Monte Carlo Strategies in Scientific Computing / Edition 1

Monte Carlo Strategies in Scientific Computing / Edition 1

by Jun S. Liu
Pub. Date:
Springer New York
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Monte Carlo Strategies in Scientific Computing / Edition 1

This book provides a self-contained and up-to-date treatment of the Monte Carlo method and develops a common framework under which various Monte Carlo techniques can be "standardized" and compared. Given the interdisciplinary nature of the topics and a moderate prerequisite for the reader, this book should be of interest to a broad audience of quantitative researchers such as computational biologists, computer scientists, econometricians, engineers, probabilists, and statisticians. It can also be used as the textbook for a graduate-level course on Monte Carlo methods. Many problems discussed in the later chapters can be potential thesis topics for master's or Ph.D. students in statistics or computer science departments.

The Monte Carlo method is a computer-based statistical sampling approach for solving numerical problems concerned with a complex system. The methodology was initially developed in the field of statistical physics during the early days of electronic computing (1945 to 1955) and has now been adopted by researchers in almost all scientific fields. The fundamental idea for constructing Markov-chain-based Monte Carlo algorithms was introduced in the 1950s. This idea was later extended to handle more and more complex physical systems. In the 1980s, statisticians and computer scientists developed Monte Carlo-based algorithms for a wide variety of integration and optimization tasks. In the 1990s, the method began to play an important role in computational biology. Over the past fifty years, researchers in diverse scientific fields have studied the Monte Carlo method and contributed to its development. Today, a large number of scientists and engineers employ Monte Carlotechniques as an essential tool in their work. For such scientists, there is a need to keep up to date with recent advances in Monte Carlo methodologies.

About the Author:
Jun S. Liu is a professor in the Department of Statistics at Harvard University

Product Details

ISBN-13: 9780387763699
Publisher: Springer New York
Publication date: 06/20/2008
Series: Springer Series in Statistics
Edition description: 1st ed. 2001, 2nd printing 2008
Pages: 344
Sales rank: 910,484
Product dimensions: 9.21(w) x 6.14(h) x 0.75(d)

Table of Contents

Preface     vii
Introduction and Examples     1
The Need of Monte Carlo Techniques     1
Scope and Outline of the Book     3
Computations in Statistical Physics     7
Molecular Structure Simulation     9
Bioinformatics: Finding Weak Repetitive Patterns     10
Nonlinear Dynamic System: Target Tracking     14
Hypothesis Testing for Astronomical Observations     16
Bayesian Inference of Multilevel Models     18
Monte Carlo and Missing Data Problems     19
Basic Principles: Rejection, Weighting, and Others     23
Generating Simple Random Variables     23
The Rejection Method     24
Variance Reduction Methods     26
Exact Methods for Chain-Structured Models     28
Dynamic programming     29
Exact simulation     30
Importance Sampling and Weighted Sample     31
An example     31
The basic idea     33
The "rule of thumb" for importance sampling     34
Concept of the weighted sample     36
Marginalization in importance sampling     37
Example: Solving a linear system     38
Example: A Bayesian missing dataproblem     40
Advanced Importance Sampling Techniques     42
Adaptive importance sampling     42
Rejection and weighting     43
Sequential importance sampling     46
Rejection control in sequential importance sampling     48
Application of SIS in Population Genetics     49
Problems     51
Theory of Sequential Monte Carlo     53
Early Developments: Growing a Polymer     55
A simple model of polymer: Self-avoid walk     55
Growing a polymer on the square lattice     56
Limitations of the growth method     59
Sequential Imputation for Statistical Missing Data Problems     60
Likelihood computation     60
Bayesian computation     62
Nonlinear Filtering     64
A General Framework     67
The choice of the sampling distribution     69
Normalizing constant     69
Pruning, enrichment, and resampling     71
More about resampling     72
Partial rejection control     75
Marginalization, look-ahead, and delayed estimate     76
Problems     77
Sequential Monte Carlo in Action     79
Some Biological Problems     79
Molecular Simulation     79
Inference in population genetics     81
Finding motif patterns in DNA sequences     84
Approximating Permanents     90
Counting 0-1 Tables with Fixed Margins     92
Bayesian Missing Data Problems     94
Murray's data     94
Nonparametric Bayes analysis of binomial data     95
Problems in Signal Processing     98
Target tracking in clutter and mixture Kalman filter     98
Digital signal extraction in fading channels     102
Problems     103
Metropolis Algorithm and Beyond     105
The Metropolis Algorithm     106
Mathematical Formulation and Hastings's Generalization     111
Why Does the Metropolis Algorithm Work?     112
Some Special Algorithms     114
Random-walk Metropolis     114
Metropolized independence sampler     115
Configurational bias Monte Carlo     116
Multipoint Metropolis Methods     117
Multiple independent proposals     118
Correlated multipoint proposals     120
Reversible Jumping Rule     122
Dynamic Weighting      124
Output Analysis and Algorithm Efficiency     125
Problems     127
The Gibbs Sampler     129
Gibbs Sampling Algorithms     129
Illustrative Examples     131
Some Special Samplers     133
Slice sampler     133
Metropolized Gibbs sampler     133
Hit-and-run algorithm     134
Data Augmentation Algorithm     135
Bayesian missing data problem     135
The original DA algorithm     136
Connection with the Gibbs sampler     137
An example: Hierarchical Bayes model     138
Finding Repetitive Motifs in Biological Sequences     139
A Gibbs sampler for detecting subtle motifs     140
Alignment and classification     141
Covariance Structures of the Gibbs Sampler     143
Data Augmentation     143
Autocovariances for the random-scan Gibbs sampler     144
More efficient use of Monte Carlo samples     146
Collapsing and Grouping in a Gibbs Sampler     146
Problems     151
Cluster Algorithms for the Ising Model     153
Ising and Potts Model Revisit     153
The Swendsen-Wang Algorithm as Data Augmentation      154
Gonvergence Analysis and Generalization     155
The Modification by Wolff     157
Further Generalization     157
Discussion     158
Problems     159
General Conditional Sampling     161
Partial Resampling     161
Case Studies for Partial Resampling     163
Gaussian random field model     163
Texture synthesis     165
Inference with multivariate t-distribution     169
Transformation Group and Generalized Gibbs     171
Application: Parameter Expansion for Data Augmentation     174
Some Examples in Bayesian Inference     176
Probit regression     176
Monte Carlo bridging for stochastic differential equation     178
Problems     181
Molecular Dynamics and Hybrid Monte Carlo     183
Basics of Newtonian Mechanics     184
Molecular Dynamics Simulation     185
Hybrid Monte Carlo     189
Algorithms Related to HMC     192
Langevin-Euler Moves     192
Generalized hybrid Monte Carlo     193
Surrogate transition method     194
Multipoint Strategies for Hybrid Monte Carlo      195
Neal's window method     195
Multipoint method     197
Application of HMC in Statistics     198
Indirect observation model     199
Estimation in the stochastic volatility model     201
Multilevel Sampling and Optimization Methods     205
Umbrella Sampling     206
Simulated Annealing     209
Simulated Tempering     210
Parallel Tempering     212
Generalized Ensemble Simulation     215
Multicanonical sampling     216
The 1/k-ensemble method     217
Comparison of algorithms     218
Tempering with Dynamic Weighting     219
Ising model simulation at sub-critical temperature     221
Neural network training     222
Population-Based Monte Carlo Methods     225
Adaptive Direction Sampling: Snooker Algorithm     226
Conjugate Gradient Monte Carlo     227
Evolutionary Monte Carlo     228
Evolutionary movements in binary-coded space     230
Evolutionary movements in continuous space     231
Some Further Thoughts     233
Numerical Examples     235
Simulating from a bimodal distribution      235
Comparing algorithms for a multimodal example     236
Variable selection with binary-coded EMC     237
Bayesian neural network training     239
Problems     242
Markov Chains and Their Convergence     245
Basic Properties of a Markov Chain     245
Chapman-Kolmogorov equation     247
Convergence to stationarity     248
Coupling Method for Card Shuffling     250
Random-to-top shuffling     250
Riffle shuffling     251
Convergence Theorem for Finite-State Markov Chains     252
Coupling Method for General Markov Chain     254
Geometric Inequalities     256
Basic setup     257
Poincare inequality     257
Example: Simple random walk on a graph     259
Cheeger's inequality     261
Functional Analysis for Markov Chains     263
Forward and backward operators     264
Convergence rate of Markov chains     266
Maximal correlation     267
Behavior of the Averages     269
Selected Theoretical Topics     271
MCMC Convergence and Convergence Diagnostics     271
Iterative Conditional Sampling      273
Data augmentation     273
Random-scan Gibbs sampler     275
Comparison of Metropolis-Type Algorithms     277
Peskun's ordering     277
Comparing schemes using Peskun's ordering     279
Eigenvalue Analysis for the Independence Sampler     281
Perfect Simulation     284
A Theory for Dynamic Weighting     287
Definitions     287
Weight behavior under different scenarios     288
Estimation with weighted samples     291
A simulation study     292
Basics in Probability and Statistics     295
Basic Probability Theory     295
Experiments, events, and probability     295
Univariate random variables and their properties     296
Multivariate random variable     298
Convergence of random variables     300
Statistical Modeling and Inference     301
Parametric statistical modeling     301
Frequentist approach to statistical inference     302
Bayesian methodology     304
Bayes Procedure and Missing Data Formalism     306
The joint and posterior distributions     306
The missing data problem     308
The Expectation-Maximization Algorithm     310
References     313
Author Index     333
Subject Index     338

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