Monte Carlo statistical methods:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2004
|
| Edition: | 2. ed. |
| Series: | Springer texts in statistics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XXX, 645 S. graph. Darst. |
| ISBN: | 0387212396 9780387212395 |
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MARC
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| 100 | 1 | |a Robert, Christian P. |d 1961- |e Verfasser |0 (DE-588)115436448 |4 aut | |
| 245 | 1 | 0 | |a Monte Carlo statistical methods |c Christian P. Robert ; George Casella |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York, NY |b Springer |c 2004 | |
| 300 | |a XXX, 645 S. |b graph. Darst. | ||
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| 650 | 4 | |a Monte-Carlo-Methode / Theorie | |
| 650 | 4 | |a Mathematical statistics | |
| 650 | 4 | |a Monte Carlo method | |
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| 655 | 4 | |a Lehrbuch - Monte-Carlo-Simulation - Markov-Ketten-Monte-Carlo-Verfahren | |
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| adam_text | Contents
Preface
to the Second Edition
.................................
IX
Preface to the First Edition
..................................XIII
1
Introduction
............................................... 1
1.1
Statistical Models
....................................... 1
1.2
Likelihood Methods
...................................... 5
1.3
Bayesian Methods
....................................... 12
1.4
Deterministic Numerical Methods
......................... 19
1.4.1
Optimization
..................................... 19
1.4.2
Integration
....................................... 21
1.4.3
Comparison
...................................... 21
1.5
Problems
............................................... 23
1.6
Notes
.................................................. 30
1.6.1
Prior Distributions
................................ 30
1.6.2
Bootstrap Methods
................................ 32
2
Random Variable Generation
.............................. 35
2.1
Introduction
............................................ 35
2.1.1
Uniform Simulation
................................ 36
2.1.2
The Inverse Transform
............................. 38
2.1.3
Alternatives
...................................... 40
2.1.4
Optimal Algorithms
............................... 41
2.2
General Transformation Methods
.......................... 42
2.3
Accept-Reject Methods
.................................. 47
2.3.1
The Fundamental Theorem of Simulation
............. 47
2.3.2
The Accept-Reject Algorithm
....................... 51
2.4
Envelope Accept-Reject Methods
.......................... 53
2.4.1
The Squeeze Principle
............................. 53
2.4.2
Log-Concave Densities
............................. 56
2.5
Problems
............................................... 62
XVIII
Contents
2.6
Notes
.................................................. 72
2.6.1
The Kiss Generator
................................ 72
2.6.2
Quasi-Monte
Carlo Methods
........................ 75
2.6.3
Mixture Representations
........................... 77
3
Monte Carlo Integration
................................... 79
3.1
Introduction
............................................ 79
3.2
Classical Monte Carlo Integration
......................... 83
3.3
Importance Sampling
.................................... 90
3.3.1
Principles
........................................ 90
3.3.2
Finite Variance Estimators
......................... 94
3.3.3
Comparing Importance Sampling with Accept-Reject
.. 103
3.4
Laplace Approximations
..................................107
3.5
Problems
...............................................110
3.6
Notes
..................................................119
3.6.1
Large Deviations Techniques
........................119
3.6.2
The Saddlepoint Approximation
.....................120
4
Controling Monte Carlo Variance
..........................123
4.1
Monitoring Variation with the CLT
........................123
4.1.1
Univariate Monitoring
.............................124
4.1.2
Multivariate Monitoring
............................128
4.2
Rao-Blackwellization
....................................130
4.3
Riemann Approximations
.................................134
4.4
Acceleration Methods
....................................140
4.4.1
Antithetic Variables
...............................140
4.4.2
Control
Variâtes
..................................145
4.5
Problems
...............................................147
4.6
Notes
..................................................153
4.6.1
Monitoring Importance Sampling Convergence
........153
4.6.2
Accept-Reject with Loose Bounds
...................154
4.6.3
Partitioning
......................................155
5
Monte Carlo Optimization
.................................157
5.1
Introduction
............................................157
5.2
Stochastic Exploration
...................................159
5.2.1
A Basic Solution
..................................159
5.2.2
Gradient Methods
.................................162
5.2.3
Simulated Annealing
...............................163
5.2.4
Prior Feedback
....................................169
5.3
Stochastic Approximation
................................174
5.3.1
Missing Data Models and Demarginalization
..........174
5.3.2
The EM Algorithm
................................176
5.3.3
Monte Carlo EM
..................................183
5.3.4
EM Standard Errors
...............................186
Contents XIX
5.4 Problems...............................................188
5.5 Notes..................................................200
5.5.1
Variations
on EM
.................................200
5.5.2
Neural
Networks..................................201
5.5.3 The Robbins-Monro
procedure
......................201
5.5.4 Monte Carlo Approximation........................203
Markov Chains............................................
205
6.1
Essentials for
MCMC
....................................206
6.2
Basic Notions
...........................................208
6.3
Irreducibility, Atoms, and Small Sets
......................213
6.3.1
Irreducibility
.....................................213
6.3.2
Atoms and Small Sets
..............................214
6.3.3
Cycles and Aperiodicity
............................217
6.4
Transience and Recurrence
...............................218
6.4.1
Classification of Irreducible Chains
..................218
6.4.2
Criteria for Recurrence
.............................221
6.4.3
Harris Recurrence
.................................221
6.5
Invariant Measures
......................................223
6.5.1
Stationary Chains
.................................223
6.5.2
Kac s Theorem
....................................224
6.5.3
Reversibility and the Detailed Balance Condition
......229
6.6
Ergodicity and Convergence
..............................231
6.6.1
Ergodicity
........................................231
6.6.2
Geometric Convergence
............................236
6.6.3
Uniform Ergodicity
................................237
6.7
Limit Theorems
.........................................238
6.7.1
Ergodic Theorems
.................................240
6.7.2
Central Limit Theorems
............................242
6.8
Problems
...............................................247
6.9
Notes
..................................................258
6.9.1
Drift Conditions
...................................258
6.9.2
Eaton s Admissibility Condition
.....................262
6.9.3
Alternative Convergence Conditions
.................263
6.9.4
Mixing Conditions and Central Limit Theorems
.......263
6.9.5
Covariance in Markov Chains
.......................265
The Metropolis-Hastings Algorithm
.......................267
7.1
The MCMC Principle
....................................267
7.2
Monte Carlo Methods Based on Markov Chains
.............269
7.3
The Metropolis-Hastings algorithm
........................270
7.3.1
Definition
........................................270
7.3.2
Convergence Properties
............................272
7.4
The Independent Metropolis-Hastings Algorithm
............276
7.4.1
Fixed Proposals
...................................276
XX
Contents
7.4.2
A Metropolis-Hastings Version of
ARS...............285
7.5
Random Walks
..........................................287
7.6
Optimization and Control
................................292
7.6.1
Optimizing the Acceptance Rate
....................292
7.6.2
Conditioning and Accelerations
.....................295
7.6.3
Adaptive Schemes
.................................299
7.7
Problems
...............................................302
7.8
Notes
..................................................313
7.8.1
Background of the Metropolis Algorithm
.............313
7.8.2
Geometric Convergence of Metropolis-Hastings
Algorithms
.......................................315
7.8.3
A Reinterpretation of Simulated Annealing
...........315
7.8.4
Reference Acceptance Rates
........................316
7.8.5
Langevin
Algorithms
...............................318
8
The Slice Sampler
.........................................321
8.1
Another Look at the Fundamental Theorem
................321
8.2
The General Slice Sampler
................................326
8.3
Convergence Properties of the Slice Sampler
................329
8.4
Problems
...............................................333
8.5
Notes
..................................................335
8.5.1
Dealing with Difficult Slices
........................335
9
The Two-Stage Gibbs Sampler
.............................337
9.1
A General Class of Two-Stage Algorithms
..................337
9.1.1
From Slice Sampling to Gibbs Sampling
..............337
9.1.2
Definition
........................................339
9.1.3
Back to the Slice Sampler
..........................343
9.1.4
The Hammersley-Clifford Theorem
..................343
9.2
Fundamental Properties
..................................344
9.2.1
Probabilistic Structures
............................344
9.2.2
Reversible and Interleaving Chains
..................349
9.2.3
The Duality Principle
..............................351
9.3
Monotone Covariance and Rao-Blackwellization
.............354
9.4
The EM-Gibbs Connection
...............................357
9.5
Transition
..............................................360
9.6
Problems
...............................................360
9.7
Notes
..................................................366
9.7.1
Inference for Mixtures
.............................366
9.7.2
ARCH Models
....................................368
10
The Multi-Stage Gibbs Sampler
...........................371
10.1
Basic Derivations
........................................371
10.1.1
Definition
........................................371
10.1.2
Completion
.......................................373
Contents XXI
10.1.3
The General Hammersley-Clifford Theorem
..........376
10.2
Theoretical Justifications
.................................378
10.2.1
Markov Properties of the Gibbs Sampler
.............378
10.2.2
Gibbs Sampling as Metropolis-Hastings
..............381
10.2.3
Hierarchical Structures
.............................383
10.3
Hybrid Gibbs Samplers
...................................387
10.3.1
Comparison with Metropolis-Hastings Algorithms
.....387
10.3.2
Mixtures and Cycles
...............................388
10.3.3
Metropolizing the Gibbs Sampler
....................392
10.4
Statistical Considerations
.................................396
10.4.1
Reparameterization
................................396
10.4.2
Rao-Blackwellization
...............................402
10.4.3
Improper Priors
...................................403
10.5
Problems
...............................................407
10.6
Notes
..................................................419
10.6.1
A Bit of Background
...............................419
10.6.2
The BUGS Software
................................420
10.6.3
Nonparametric Mixtures
...........................420
10.6.4
Graphical Models
.................................422
11
Variable Dimension Models and Reversible Jump
Algorithms
................................................425
11.1
Variable Dimension Models
...............................425
11.1.1
Bayesian Model Choice
.............................426
11.1.2
Difficulties in Model Choice
.........................427
11.2
Reversible Jump Algorithms
..............................429
11.2.1
Green s Algorithm
.................................429
11.2.2
A Fixed Dimension Reassessment
...................432
11.2.3
The Practice of Reversible Jump MCMC
.............433
11.3
Alternatives to Reversible Jump MCMC
....................444
11.3.1
Saturation
........................................444
11.3.2
Continuous-Time Jump Processes
...................446
11.4
Problems
...............................................449
11.5
Notes
..................................................458
11.5.1
Occam s Razor
....................................458
12
Diagnosing Convergence
...................................459
12.1
Stopping the Chain
......................................459
12.1.1
Convergence Criteria
..............................461
12.1.2
Multiple Chains
...................................464
12.1.3
Monitoring Reconsidered
...........................465
12.2
Monitoring Convergence to the Stationary Distribution
.......465
12.2.1
A First Illustration
................................465
12.2.2
Nonparametric Tests of Stationarity
.................466
12.2.3
Renewal Methods
.................................470
XXII Contents
12.2.4
Missing Mass
.....................................474
12.2.5
Distance Evaluations
..............................478
12.3
Monitoring Convergence of Averages
.......................480
12.3.1
A First Illustration
................................480
12.3.2
Multiple Estimates
................................483
12.3.3
Renewal Theory
...................................490
12.3.4
Within and Between Variances
......................497
12.3.5
Effective Sample Size
..............................499
12.4
Simultaneous Monitoring
.................................500
12.4.1
Binary Control
....................................500
12.4.2
Valid Discretization
................................503
12.5
Problems
...............................................504
12.6
Notes
..................................................508
12.6.1
Spectral Analysis
..................................508
12.6.2
The CODA Software
................................509
13
Perfect Sampling
..........................................511
13.1
Introduction
............................................511
13.2
Coupling from the Past
..................................513
13.2.1
Random Mappings and Coupling
....................513
13.2.2
Propp
and Wilson s Algorithm
......................516
13.2.3
Monotonicity
and Envelopes
........................518
13.2.4
Continuous States Spaces
...........................523
13.2.5
Perfect Slice Sampling
.............................526
13.2.6
Perfect Sampling via Automatic Coupling
............530
13.3
Forward Coupling
.......................................532
13.4
Perfect Sampling in Practice
..............................535
13.5
Problems
...............................................536
13.6
Notes
..................................................539
13.6.1
History
..........................................539
13.6.2
Perfect Sampling and Tempering
....................540
14
Iterated and Sequential Importance Sampling
.............545
14.1
Introduction
............................................545
14.2
Generalized Importance Sampling
.........................546
14.3
Particle Systems
........................................547
14.3.1
Sequential Monte Carlo
............................547
14.3.2
Hidden Markov Models
............................549
14.3.3
Weight Degeneracy
................................551
14.3.4
Particle Filters
....................................552
14.3.5
Sampling Strategies
................................554
14.3.6
Fighting the Degeneracy
...........................556
14.3.7
Convergence of Particle Systems
.....................558
14.4
Population Monte Carlo
..................................559
14.4.1
Sample Simulation
.................................560
Contents XXIII
14.4.2 General Iterative
Importance
Sampling
...............560
14.4.3 Population Monte Carlo............................562
14.4.4 An Illustration
for the Mixture
Model................563
14.4.5 Adaptativity in
Sequential Algorithms...............
565
14.5 Problems...............................................570
14.6 Notes..................................................577
14.6.1
A
Brief
History of Particle
Systems..................577
14.6.2 Dynamic
Importance Sampling
......................577
14.6.3
Hidden Markov Models
............................579
A Probability Distributions
..................................581
В
Notation
...................................................585
B.I Mathematical
...........................................585
B.2 Probability
.............................................586
B.3 Distributions
............................................586
B.4 Markov Chains
..........................................587
B.5 Statistics
...............................................588
B.6 Algorithms
.............................................588
References
.....................................................591
Index of Names
................................................623
Index of Subjects
..............................................631
Monte Carlo
statistical methods, particularly those based on Markov chains, are now
an essential component of the standard set of techniques used by statisticians. This new
edition has been revised towards a coherent and flowing coverage of these simulation
techniques, with incorporation of the most recent developments in the field. In particu¬
lar, the introductory coverage of random variable generation has been totally revised,
with many concepts being unified through a fundamental theorem of simulation.
There are five completely new chapters that cover Monte Carlo control, reversible
jump, slice sampling, sequential Monte Carlo, and perfect sampling. There is a more
¡η
-depth coverage of Gibbs sampling, which is now contained in three consecutive
chapters. The development of Gibbs sampling starts with slice sampling and its con¬
nection with the fundamental theorem of simulation, and builds up to two-stage Gibbs
sampling and its theoretical properties. A third chapter covers the multi-stage Gibbs
sampler and its variety of applications. Lastly, chapters from the previous edition have
been revised towards easier access, with the examples getting more detailed cover¬
age.
This textbook is intended for a second year graduate course, but will also be useful to
someone who either wants to apply simulation techniques for the resolution of practi¬
cal problems or wishes to grasp the fundamental principles behind those methods. The
authors do not assume familiarity with Monte Carlo techniques (such as random vari¬
able generation), with computer programming, or with any Markov chain theory (the
necessary concepts are developed in Chapter
ó). A
solutions manual, which covers
approximately
40%
of the problems, is available for instructors who require the book
for a course.
Christian P. Robert is Professor of Statistics in the Applied Mathematics Department at
Université
Paris
Dauphine,
France. He is also Head of the Statistics Laboratory at the
Center for Research in Economics and Statistics (CREST) of the National Institute for
Statistics and Economic Studies (INSEE) in Paris, and Adjunct Professor at
Ecole
Polytechnique.
He has written three other books and won the
2004
DeGroot Prize for
The Bayesian Choice, Second Edition, Springer
2001.
He also edited Discretization
and MCMC Convergence Assessment, Springer
1998.
He has served as associate edi¬
tor for the Annals of Statistics, Statistical Science and the Journal of the American
Statistical Association. He is a fellow of the Institute of Mathematical Statistics, and a
winner of the Young Statistician Award of the
Société de Statistique de Paris in
1995.
George
Casella
is Distinguished Professor and Chair,
Department
of Statistics,
University of Florida. He has served as the Theory and Methods Editor of the Journal
of the American Statistical Association and Executive Editor of Statistical Science. He
has authored three other textbooks: Statistical Inference, Second Edition,
2001,
with
Roger L.
Berger;
Theory of Point Estimation,
1998,
with
Erich Lehmann;
and Variance
Components,
1992,
with Shayle R. Searle and Charles E. McCulloch. He is a fellow
of the Institute of Mathematical Statistics and the American Statistical Association, and
an elected fellow of the International Statistical Institute.
|
| any_adam_object | 1 |
| author | Robert, Christian P. 1961- Casella, George 1951-2012 |
| author_GND | (DE-588)115436448 (DE-588)170529525 |
| author_facet | Robert, Christian P. 1961- Casella, George 1951-2012 |
| author_role | aut aut |
| author_sort | Robert, Christian P. 1961- |
| author_variant | c p r cp cpr g c gc |
| building | Verbundindex |
| bvnumber | BV019407317 |
| callnumber-first | Q - Science |
| callnumber-label | QA276 |
| callnumber-raw | QA276 |
| callnumber-search | QA276 |
| callnumber-sort | QA 3276 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | QH 239 SK 820 SK 830 |
| classification_tum | MAT 629f |
| ctrlnum | (OCoLC)249801921 (DE-599)BVBBV019407317 |
| dewey-full | 519.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.5 |
| dewey-search | 519.5 |
| dewey-sort | 3519.5 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 2. ed. |
| format | Book |
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| genre | Lehrbuch - Monte-Carlo-Simulation - Markov-Ketten-Monte-Carlo-Verfahren |
| genre_facet | Lehrbuch - Monte-Carlo-Simulation - Markov-Ketten-Monte-Carlo-Verfahren |
| id | DE-604.BV019407317 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:59:36Z |
| institution | BVB |
| isbn | 0387212396 9780387212395 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-012869394 |
| oclc_num | 249801921 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-473 DE-BY-UBG DE-1049 DE-20 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-706 DE-945 DE-739 DE-188 DE-578 DE-824 DE-91 DE-BY-TUM DE-384 DE-573 |
| owner_facet | DE-91G DE-BY-TUM DE-473 DE-BY-UBG DE-1049 DE-20 DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-706 DE-945 DE-739 DE-188 DE-578 DE-824 DE-91 DE-BY-TUM DE-384 DE-573 |
| physical | XXX, 645 S. graph. Darst. |
| publishDate | 2004 |
| publishDateSearch | 2004 |
| publishDateSort | 2004 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer texts in statistics |
| spelling | Robert, Christian P. 1961- Verfasser (DE-588)115436448 aut Monte Carlo statistical methods Christian P. Robert ; George Casella 2. ed. New York, NY Springer 2004 XXX, 645 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer texts in statistics Lehrbuch / Textbook - 28 Metodo Monte Carlo sbt Monte-Carlo-Methode / Theorie Mathematical statistics Monte Carlo method Markov-Kette (DE-588)4037612-6 gnd rswk-swf Monte-Carlo-Simulation (DE-588)4240945-7 gnd rswk-swf Lehrbuch - Monte-Carlo-Simulation - Markov-Ketten-Monte-Carlo-Verfahren Monte-Carlo-Simulation (DE-588)4240945-7 s Markov-Kette (DE-588)4037612-6 s DE-604 Casella, George 1951-2012 Verfasser (DE-588)170529525 aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012869394&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012869394&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Robert, Christian P. 1961- Casella, George 1951-2012 Monte Carlo statistical methods Lehrbuch / Textbook - 28 Metodo Monte Carlo sbt Monte-Carlo-Methode / Theorie Mathematical statistics Monte Carlo method Markov-Kette (DE-588)4037612-6 gnd Monte-Carlo-Simulation (DE-588)4240945-7 gnd |
| subject_GND | (DE-588)4037612-6 (DE-588)4240945-7 |
| 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 | Lehrbuch / Textbook - 28 Metodo Monte Carlo sbt Monte-Carlo-Methode / Theorie Mathematical statistics Monte Carlo method Markov-Kette (DE-588)4037612-6 gnd Monte-Carlo-Simulation (DE-588)4240945-7 gnd |
| topic_facet | Lehrbuch / Textbook - 28 Metodo Monte Carlo Monte-Carlo-Methode / Theorie Mathematical statistics Monte Carlo method Markov-Kette Monte-Carlo-Simulation Lehrbuch - Monte-Carlo-Simulation - Markov-Ketten-Monte-Carlo-Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012869394&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012869394&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT robertchristianp montecarlostatisticalmethods AT casellageorge montecarlostatisticalmethods |