Monte Carlo Statistical Methods

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