Table of contents for An introduction to Bayesian analysis : theory and methods / Jayanta K. Ghosh, Mohan Delampady, Tapas Samanta.


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1   Statistical Prelim inaries  . ................... .............
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
L4 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
"2  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
,28  Prediction o  a Fuiture Observation  ............ ..... ...  50
2.9 Examples of Cox and Welch Revisited. . . .               51.......... . 51
2.10 Elimination of Nuisance Parametersl .  ......           5
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  .............................. .
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.   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
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
1.3.2 Tiern-ey-Kadane-Kass Refinements , . . . . .......... . 115
4.4  Exercises  .. .. ...... . .. ... .. .. . . . .. ...... ...... .  .. 119
5   Choice of Priors for Low-dimensional Parameters . .. ....... 121
5,1. Different Methods of Construction of Objective Priors ....... 122
5.1.1 Uniform Distribution and Its Criticisms .............. 123
51.2 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.,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  .. .. i55
5.6  Exercises  ....... .... ... ... .. .. ................... . . .. ..  156
6   Hypothesis Testing and Model Selection . . . .      . . .. . .. 159
6.   Prelim inaries  .  .... ..... ............... ... .. .  .... .  159
6.1.1  BIC  Revisited  ..... ...... ...... .......... ....... 161
6.2 P-value and Posterior Probability of Ha as Measures of
Evidence Against the Null ............................... 163
6.3 Bounds on Bayes Factors and Posterior Probabilities ..1..... 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 Resealed 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
"7  Bayesian Computations......................         .. 20
7.1 Analytic Approximation      .........                 . .207
.2  The E-M Algorithm   .........     ...    ..   ....... 208
7.3  Monte Carlo  Sampling  ........  ............           . 211
7.4 Markov Chain Monte Carlo Methods......      .......  ...... 215
7 .1  Introduction  .. .... ...... ..... ... ..........    1215
7.4.2  Markov Chains in MCMC      ... .. .... ..... .....216
7.4.3  Metropolis-Hastings Algorithm. .. .. . ........ .. . 218
7.4   Gibbs  Sampling  .............    ....  .... ........  220
7.4.5  RaoBlackwellization ..  ............              223
"7.4.6  Examples  .. ........ .          ........... ..... 225
7.4.7  Convergence Issues  ......... ......... ..  ........231
7.  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 ...............59
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 Highg-dimensional Estimation and Prediction Based on Model
Selection or Model Averaging . ....   .....   ....... 276
9,9  Discussion  .,.....  .....  ..  ......... ...... . . ...... . 284
9.10  Exercises .. .           . . ..........                . 285
10 Some Applications ...........        ......  .......   .. ..... 289
10.  Disease Mapping  ... ...  ..  .  .. ..  .......        289
10,2 Bayesian Nonparametric Regression Using Wavvelets . ..... .292
10.2.1 A Brief Overview of Wavelets . ..... .....  ...  293
10.2.2 Hierarlchical Prior Structure and Posterior
Computations. .  .. .....    ..  .......     ... - 296
10.3 Estimation of Regression Function Using Dirichlet
Multinornial Allocation ....     . .. .. .... . ........  299
10.4  Exercises         ... ...   . .. ...  ......  ..  ... ...302
A   Common Statistical Densities            ....           ....303
A.1 Continuous Models ....     .. . . . . . .... 303
A.2  Discrete  Models  .......... .... . .. .....     . ...  306
B   Birnbaum's Theorem on Likelihood Principle ...       ..... 307
C   Coherence      . . .                                  .... ... 311
D   Microarray     ..   . ............  ..13... .... .          3
E   Bayes Sufficiency .          .. ...       ..   ........315



Library of Congress subject headings for this publication: Bayesian statistical decision theory