An Introduction to Bayesian Analysis: Theory and Methods (Springer Texts in Statistics) (Hardcover)
Jayanta K. Ghosh (Author), Mohan Delampady (Author), Tapas Samanta (Author)



Product Details

• Hardcover: 352 pages
• Publisher: Springer; 1 edition (July 27, 2006)
• Language: English
• ISBN-10: 0387400842
• ISBN-13: 978-0387400846

Contents
Statistical Preliminaries 1
1.1 Common Models 1
1.1.1 Exponential Families 4
1.1.2 Location-Scale Families 5
1.1.3 Regular Family 6
1.2 Likelihood Function 7
1.3 Sufficient Statistics and Ancillary Statistics 9
1.4 Three Basic Problems of Inference in Classical Statistics 11
1.4.1 Point Estimates 11
1.4.2 Testing Hypotheses 16
1.4.3 Interval Estimation 20
1.5 Inference as a Statistical Decision Problem 21
1.6 The Changing Face of Classical Inference 23
1.7 Exercises 24
Bayesian Inference and Decision Theory 29
2.1 Subjective and Frequentist Probability 29
2.2 Bayesian Inference 30
2.3 Advantages of Being a Bayesian 35
2.4 Paradoxes in Classical Statistics 37
2.5 Elements of Bayesian Decision Theory 38
2.6 Improper Priors 40
2.7 Common Problems of Bayesian Inference 41
2.7.1 Point Estimates 41
2.7.2 Testing 42
2.7.3 Credible Intervals 48
2.7.4 Testing of a Sharp Null Hypothesis Through Credible
Intervals 49
2.8 Prediction of a Future Observation 50
2.9 Examples of Cox and Welch Revisited 51
2.10 Elimination of Nuisance Parameters 51
X Contents
2.11 A High-dimensional Example 53
2.12 Exchangeability 54
2.13 Normative and Descriptive Aspects of Bayesian Analysis,
Elicitation of Probability 55
2.14 Objective Priors and Objective Bayesian Analysis 55
2.15 Other Paradigms 57
2.16 Remarks 57
2.17 Exercises 58
3 Utility, Prior, and Bayesian Robustness 65
3.1 Utility, Prior, and Rational Preference 65
3.2 Utility and Loss 67
3.3 Rationality Axioms Leading to the Bayesian Approach 68
3.4 Coherence 70
3.5 Bayesian Analysis with Subjective Prior 71
3.6 Robustness and Sensitivity 72
3.7 Classes of Priors 74
3.7.1 Conjugate Class 74
3.7.2 Neighborhood Class 75
3.7.3 Density Ratio Class 75
3.8 Posterior Robustness: Measures and Techniques 76
3.8.1 Global Measures of Sensitivity 76
3.8.2 Belief Functions 81
3.8.3 Interactive Robust Bayesian Analysis 83
3.8.4 Other Global Measures 84
3.8.5 Local Measures of Sensitivity 84
3.9 Inherently Robust Procedures 91
3.10 Loss Robustness 92
3.11 Model Robustness 93
3.12 Exercises 94
4 Large Sample Methods 99
4.1 Limit of Posterior Distribution 100
4.1.1 Consistency of Posterior Distribution 100
4.1.2 Asymptotic Normality of Posterior Distribution 101
4.2 Asymptotic Expansion of Posterior Distribution 107
4.2.1 Determination of Sample Size in Testing 109
4.3 Laplace Approximation 113
4.3.1 Laplace's Method 113
4.3.2 Tierney-Kadane-Kass Refinements 115
4.4 Exercises 119
Contents XI
Choice of Priors for Low-dimensional Parameters 121
5.1 DiflFerent Methods of Construction of Objective Priors 122
5.1.1 Uniform Distribution and Its Criticisms 123
5.1.2 Jeffreys Prior as a Uniform Distribution 125
5.1.3 Jeffreys Prior as a Minimizer of Information 126
5.1.4 Jeffreys Prior as a Probability Matching Prior 129
5.1.5 Conjugate Priors and Mixtures 132
5.1.6 Invariant Objective Priors for Location-Scale Families . . 135
5.1.7 Left and Right Invariant Priors 136
5.1.8 Properties of the Right Invariant Prior for
Location-Scale Families 138
5.1.9 General Group Families 139
5.1.10 Reference Priors 140
5.1.11 Reference Priors Without Entropy Maximization 145
5.1.12 Objective Priors with Partial Information 146
5.2 Discussion of Objective Priors 147
5.3 Exchangeability 149
5.4 Elicitation of Hyperparameters for Prior 149
5.5 A New Objective Bayes Methodology Using Correlation 155
5.6 Exercises 156
Hypothesis Testing and Model Selection 159
6.1 Preliminaries 159
6.1.1 BIG Revisited 161
6.2 P-value and Posterior Probability of HQ as Measures of
Evidence Against the Null 163
6.3 Bounds on Bayes Factors and Posterior Probabilities 164
6.3.1 Introduction 164
6.3.2 Choice of Classes of Priors 165
6.3.3 Multiparameter Problems 168
6.3.4 Invariant Tests 172
6.3.5 Interval Null Hypotheses and One-sided Tests 176
6.4 Role of the Choice of an Asymptotic Framework 176
6.4.1 Comparison of Decisions via P-values and Bayes
Factors in Bahadur's Asymptotics 178
6.4.2 Pitman Alternative and Rescaled Priors 179
6.5 Bayesian P-value 179
6.6 Robust Bayesian Outlier Detection 185
6.7 Nonsubjective Bayes Factors 188
6.7.1 The Intrinsic Bayes Factor 190
6.7.2 The Fractional Bayes Factor 191
6.7.3 Intrinsic Priors 194
6.8 Exercises 199
XII Contents
7 Bayesian Computations 205
7.1 Analytic Approximation 207
7.2 The E-M Algorithm 208
7.3 Monte Carlo Sampling 211
7.4 Markov Chain Monte Carlo Methods 215
7.4.1 Introduction 215
7.4.2 Markov Chains in MCMC 216
7.4.3 Metropolis-Hastings Algorithm 218
7.4.4 Gibbs Sampling 220
7.4.5 Rao-Blackwellization 223
7.4.6 Examples 225
7.4.7 Convergence Issues 231
7.5 Exercises 233
8 Some Common Problems in Inference 239
8.1 Comparing Two Normal Means 239
8.2 Linear Regression 241
8.3 Logit Model, Probit Model, and Logistic Regression 245
8.3.1 The Logit Model 246
8.3.2 The Probit Model 251
8.4 Exercises 252
9 High-dimensional Problems 255
9.1 Exchangeability, Hierarchical Priors, Approximation to
Posterior for Large p, and MCMC 256
9.1.1 MCMC and E-M Algorithm 259
9.2 Parametric Empirical Bayes 260
9.2.1 PEB and HB Interval Estimates 262
9.3 Linear Models for High-dimensional Parameters 263
9.4 Stein's Frequentist Approach to a High-dimensional Problem.. 264
9.5 Comparison of High-dimensional and Low-dimensional
Problems 268
9.6 High-dimensional Multiple Testing (PEB) 269
9.6.1 Nonparametric Empirical Bayes Multiple Testing 271
9.6.2 False Discovery Rate (FDR) 272
9.7 Testing of a High-dimensional Null as a Model Selection
Problem 273
9.8 High-dimensional Estimation and Prediction Based on Model
Selection or Model Averaging 276
9.9 Discussion 284
9.10 Exercises 285
Contents XIII
10 Some Applications 289
10.1 Disease Mapping 289
10.2 Bayesian Nonparametric Regression Using Wavelets 292
10.2.1 A Brief Overview of Wavelets 293
10.2.2 Hierarchical Prior Structure and Posterior
Computations 296
10.3 Estimation of Regression Function Using Dirichlet
Multinomial Allocation 299
10.4 Exercises 302
A Common Statistical Densities 303
A.l Continuous Models 303
A.2 Discrete Models 306
B Birnbaum's Theorem on Likelihood Principle 307
C Coherence 311
D Microarray 313
E Bayes Sufficiency 315
References 317
Author Index 339
Subject Index 345