Holdings: Monte Carlo statistical methods

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Bibliographic Details
Title:	Monte Carlo statistical methods
From:	Christian P. Robert ; George Casella
Person:	Robert, Christian P. 1961- Verfasser aut Casella, George 1951-2012
Main Authors:	Robert, Christian P. 1961- (Author), Casella, George 1951-2012 (Author)
Format:	Book
Language:	English
Published:	New York, NY Springer 2004
Edition:	2. ed.
Series:	Springer texts in statistics
Notation:	QH 239 SK 820 SK 830
Subjects:	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
Online Access:	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
Physical Description:	XXX, 645 S. graph. Darst.
ISBN:	0387212396 9780387212395

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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.
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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	2024-12-20T12:00:03Z
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
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physical	XXX, 645 S. graph. Darst.
publishDate	2004
publishDateSearch	2004
publishDateSort	2004
publisher	Springer
record_format	marc
series2	Springer texts in statistics
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

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