Monte Carlo Statistical Methods

by Robert, Christian P.; Casella, George

• ISBN13: 9780387212395
• ISBN10: 0387212396
• Edition: 2nd
• Format: Hardcover
• Copyright: 2004-10-30
• Publisher: Springer Nature

Summary

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 particular, 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 in-depth coverage of Gibbs sampling, which is now contained in three consecutive chapters. The development of Gibbs sampling starts with slice sampling and its connection 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 coverage. 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 practical problems or wishes to grasp the fundamental principles behind those methods. The authors do not assume familiarity with Monte Carlo techniques (such as random variable generation), with computer programming, or with any Markov chain theory (the necessary concepts are developed in Chapter 6). 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 Universit?? 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, including The Bayesian Choice, Second Edition, Springer 2001. He also edited Discretization and MCMC Convergence Assessment, Springer 1998. He has served as associate editor for the Annals of Statistics 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 Societi?? 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.

Author Biography

Christian P. Robert is Professor of Statistics in the Applied Mathematics Department at Universite Paris Dauphine France.

Table of Contents

Preface to the Second Edition	ix
Preface to the First Edition	xiii
1 Introduction	1	(34)
1.1 Statistical Models	1	(4)
1.2 Likelihood Methods	5	(7)
1.3 Bayesian Methods	12	(7)
1.4 Deterministic Numerical Methods	19	(4)
1.4.1 Optimization	19	(2)
1.4.2 Integration	21	(1)
1.4.3 Comparison	21	(2)
1.5 Problems	23	(7)
1.6 Notes	30	(5)
1.6.1 Prior Distributions	30	(2)
1.6.2 Bootstrap Methods	32	(3)
2 Random Variable Generation	35	(44)
2.1 Introduction	35	(7)
2.1.1 Uniform Simulation	36	(2)
2.1.2 The Inverse Transform	38	(2)
2.1.3 Alternatives	40	(1)
2.1.4 Optimal Algorithms	41	(1)
2.2 General Transformation Methods	42	(5)
2.3 Accept-Reject Methods	47	(6)
2.3.1 The Fundamental Theorem of Simulation	47	(4)
2.3.2 The Accept-Reject Algorithm	51	(2)
2.4 Envelope Accept-Reject Methods	53	(9)
2.4.1 The Squeeze Principle	53	(3)
2.4.2 Log-Concave Densities	56	(6)
2.5 Problems	62	(10)
2.6 Notes	72	(7)
2.6.1 The Kiss Generator	72	(3)
2.6.2 Quasi-Monte Carlo Methods	75	(2)
2.6.3 Mixture Representations	77	(2)
3 Monte Carlo Integration	79	(44)
3.1 Introduction	79	(4)
3.2 Classical Monte Carlo Integration	83	(7)
3.3 Importance Sampling	90	(17)
3.3.1 Principles	90	(4)
3.3.2 Finite Variance Estimators	94	(9)
3.3.3 Comparing Importance Sampling with Accept-Reject	103	(4)
3.4 Laplace Approximations	107	(3)
3.5 Problems	110	(9)
3.6 Notes	119	(4)
3.6.1 Large Deviations Techniques	119	(1)
3.6.2 The Saddlepoint Approximation	120	(3)
4 Controling Monte Carlo Variance	123	(34)
4.1 Monitoring Variation with the CLT	123	(7)
4.1.1 Univariate Monitoring	124	(4)
4.1.2 Multivariate Monitoring	128	(2)
4.2 Rao-Blackwellization	130	(4)
4.3 Riemann Approximations	134	(6)
4.4 Acceleration Methods	140	(7)
4.4.1 Antithetic Variables	140	(5)
4.4.2 Control Variates	145	(2)
4.5 Problems	147	(6)
4.6 Notes	153	(4)
4.6.1 Monitoring Importance Sampling Convergence	153	(1)
4.6.2 Accept-Reject with Loose Bounds	154	(1)
4.6.3 Partitioning	155	(2)
5 Monte Carlo Optimization	157	(48)
5.1 Introduction	157	(2)
5.2 Stochastic Exploration	159	(15)
5.2.1 A Basic Solution	159	(3)
5.2.2 Gradient Methods	162	(1)
5.2.3 Simulated Annealing	163	(6)
5.2.4 Prior Feedback	169	(5)
5.3 Stochastic Approximation	174	(14)
5.3.1 Missing Data Models and Demarginalization	174	(2)
5.3.2 The EM Algorithm	176	(7)
5.3.3 Monte Carlo EM	183	(3)
5.3.4 EM Standard Errors	186	(2)
5.4 Problems	188	(12)
5.5 Notes	200	(5)
5.5.1 Variations on EM	200	(1)
5.5.2 Neural Networks	201	(1)
5.5.3 The Robbins-Monro procedure	201	(2)
5.5.4 Monte Carlo Approximation	203	(2)
6 Markov Chains	205	(62)
6.1 Essentials for MCMC	206	(2)
6.2 Basic Notions	208	(5)
6.3 Irreducibility, Atoms, and Small Sets	213	(5)
6.3.1 Irreducibility	213	(1)
6.3.2 Atoms and Small Sets	214	(3)
6.3.3 Cycles and Aperiodicity	217	(1)
6.4 Transience and Recurrence	218	(5)
6.4.1 Classification of Irreducible Chains	218	(3)
6.4.2 Criteria for Recurrence	221	(1)
6.4.3 Harris Recurrence	221	(2)
6.5 Invariant Measures	223	(8)
6.5.1 Stationary Chains	223	(1)
6.5.2 Kac's Theorem	224	(5)
6.5.3 Reversibility and the Detailed Balance Condition	229	(2)
6.6 Ergodicity and Convergence	231	(7)
6.6.1 Ergodicity	231	(5)
6.6.2 Geometric Convergence	236	(1)
6.6.3 Uniform Ergodicity	237	(1)
6.7 Limit Theorems	238	(9)
6.7.1 Ergodic Theorems	240	(2)
6.7.2 Central Limit Theorems	242	(5)
6.8 Problems	247	(11)
6.9 Notes	258	(9)
6.9.1 Drift Conditions	258	(4)
6.9.2 Eaton's Admissibility Condition	262	(1)
6.9.3 Alternative Convergence Conditions	263	(1)
6.9.4 Mixing Conditions and Central Limit Theorems	263	(2)
6.9.5 Covariance in Markov Chains	265	(2)
7 The Metropolis-Hastings Algorithm	267	(54)
7.1 The MCMC Principle	267	(2)
7.2 Monte Carlo Methods Based on Markov Chains	269	(1)
7.3 The Metropolis-Hastings algorithm	270	(6)
7.3.1 Definition	270	(2)
7.3.2 Convergence Properties	272	(4)
7.4 The Independent Metropolis-Hastings Algorithm	276	(11)
7.4.1 Fixed Proposals	276	(9)
7.4.2 A Metropolis-Hastings Version of ARS	285	(2)
7.5 Random Walks	287	(5)
7.6 Optimization and Control	292	(10)
7.6.1 Optimizing the Acceptance Rate	292	(3)
7.6.2 Conditioning and Accelerations	295	(4)
7.6.3 Adaptive Schemes	299	(3)
7.7 Problems	302	(11)
7.8 Notes	313	(8)
7.8.1 Background of the Metropolis Algorithm	313	(2)
7.8.2 Geometric Convergence of Metropolis-Hastings Algorithms	315	(1)
7.8.3 A Reinterpretation of Simulated Annealing	315	(1)
7.8.4 Reference Acceptance Rates	316	(2)
7.8.5 Langevin Algorithms	318	(3)
8 The Slice Sampler	321	(16)
8.1 Another Look at the Fundamental Theorem	321	(5)
8.2 The General Slice Sampler	326	(3)
8.3 Convergence Properties of the Slice Sampler	329	(4)
8.4 Problems	333	(2)
8.5 Notes	335	(2)
8.5.1 Dealing with Difficult Slices	335	(2)
9 The Two-Stage Gibbs Sampler	337	(34)
9.1 A General Class of Two-Stage Algorithms	337	(7)
9.1.1 From Slice Sampling to Gibbs Sampling	337	(2)
9.1.2 Definition	339	(4)
9.1.3 Back to the Slice Sampler	343	(1)
9.1.4 The Hammersley-Clifford Theorem	343	(1)
9.2 Fundamental Properties	344	(10)
9.2.1 Probabilistic Structures	344	(5)
9.2.2 Reversible and Interleaving Chains	349	(2)
9.2.3 The Duality Principle	351	(3)
9.3 Monotone Covariance and Rao-Blackwellization	354	(3)
9.4 The EM-Gibbs Connection	357	(3)
9.5 Transition	360	(1)
9.6 Problems	360	(6)
9.7 Notes	366	(5)
9.7.1 Inference for Mixtures	366	(2)
9.7.2 ARCH Models	368	(3)
10 The Multi-Stage Gibbs Sampler	371	(54)
10.1 Basic Derivations	371	(7)
10.1.1 Definition	371	(2)
10.1.2 Completion	373	(3)
10.1.3 The General Hammersley-Clifford Theorem	376	(2)
10.2 Theoretical Justifications	378	(9)
10.2.1 Markov Properties of the Gibbs Sampler	378	(3)
10.2.2 Gibbs Sampling as Metropolis-Hastings	381	(2)
10.2.3 Hierarchical Structures	383	(4)
10.3 Hybrid Gibbs Samplers	387	(9)
10.3.1 Comparison with Metropolis-Hastings Algorithms	387	(1)
10.3.2 Mixtures and Cycles	388	(4)
10.3.3 Metropolizing the Gibbs Sampler	392	(4)
10.4 Statistical Considerations	396	(11)
10.4.1 Reparameterization	396	(6)
10.4.2 Rao-Blackwellization	402	(1)
10.4.3 Improper Priors	403	(4)
10.5 Problems	407	(12)
10.6 Notes	419	(6)
10.6.1 A Bit of Background	419	(1)
10.6.2 The BUGS Software	420	(1)
10.6.3 Nonparametric Mixtures	420	(2)
10.6.4 Graphical Models	422	(3)
11 Variable Dimension Models and Reversible Jump Algorithms	425	(34)
11.1 Variable Dimension Models	425	(4)
11.1.1 Bayesian Model Choice	426	(1)
11.1.2 Difficulties in Model Choice	427	(2)
11.2 Reversible Jump Algorithms	429	(15)
11.2.1 Green's Algorithm	429	(3)
11.2.2 A Fixed Dimension Reassessment	432	(1)
11.2.3 The Practice of Reversible Jump MCMC	433	(11)
11.3 Alternatives to Reversible Jump MCMC	444	(5)
11.3.1 Saturation	444	(2)
11.3.2 Continuous-Time Jump Processes	446	(3)
11.4 Problems	449	(9)
11.5 Notes	458	(1)
11.5.1 Occam's Razor	458	(1)
12 Diagnosing Convergence	459	(52)
12.1 Stopping the Chain	459	(6)
12.1.1 Convergence Criteria	461	(3)
12.1.2 Multiple Chains	464	(1)
12.1.3 Monitoring Reconsidered	465	(1)
12.2 Monitoring Convergence to the Stationary Distribution	465	(15)
12.2.1 A First Illustration	465	(1)
12.2.2 Nonparametric Tests of Stationarity	466	(4)
12.2.3 Renewal Methods	470	(4)
12.2.4 Missing Mass	474	(4)
12.2.5 Distance Evaluations	478	(2)
12.3 Monitoring Convergence of Averages	480	(20)
12.3.1 A First Illustration	480	(3)
12.3.2 Multiple Estimates	483	(7)
12.3.3 Renewal Theory	490	(7)
12.3.4 Within and Between Variances	497	(2)
12.3.5 Effective Sample Size	499	(1)
12.4 Simultaneous Monitoring	500	(4)
12.4.1 Binary Control	500	(3)
12.4.2 Valid Discretization	503	(1)
12.5 Problems	504	(4)
12.6 Notes	508	(3)
12.6.1 Spectral Analysis	508	(1)
12.6.2 The CODA Software	509	(2)
13 Perfect Sampling	511	(34)
13.1 Introduction	511	(2)
13.2 Coupling from the Past	513	(19)
13.2.1 Random Mappings and Coupling	513	(3)
13.2.2 Propp and Wilson's Algorithm	516	(2)
13.2.3 Monotonicity and Envelopes	518	(5)
13.2.4 Continuous States Spaces	523	(3)
13.2.5 Perfect Slice Sampling	526	(4)
13.2.6 Perfect Sampling via Automatic Coupling	530	(2)
13.3 Forward Coupling	532	(3)
13.4 Perfect Sampling in Practice	535	(1)
13.5 Problems	536	(3)
13.6 Notes	539	(6)
13.6.1 History	539	(1)
13.6.2 Perfect Sampling and Tempering	540	(5)
14 Iterated and Sequential Importance Sampling	545	(36)
14.1 Introduction	545	(1)
14.2 Generalized Importance Sampling	546	(1)
14.3 Particle Systems	547	(12)
14.3.1 Sequential Monte Carlo	547	(2)
14.3.2 Hidden Markov Models	549	(2)
14.3.3 Weight Degeneracy	551	(1)
14.3.4 Particle Filters	552	(2)
14.3.5 Sampling Strategies	554	(2)
14.3.6 Fighting the Degeneracy	556	(2)
14.3.7 Convergence of Particle Systems	558	(1)
14.4 Population Monte Carlo	559	(11)
14.4.1 Sample Simulation	560	(1)
14.4.2 General Iterative Importance Sampling	560	(2)
14.4.3 Population Monte Carlo	562	(1)
14.4.4 An Illustration for the Mixture Model	563	(2)
14.4.5 Adaptativity in Sequential Algorithms	565	(5)
14.5 Problems	570	(7)
14.6 Notes	577	(8)
14.6.1 A Brief History of Particle Systems	577	(1)
14.6.2 Dynamic Importance Sampling	577	(2)
14.6.3 Hidden Markov Models	579	(2)
A Probability Distributions	581	(4)
B Notation	585	(6)
B.1 Mathematical	585	(1)
B.2 Probability	586	(1)
B.3 Distributions	586	(1)
B.4 Markov Chains	587	(1)
B.5 Statistics	588	(1)
B.6 Algorithms	588	(3)
References	591	(32)
Index of Names	623	(8)
Index of Subjects	631