.2.3 The Case with Nuisance Parameters 75
2.2.4 Conclusions 77
2.3 Reference Priors for Empirical Likelihoods 78
Bertrand Clarke and Ao Yuan 78
2.3.1 Empirical Likelihood 79
2.3.2 Reference Priors 80
2.3.3 Relative Entropy Reference Priors 83
2.3.4 Hellinger Reference Prior 87
2.3.5 Chi-square Reference Prior 88
2.3.6 Discussion 90
3 Bayesian Decision Based Estimation and Predictive Inference 91
3.1 Bayesian Shrinkage Estimation 91
William E. Strawderman 91
3.1.1 Some Intuition into Shrinkage Estimation 92
3.1.2 Some Theory for the Normal Case with Covariance 2I 94
3.1.3 Results for Known and General Quadratic Loss 99
3.1.4 Conclusion and Extensions 104
3.2 Bayesian Predictive Density Estimation 105
Edward I. George and Xinyi Xu 105
3.2.1 Prediction for the Multivariate Normal Distribution 107
3.2.2 Predictive Density Estimation for Linear Regression 110
3.2.3 Multiple Shrinkage Predictive Density Estimation 112
3.2.4 Simulation Studies 113
3.2.5 Concluding Remarks 117
3.3 Automated Bias-variance Trade-off: Intuitive Inadmissibility or Inadmissible Intuition? 117
Xiao-Li Meng 117
3.3.1 Always a Good Question ... 118
3.3.2 Gene-Environment Interaction and a Misguided Insight 119
3.3.3 Understanding Partially Bayes Methods 122
3.3.4 Completing M& C's Argument
3.3.5 Learning through Exam: The Actual Qualifying Exam Problem 127
3.3.6 Interweaving Research and Pedagogy: The Actual Annotated Solution 129
3.3.7 A Piece of Inadmissible Cake? 133
4 Bayesian Model Selection and Hypothesis Tests 135
4.1 Performance of Bayesian Model Selection Criteria for Gaussian Mixture Models 135
Russell J. Steele and Adrian E. Raftery 135
4.1.1 Bayesian Model Selection for Mixture Models 136
4.1.2 A Unit Information Prior for Mixture Models 140
4.1.3 Examples 144
4.1.4 Simulation Study 147
4.1.5 Discussion 151
4.2 How Large Should the Training Sample Be? 152
Luis Pericchi 152
4.2.1 General Methodology 153
4.2.2 An Exact Calculation 157
4.2.3 Discussion of the FivePercent-Cubic-Root Rule 164
4.3 A Conservative Property of Bayesian Hypothesis Tests 164
Valen E. Johnson 164
4.3.1 An Inequality 165
4.3.2 Discussion 167
4.4 An Assessment of the Performance of Bayesian Model Averaging in the Linear Model 168
Ilya Lipkovich, Keying Ye, and Eric P. Smith 168
4.4.1 Assessment of BMA Performance 170
4.4.2 A Simulation Study of BMA Performance 171
4.4.3 Summary 177
5 Bayesian Inference for Complex Computer Models 178
5.1 A Methodological Review of Computer Models 178
M.J. Bayarri 178
5.1.1 Computer Models and Emulators 179
5.1.2 The Discrepancy (Bias) Function 180
5.1.3 Confounding of Tuning and Bias 184
5.1.4 Modularization 185
5.1.5 Additional Issues 188
5.1.6 Summary 189
5.2 Computer Model Calibration with Multivariate Spatial Output 189
K. Sham Bhat, Murali Haran, and Marlos Goes 189
5.2.1 Computer Model Calibration with Spatial Output 191
5.2.2 Calibration with Multivariate Spatial Output 193
5.2.3 Application to Climate Parameter Inference 197
5.2.4 Results 200
5.2.5 Summary 205
6 Bayesian Nonparametrics and Semi-parametrics 206
6.1 Bayesian Nonparametric Goodness of Fit Tests 206
Surya T. Tokdar, Arijit Chakrabarti, and Jayanta K. Ghosh 206
6.1.1 An Early Application of Bayesian Ideas in Goodness of Fit Problems 208
6.1.2 Testing a Point Null versus Non-parametric Alternatives 208
6.1.3 Posterior Consistency for a Composite Goodness of Fit Test 210
6.1.4 Bayesian Goodness of Fit Tests 213
6.2 Species Sampling Model and Its Application to Bayesian Statistics 215
Jaeyong Lee 215
6.2.1 Basic Theory 217
6.2.2 Construction Methods for EPPFs 222
6.2.3 Statistical Applications 225
6.2.4 Discussion 227
6.3 Hierarchical Models, Nested Models, and Completely Random Measures 228
Michael I. Jordan 228
6.3.1 Completely Random Measures 229
6.3.2 Marginal Probabilities 231
6.3.3 Hierarchical Models 233
6.3.4 Nested Models 235
6.3.5 Discussion 237
7 Bayesian Influence and Frequentist Interface 239
7.1 Bayesian Influence Methods 239
Hongtu Zhu, Joseph G. Ibrahim, Hyunsoon Cho, and Niansheng Tang 239
7.1.1 Bayesian Case Influence Measures 241
7.1.2 Bayesian Global and Local Robustness 246
7.1.3 An Illustrative Example 253
7.2 The Choice of Nonsubjective Priors on Hyperparameters for Hierarchical Bayes Models 257
Gauri S. Datta and J.N.K. Rao 257
7.2.1 Probability Matching in Small Area Estimation 260
7.2.2 Frequentist Evaluation of Posterior Variance 262
7.2.3 Discussion 265
7.3 Exact Matching Inference for a Multivariate Normal Model 267
Luyan Dai and Dongchu Sun 267
7.3.1 The Background 269
7.3.2 Main Results 272
8 Bayesian Clinical Trials 277
8.1 Application of a Bayesian Doubly Optimal Group Sequential Design for Clinical Trials 277
J. Kyle Wathen and Peter F. Thall 277
8.1.1 A Non-Small Cell Lung Cancer Trial 277
8.1.2 Bayesian Doubly Optimal Group Sequential Designs 279
8.1.3 Application of BDOGS to the Lung Cancer Trial 282
8.1.4 Discussion 289
8.2 Experimental Design and Sample Size Computations for Longitudinal Models 290
Robert E. Weiss and Yan Wang 290
8.2.1 Covariates and Missing Data 291
8.2.2 Simulating the Predictive Distributions of the Bayes Factor 291
8.2.3 Sample Size for a New Repeated Measures Pediatric Pain Study 292
8.3 A Bayes Rule for Subgroup Reporting 297
Peter Müller, Siva Sivaganesan, and Purushottam W. Laud 297
8.3.1 The Model Space 297
8.3.2 Subgroup Selection as a Decision Problem 298
8.3.3 Probability Model 301
8.3.4 A Dementia Trial 302
8.3.5 Discussion 304
9 Bayesian Methods for Genomics, Molecular and Systems Biology 305
9.1 Bayesian Modelling for Biological Annotation of Gene Expression Pathway Signatures 305
Haige Shen and Mike West 305
9.1.1 Context and Models 307
9.1.2 Computation 310
9.1.3 Evaluation and Illustrations 313
9.1.4 Applications to Hormonal Pathways in Breast Cancer 316
9.1.5 Theoretical and Algorithmic Details 320
9.1.6 Summary Comments 322
9.2 Bayesian Methods for Network-Structured Genomics Data 323
Stefano Monni and Hongzhe Li 323
9.2.1 Bayesian Variable Selection with a Markov Random Field Prior 324
9.2.2 Numerical Examples 329
9.2.3 Discussion and Future Direction 335
9.3 Bayesian Phylogenetics 336
Erik W. Bloomquist and Marc A. Suchard 336
9.3.1 Statistical Phyloalignment 339
9.3.2 Multilocus Data 341
9.3.3 Looking Ahead 344
10 Bayesian Data Mining and Machine Learning 346
10.1 Bayesian Model-based Principal Component Analysis 346
Bani K. Mallick, Shubhankar Ray, and Soma Dhavala 346
10.1.1 Random Principal Components 348
10.1.2 Piecewise RPC Models 350
10.1.3 Principal Components Clustering 353
10.1.4 Reversible Jump Proposals 356
10.1.5 Experimental Results 359
10.2 Priors on the Variance in Sparse Bayesian Learning: the demi-Bayesian Lasso 365
Suhrid Balakrishnan and David Madigan 365
10.2.1 Background and Notation 366
10.2.2 The demi-Bayesian Lasso 369
10.2.3 Experiments and Results 373
10.2.4 Discussion 378
10.3 Hierarchical Bayesian Mixed-Membership Models and Latent Pattern Discovery 379
Edoardo M. Airoldi, Stephen E. Fienberg, Cyrille J. Joutard, and Tanzy M. Love 379
10.3.1 Characterizing HBMM Models 382
10.3.2 Strategies for Model Choice 383
10.3.3 Case Study: PNAS 1997--2001 384
10.3.4 Case Study: Disability Profiles 388
10.3.5 Summary 393
11 Bayesian Inference in Political Science, Finance, and Marketing Research 395
11.1 Prior Distributions for Bayesian Data Analysis in Political Science 395
Andrew Gelman 395
11.1.1 Statistics in Political Science 396
11.1.2 Mixture Models and Different Ways of Encoding Prior Information 397
11.1.3 Incorporating Extra Information Using Poststratification 398
11.1.4 Prior Distributions for Varying-Intercept, Varying-Slope Multilevel Regressions 399
11.1.5 Summary 400
11.2 Bayesian Computation in Finance 401
Satadru Hore, Michael Johannes, Hedibert Lopes, Robert E. McCulloch, and Nicholas G. Polson 401
11.2.1 Empirical Bayesian Asset Pricing 402
11.2.2 Bayesian Inference via SMC 403
11.2.3 Bayesian Inference via MCMC 406
11.2.4 Conclusion 414
11.3 Simulation-based-Estimation in Portfolio Selection 414
Eric Jacquier and Nicholas G. Polson 414
11.3.1 Basic Asset Allocation 416
11.3.2 Optimum Portfolios by MCMC 423
11.3.3 Discussion 427
11.4 Bayesian Multidimensional Scaling and Its Applications in Marketing Research 428
Duncan K.H. Fong 428
11.4.1 Bayesian Vector MDS Models 430
11.4.2 A Marketing Application 432
11.4.3 Discussion and Future Research 434
12 Bayesian Categorical Data Analysis 436
12.1 Good Smoothing 436
James H. Albert 436
12.1.1 Good's 1967 Paper 437
12.1.2 Examples of Good Smoothing 443
12.1.3 Smoothing Hitting Rates in Baseball 447
12.1.4 Closing Comments 452
12.2 Bayesian Analysis of Matched Pair Data 453
Malay Ghosh and Bhramar Mukherjee 453
12.2.1 Item Response Models 454
12.2.2 Bayesian Analysis of Matched Case-Control Data 456
12.2.3 Some Equivalence Results in Matched Case-Control Studies 462
12.2.4 Other Work 465
12.2.5 Conclusion 466
12.3 Bayesian Choice of Links and Computation for Binary Response Data 468
Ming-Hui Chen, Sungduk Kim, Lynn Kuo, and Wangang Xie 468
12.3.1 The Binary Regression Models 470
12.3.2 Prior and Posterior Distributions 471
12.3.3 Computational Development 473
12.3.4 A Case Study 478
12.3.5 Discussion 481
13 Bayesian Geophysical, Spatial and Temporal Statistics 484
13.1 Modeling Spatial Gradients on Response Surfaces 484
Sudipto Banerjee and Alan E. Gelfand 484
13.1.1 Directional Derivative Processes 486
13.1.2 Mean Surface Gradients 488
13.1.3 Posterior Inference for Gradients 490
13.1.4 Gradients under Spatial Dirichlet Processes 492
13.1.5 Illustration 494
13.1.6 Concluding Remarks 500
13.2 Non-Gaussian Hierarchical Generalized Linear Geostatistical Model Selection 501
Xia Wang, Dipak K. Dey, and Sudipto Banerjee 501
13.2.1 A Review on the Generalized Linear Geostatistical Model 503
13.2.2 Generalized Extreme Value Link Model 504
13.2.3 Prior and Posterior Distributions for the GLGM Model under Different Links 506
13.2.4 A Simulated Data Example 507
13.2.5 Analysis of Celastrus Orbiculatus Data 509
13.2.6 Discussion 513
13.3 Objective Bayesian Analysis for Gaussian Random Fields 514
Victor De Oliveira 514
13.3.1 Gaussian Random Field Models 515
13.3.2 Integrated Likelihoods 516
13.3.3 Reference Priors 517
13.3.4 Jeffreys Priors 520
13.3.5 Other Spatial Models 522
13.3.6 Further Properties 524
13.3.7 Multi-Parameter Cases 525
13.3.8 Discussion and Some Open Problems 528
14 Posterior Simulation and Monte Carlo Methods 529
14.1 Importance Sampling Methods for Bayesian Discrimination between Embedded Models 529
Jean-Michel Marin and Christian P. Robert 529
14.1.1 The Pima Indian Benchmark Model 530
14.1.2 The Basic Monte Carlo Solution 533
14.1.3 Usual Importance Sampling Approximations 534
14.1.4 Bridge Sampling Methodology 536
14.1.5 Harmonic Mean Approximations 539
14.1.6 Exploiting Functional Equalities 541
14.1.7 Conclusion 543
14.2 Bayesian Computation and the Linear Model 543
Matthew J. Heaton and James G. Scott 543
14.2.1 Bayesian Linear Models 545
14.2.2 Algorithms for Variable Selection and Shrinkage 547
14.2.3 Examples 553
14.2.4 Final Remarks 561
14.3 MCMC for Constrained Parameter and Sample Spaces 561
Merrill W. Liechty, John C. Liechty, and Peter Müller 561
14.3.1 The Shadow Prior 563
14.3.2 Example: Modeling Correlation Matrices 565
14.3.3 Simulation Study 566
14.3.4 Classes of Models Suitable for Shadow Prior Augmentations 567
14.3.5 Conclusion 568
References 570
Author Index 630
Subject Index 641