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[书籍] An introduction to bayesian analysis

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    2015-4-12 14:55
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    [LV.3]偶尔看看II

    发表于 2015-3-22 10:00:07 | 显示全部楼层 |阅读模式 |          
    基础入门教材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
    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
    2.8 Prediction of a Future Observation . . . . . . . . . . . . . . . . . . . . . . . . 50
    2.9 Examples of Cox and Welch Revisited. . . . . . . . . . . . . . . . . . . . . . 51
    2.10 Elimination of Nuisance Parameters ....................... 51

    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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4
    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 Saunple ~ethods .................................... 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

    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, Pro bit 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

    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.1 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

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