Monte Carlo statistical methods:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY [u.a.]
Springer
2000
|
| Edition: | Corr. 2. print. |
| Series: | Springer texts in statistics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXI, 507 S. graph. Darst. |
| ISBN: | 038798707X |
Staff View
MARC
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Record in the Search Index
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|---|---|
| adam_text | Contents
List of Tables xvii
List of Figures xix
1 Introduction 1
1.1 Statistical Models 1
1.2 Likelihood Methods 5
1.3 Bayesian Methods 11
1.4 Deterministic Numerical Methods 17
1.5 Problems 19
1.6 Notes 31
2 Random Variable Generation 35
2.1 Basic Methods 35
2.1.1 Introduction 35
2.1.2 The Kiss Generator 39
2.1.3 Beyond Uniform Distributions 43
2.2 Transformation Methods 45
2.3 Accept Reject Methods 49
2.3.1 General Principle 49
2.3.2 Envelope Accept Reject Methods 53
2.3.3 Log Concave Densities 56
2.4 Problems 60
2.5 Notes 69
3 Monte Carlo Integration 71
3.1 Introduction 71
3.2 Classical Monte Carlo Integration 74
3.3 Importance Sampling 80
3.3.1 Principles 80
3.3.2 Finite Variance Estimators 83
3.3.3 Comparing Importance Sampling with Accept Reject 92
3.4 Riemann Approximations 96
3.5 Laplace Approximations 103
xiv CONTENTS
3.6 The Saddlepoint Approximation 104
3.6.1 An Edgeworth Derivation 105
3.6.2 Tail Areas 108
3.7 Acceleration Methods 109
3.7.1 Antithetic Variables 109
3.7.2 Control Variates 113
3.7.3 Conditional Expectations 116
3.8 Problems 119
3.9 Notes 131
4 Markov Chains 139
4.1 Essentials for MCMC 140
4.2 Basic Notions 141
4.3 Irreducibility, Atoms, and Small Sets 145
4.3.1 Irreducibility 145
4.3.2 Atoms and Small Sets 147
4.3.3 Cycles and Aperiodicity 150
4.4 Transience and Recurrence 151
4.4.1 Classification of Irreducible Chains 151
4.4.2 Criteria for Recurrence 153
4.4.3 Harris Recurrence 154
4.5 Invariant Measures 155
4.6 Ergodicity and Convergence 159
4.6.1 Ergodicity 159
4.6.2 Geometric Convergence 164
4.6.3 Uniform Ergodicity 166
4.7 Limit Theorems 167
4.7.1 Ergodic Theorems 168
4.7.2 Central Limit Theorems 170
4.8 Covariance in Markov Chains 172
4.9 Problems 174
4.10 Notes 185
5 Monte Carlo Optimization 193
5.1 Introduction 193
5.2 Stochastic Exploration 194
5.2.1 A Basic Solution 194
5.2.2 Gradient Methods 196
5.2.3 Simulated Annealing 199
5.2.4 Prior Feedback 202
5.3 Stochastic Approximation 206
5.3.1 Missing Data Models and Demarginalization 207
5.3.2 Monte Carlo Approximation 208
5.3.3 The EM Algorithm 213
5.3.4 Monte Carlo EM 216
5.4 Problems 218
CONTENTS xv
5.5 Notes 227
6 The Metropolis Hastings Algorithm 231
6.1 Monte Carlo Methods Based on Markov Chains 231
6.2 The Metropolis Hastings algorithm 233
6.2.1 Definition 233
6.2.2 Convergence Properties 236
6.3 A Collection of Metropolis Hastings Algorithms 238
6.3.1 The Independent Case 239
6.3.2 Random Walks 245
6.3.3 ARMS: A General Metropolis Hastings Algorithm 249
6.4 Optimization and Control 251
6.4.1 Optimizing the Acceptance Rate 252
6.4.2 Conditioning and Accelerations 255
6.5 Further Topics 259
6.5.1 Reversible Jumps 259
6.5.2 Langevin Algorithms 264
6.6 Problems 268
6.7 Notes 279
7 The Gibbs Sampler 285
7.1 General Principles 285
7.1.1 Definition 285
7.1.2 Completion 288
7.1.3 Convergence Properties 293
7.1.4 Gibbs Sampling and Metropolis Hastings 296
7.1.5 The Hammersley Clifford Theorem 297
7.1.6 Hierarchical Structures 299
7.2 The Two Stage Gibbs Sampler 303
7.2.1 Dual Probability Structures 303
7.2.2 Reversible and Interleaving Chains 308
7.2.3 Monotone Covariance and Rao Blackwellization 310
7.2.4 The Duality Principle 315
7.3 Hybrid Gibbs Samplers 317
7.3.1 Comparison with Metropolis Hastings Algorithms 317
7.3.2 Mixtures and Cycles 319
7.3.3 Metropolizing the Gibbs Sampler 322
7.3.4 Reparameterization 326
7.4 Improper Priors 328
7.5 Problems 332
7.6 Notes 348
8 Diagnosing Convergence 363
8.1 Stopping the Chain 363
8.1.1 Convergence Criteria 363
8.1.2 Multiple Chains 366
xvi CONTENTS
8.1.3 Conclusions 367
8.2 Monitoring Convergence to the Stationary Distribution 368
8.2.1 Graphical Methods 368
8.2.2 Nonparametric Tests of Stationarity 370
8.2.3 Renewal Methods 371
8.2.4 Distance Evaluations 375
8.3 Monitoring Convergence of Averages 380
8.3.1 Graphical Methods 380
8.3.2 Multiple Estimates 381
8.3.3 Renewal Theory 388
8.3.4 Within and Between Variances 394
8.4 Simultaneous Monitoring 397
8.4.1 Binary Control 397
8.4.2 Valid Discretization 401
8.5 Problems 402
8.6 Notes 407
9 Implementation in Missing Data Models 415
9.1 Introduction 415
9.2 First Examples 416
9.2.1 Discrete Data Models 416
9.2.2 Data Missing at Random 419
9.3 Finite Mixtures of Distributions 422
9.4 A Reparameterization of Mixtures 425
9.5 Extensions 429
9.5.1 Hidden Markov Chains 430
9.5.2 Changepoint models 432
9.5.3 Stochastic Volatility 435
9.6 Problems 438
9.7 Notes 448
A Probability Distributions 451
B Notation 455
B.I Mathematical 455
B.2 Probability 455
B.3 Distributions 456
B.4 Markov Chains 457
B.5 Statistics 457
B.6 Algorithms 458
C References 459
Author Index 488
Subject Index 495
|
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| indexdate | 2024-07-09T18:50:36Z |
| institution | BVB |
| isbn | 038798707X |
| language | English |
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| physical | XXI, 507 S. graph. Darst. |
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| spelling | Robert, Christian P. 1961- Verfasser (DE-588)115436448 aut Monte Carlo statistical methods Christian P. Robert ; George Casella Corr. 2. print. New York, NY [u.a.] Springer 2000 XXI, 507 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer texts in statistics Monte-Carlo-Simulation (DE-588)4240945-7 gnd rswk-swf Markov-Kette (DE-588)4037612-6 gnd rswk-swf Monte-Carlo-Simulation (DE-588)4240945-7 s Markov-Kette (DE-588)4037612-6 s 1\p DE-604 DE-604 Casella, George Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009368040&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Robert, Christian P. 1961- Casella, George Monte Carlo statistical methods Monte-Carlo-Simulation (DE-588)4240945-7 gnd Markov-Kette (DE-588)4037612-6 gnd |
| subject_GND | (DE-588)4240945-7 (DE-588)4037612-6 |
| title | Monte Carlo statistical methods |
| title_auth | Monte Carlo statistical methods |
| title_exact_search | Monte Carlo statistical methods |
| title_full | Monte Carlo statistical methods Christian P. Robert ; George Casella |
| title_fullStr | Monte Carlo statistical methods Christian P. Robert ; George Casella |
| title_full_unstemmed | Monte Carlo statistical methods Christian P. Robert ; George Casella |
| title_short | Monte Carlo statistical methods |
| title_sort | monte carlo statistical methods |
| topic | Monte-Carlo-Simulation (DE-588)4240945-7 gnd Markov-Kette (DE-588)4037612-6 gnd |
| topic_facet | Monte-Carlo-Simulation Markov-Kette |
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