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(s): George Casella
Edition: Hardcover
Publisher: Springer
Year: 2005
Language: English
Pages: 649
Cover Page
Title Page
Copyright Page
Dedication Page
Preface to Second Edition
Preface to First Edition
Table of Contents
List of Tables
List of Figures
1 Introduction
1.1 Statistical Models
1.2 Likelihood Methods
1.3 Bayesian Methods
1.4 Deterministic Numerical Methods
1.4.1 Optimization
1.4.2 Integration
1.4.3 Comparison
1.5 Problems
1.6 Notes
1.6.1 Prior Distributions
1.6.2 Bootstrap Methods
2 Random V^ariable Generation
2.1 Introduction
2.1.1 Uniform Simulation
2.1.2 The Inverse Transform
2.1.3 Alternatives
2.1.4 Optimal Algorithms
2.2 General Transformation Methods
2.3 Accept-Reject Methods
2.3.1 The Fundamental Theorem of Simulation
2.3.2 The Accept-Reject Algorithm
2.4 Envelope Accept-Reject Methods
2.4.1 The Squeeze Principle
2.4.2 Log-Concave Densities
2.5 Problems
2.6 Notes
2.6.1 The Kiss Generator
2.6.2 Quasi-Monte Carlo Methods
2.6.3 Mixture Representations
3 Monte Carlo Integration
3.1 Introduction
3.2 Classical Monte Carlo Integration
3.3 Importance Sampling
3.3.1 Principles
3.3.2 Finite Variance Estimators
3.3.3 Comparing Importance Sampling with Accept-Reject . .
3.4 Laplace Approximations
3.5 Problems
3.6 Notes
3.6.1 Large Deviations Techniques
3.6.2 The Saddlepoint Approximation
4 Controling Monte Carlo V^ariance
4.1 Monitoring Variation with the CLT
4.1.1 Univariate Monitoring
4.1.2 Multivariate Monitoring
4.2 Rao-Blackwellization
4.3 Riemann Approximations
4.4 Acceleration Methods
4.4.1 Antithetic Variables
4.4.2 Control Variates
4.5 Problems
4.6 Notes
4.6.1 Monitoring Importance Sampling Convergence
4.6.2 Accept-Reject with Loose Bounds
4.6.3 Partitioning
5 Monte Carlos Optimization
5.1 Introduction
5.2 Stochastic Exploration
5.2.1 A Basic Solution
5.2.2 Gradient Methods
5.2.3 Simulated Annealing
5.2.4 Prior Feedback
5.3 Stochastic Approximation
5.3.1 Missing Data Models and Demarginalization
5.3.2 The EM Algorithm
5.3.3 Monte Carlo EM
5.3.4 EM Standard Errors
5.4 Problems
5.5 Notes
5.5.1 Variations on EM
5.5.2 Neural Networks
5.5.3 The Robbins-Monro procedure
5.5.4 Monte Carlo Approximation
6 Markov Chains
6.1 Essentials for MCMC
6.2 Basic Notions
6.3 Irreducibility, Atoms, and Small Sets
6.3.1 Irreducibility
6.3.2 Atoms and Small Sets
6.3.3 Cycles and Aperiodicity
6.4 Transience and Recurrence
6.4.1 Classification of Irreducible Chains
6.4.2 Criteria for Recurrence
6.4.3 Harris Recurrence
6.5 Invariant Measures
6.5.1 Stationary Chains
6.5.2 Kacs Theorem
6.5.3 Reversibility and the Detailed Balance Condition
6.6 Ergodicity and Convergence
6.6.1 Ergodicity
6.6.2 Geometric Convergence
6.6.3 Uniform Ergodicity
6.7 Limit Theorems
6.7.1 Ergodic Theorems
6.7.2 Central Limit Theorems
6.8 Problems
6.9 Notes
6.9.1 Drift Conditions
6.9.2 Eatons Admissibility Condition
6.9.3 Alternative Convergence Conditions
6.9.4 Mixing Conditions and Central Limit Theorems
6.9.5 Covariance in Markov Chains
7 The Metropolis-Hastings Algorithm
7.1 The MCMC Principle
7.2 Monte Carlo Methods Based on Markov Chains
7.3 The Metropolis-Hastings algorithm
7.3.1 Definition
7.3.2 Convergence Properties
7.4 The Independent Metropolis-Hastings Algorithm
7.4.1 Fixed Proposals
7.4.2 A Metropolis-Hastings Version of ARS
7.5 Random Walks
7.6 Optimization and Control
7.6.1 Optimizing the Acceptance Rate
7.6.2 Conditioning and Accelerations
7.6.3 Adaptive Schemes
7.7 Problems
7.8 Notes
7.8.1 Background of the Metropolis Algorithm
7.8.2 Geometric Convergence of Metropolis-Hastings Algorithms
7.8.3 A Reinterpretation of Simulated Annealing
7.8.4 Reference Acceptance Rates
7.8.5 Langevin Algorithms
8 The Slice Sampler
8.1 Another Look at the Fundamental Theorem
8.2 The General Slice Sampler
8.3 Convergence Properties of the Slice Sampler
8.4 Problems
8.5 Notes
8.5.1 Dealing with Difficult Slices
9 The Two-Stage Gibbs Sampler
9.1 A General Class of Two-Stage Algorithms
9.1.1 Prom Slice Sampling to Gibbs Sampling
9.1.2 Definition
9.1.3 Back to the Slice Sampler
9.1.4 The Hammersley-Clifford Theorem
9.2 Fundamental Properties
9.2.1 Probabilistic Structures
9.2.2 Reversible and Interleaving Chains
9.2.3 The Duality Principle
9.3 Monotone Covariance and Rao-Blackwellization
9.4 The EM-Gibbs Connection
9.5 Transition
9.6 Problems
9.7 Notes
9.7.1 Inference for Mixtures
9.7.2 ARCH Models
10 The Multi-Stage Gibbs Sampler
10.1 Basic Derivations
10.1.1 Definition
10.1.2 Completion
10.1.3 The General Hammersley-Clifford Theorem
10.2 Theoretical Justifications
10.2.1 Markov Properties of the Gibbs Sampler
10.2.2 Gibbs Sampling as Metropolis-Hastings
10.2.3 Hierarchical Structures
10.3 Hybrid Gibbs Samplers
10.3.1 Comparison with Metropolis-Hastings Algorithms
10.3.2 Mixtures and Cycles
10.3.3 Metropolizing the Gibbs Sampler
10.4 Statistical Considerations
10.4.1 Reparameterization
10.4.2 Rao-Blackwellization
10.4.3 Improper Priors
10.5 Problems
10.6 Notes
10.6.1 A Bit of Background
10.6.2 The BUGS Software
10.6.3 Nonparametric Mixtures
10.6.4 Graphical Models
11 Variable Dimension Models and Reversible Jump Algorithms
11.1 Variable Dimension Models
11.1.1 Bayesian Model Choice
11.1.2 Difficulties in Model Choice
11.2 Reversible Jump Algorithms
11.2.1 Greens Algorithm
11.2.2 A Fixed Dimension Reassessment
11.2.3 The Practice of Reversible Jump MCMC
11.3 Alternatives to Reversible Jump MCMC
11.3.1 Saturation
11.3.2 Continuous-Time Jump Processes
11.4 Problems
11.5 Notes
11.5.1 Occams Razor
12 Diagnosing Convergence
12.1 Stopping the Chain
12.1.1 Convergence Criteria
12.1.2 Multiple Chains
12.1.3 Monitoring Reconsidered
12.2 Monitoring Convergence to the Stationary Distribution
12.2.1 A First Illustration
12.2.2 Nonparametric Tests of Stationarity
12.2.3 Renewal Methods
12.2.4 Missing Mass
12.2.5 Distance Evaluations
12.3 Monitoring Convergence of Averages
12.3.1 A First Illustration
12.3.2 Multiple Estimates
12.3.3 Renewal Theory
12.3.4 Within and Between Variances
12.3.5 Effective Sample Size
12.4 Simultaneous Monitoring
12.4.1 Binary Control
12.4.2 Valid Discretization
12.5 Problems
12.6 Notes
12.6.1 Spectral Analysis
12.6.2 The CODA Software
13 Perfect Sampling
13.1 Introduction
13.2 Coupling from the Past
13.2.1 Random Mappings and Coupling
13.2.2 Propp and Wilsons Algorithm
13.2.3 Monotonicity and Envelopes
13.2.4 Continuous States Spaces
13.2.5 Perfect Slice Sampling
13.2.6 Perfect Sampling via Automatic Coupling
13.3 Forward Coupling
13.4 Perfect Sampling in Practice
13.5 Problems
13.6 Notes
13.6.1 History
13.6.2 Perfect Sampling and Tempering
14 Iterated and Sequential Importance Sampling
14.1 Introduction
14.2 Generalized Importance Sampling
14.3 Particle Systems
14.3.1 Sequential Monte Carlo
14.3.2 Hidden Markov Models
14.3.3 Weight Degeneracy
14.3.4 Particle Filters
14.3.5 Sampling Strategies
14.3.6 Fighting the Degeneracy
14.3.7 Convergence of Particle Systems
14.4 Population Monte Carlo
14.4.1 Sample Simulation
14.4.2 General Iterative Importance Sampling
14.4.3 Population Monte Carlo
14.4.4 An Illustration for the Mixture Model
14.4.5 Adaptativity in Sequential Algorithms
14.5 Problems
14.6 Notes
14.6.1 A Brief History of Particle Systems
14.6.2 Dynamic Importance Sampling
14.6.3 Hidden Markov Models
A Probability Distributions
B Notation
B.1 Mathematical
B.2 Probability
B.3 Distributions
B.4 Markov Chains
B.5 Statistics
B.6 Algorithms
References
Index of Names
Index of Subjects
