Markov Chain Monte Carlo: Innovations and Applications (Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore)

Markov Chain Monte Carlo (MCMC) originated in statistical physics, but has spilled over into various application areas, leading to a corresponding variety of techniques and methods. That variety stimulates new ideas and developments from many different places, and there is much to be gained from cross-fertilization. This book presents five expository essays by leaders in the field, drawing from perspectives in physics, statistics and genetics, and showing how different aspects of MCMC come to the fore in different contexts. The essays derive from tutorial lectures at an interdisciplinary program at the Institute for Mathematical Sciences, Singapore, which exploited the exciting ways in which MCMC spreads across different disciplines. Cover ......Page 1 MARKOV CHAIN MONTE CARLO - Innovations and Applications......Page 4 ISBN 9812564276......Page 5 CONTENTS......Page 6 FOREWORD......Page 8 PREFACE......Page 10 1. Probability......Page 14 2. Statistical Physics......Page 16 3. Mathematical Genetics......Page 18 INTRODUCTION TO MARKOV CHAIN MONTE CARLO SIMULATIONS AND THEIR STATISTICAL ANALYSIS......Page 20 1. Introduction......Page 21 2. Probability Distributions and Sampling......Page 22 3. Random Numbers and Fortran Code......Page 23 3.1. How to Get and Run the Fortran Code......Page 24 4. Confdence Intervals and Heapsort......Page 25 5. The Central Limit Theorem and Binning......Page 27 6. Gaussian Error Analysis for Large and Small Samples......Page 30 6.1. Chi-squared Distribution, Error of the Error Bar, F-Test......Page 34 6.2. The Jackknife Approach......Page 35 7. Statistical Physics and Potts Models......Page 36 8. Sampling and Re-weighting......Page 38 9. Importance Sampling and Markov Chain Monte Carlo......Page 40 9.1. Metropolis and Heat Bath Algorithm for Potts Models......Page 42 9.2. The O(3) sigma Model and the Heat Bath Algorithm......Page 44 9.3. Example Runs ......Page 45 10. Statistical Errors of Markov Chain Monte Carlo Data......Page 49 10.1. Autocorrelations......Page 50 10.2. Integrated Autocorrelation Time and Binning......Page 52 10.3. Illustration: Metropolis Generation of Normally......Page 53 11. Self-Consistent versus Reasonable Error Analysis......Page 56 13. Multicanonical Simulations......Page 59 13.1. How to Get the Weights?......Page 62 14. Multicanonical Example Runs (2d Ising and Potts Models)......Page 63 14.1. Energy and Specifc Heat Calculation......Page 65 14.2. Free Energy and Entropy Calculation......Page 67 14.3. Time Series Analysis......Page 68 AN INTRODUCTION TO MONTE CARLO METHODS IN STATISTICAL PHYSICS......Page 72 1. Introduction......Page 73 2.1. The “Classical” Method: Metropolis Sampling......Page 74 2.2. Choice of Boundary Conditions......Page 76 2.3. Random Number Generators!......Page 78 3.1. Finite Size Effects......Page 79 3.2. Finite Sampling Time Effects......Page 82 3.3. Histogram Reweighting......Page 87 3.4. How to Find the Free Energy......Page 88 4.1. “Optimized” Metropolis......Page 91 4.2. Cluster Flipping Algorithms......Page 92 4.3. Probability Changing Cluster Algorithm......Page 93 4.4. The N-Fold Way and Extensions......Page 94 4.5. Phase Switch Monte Carlo......Page 96 4.6. Multicanonical Monte Carlo......Page 100 4.7. “Wang-Landau” Sampling......Page 101 5. Summary and Perspective......Page 108 NOTES ON PERFECT SIMULATION......Page 112 1. CFTP: The Classic Case......Page 115 1.1. Coupling and Convergence: The Binary Switch......Page 117 1.2. Random Walk CFTP......Page 119 1.3. The CFTP Theorem......Page 122 1.4. The Falling Leaves of Fontainebleau......Page 123 1.5. Ising CFTP......Page 124 1.6. Point Process CFTP......Page 126 1.7. CFTP in Space and Time......Page 128 1.8. Some Complements......Page 129 2.1. Small Sets......Page 130 2.2. Murdoch-Green Small-set CFTP......Page 133 2.3. Slice Sampler CFTP......Page 134 2.4. Multi-shift Sampler......Page 136 2.6. Read-once CFTP......Page 137 2.7. Some Complements......Page 138 3.1. Queues......Page 139 3.3. Classic CFTP and Uniform Ergodicity......Page 141 3.4. Dominated CFTP......Page 143 3.5. Non-linear Birth-death Processes......Page 145 3.6. Point Processes......Page 147 3.8. Some Complements......Page 149 4.1. Siegmund Duality......Page 150 4.2. Fill’s Method......Page 151 4.3. FMMR and CFTP......Page 152 4.4. Eÿciency and the Price of Perfection......Page 153 4.5. Dominated CFTP and Foster-Lyapunov Conditions......Page 155 4.6. Backward-forward Algorithm......Page 157 4.7. Some Complements......Page 159 SEQUENTIAL MONTE CARLO METHODS AND THEIR APPLICATIONS......Page 166 1. Introduction......Page 167 2. Stochastic Dynamic Systems......Page 168 2.1. Generalized State Space Models......Page 169 2.2. The Growth Principle......Page 171 3.1. Importance Sampling......Page 172 3.2. The SMC Framework......Page 174 4.1. Propagation in State Space Models......Page 176 4.2. Delay Strategy (Look Ahead)......Page 179 5.1. The Priority Score......Page 182 2. Residual sampling [56]:......Page 183 5.4. Rejection Control......Page 184 6.2. Mixture Kalman Filters (MKF)......Page 185 7. Design Issues (IV): Inferences......Page 187 8.1. Target Tracking......Page 188 8.2. Signal Processing......Page 191 8.3. Stochastic Volatility Models......Page 193 8.4. Self-Avoiding Walks on Lattice......Page 194 8.6. Other Applications......Page 195 MCMC IN THE ANALYSIS OF GENETIC DATA ON PEDIGREES......Page 202 2. Pedigrees, Inheritance, and Genetic Models......Page 203 3. The Structure of a Genetic Model......Page 207 4. Exact Computations on Pedigrees: Peeling Algorithms......Page 211 5. MCMC on Pedigree Structures......Page 218 6. Genetic Mapping and the Location Lod Score......Page 221 7. Monte Carlo Likelihood on Pedigrees......Page 224 8. An Illustrative Example......Page 229 9. Conclusion......Page 233 INDEX......Page 236 Markov Chain Monte Carlo (mcmc) Originated In Statistical Physics, But Has Spilled Over Into Various Application Areas, Leading To A Corresponding Variety Of Techniques And Methods. That Variety Stimulates New Ideas And Developments From Many Different Places, And There Is Much To Be Gained From Cross-fertilization. This Book Presents Five Expository Essays By Leaders In The Field, Drawing From Perspectives In Physics, Statistics And Genetics, And Showing How Different Aspects Of Mcmc Come To The Fore In Different Contexts.--book Jacket. Introduction To Markov Chain Monte Carlo Simulations And Their Statistical Analysis / B.a. Berg -- An Introduction To Monte Carlo Methods In Statistical Physics / D.p. Landau -- Notes On Perfect Simulation / W.s. Kendall -- Sequential Monte Carlo Methods And Their Applications / R. Chen -- Mcmc In The Analysis Of Genetic Data On Pedigrees / E.a. Thompson. Editors, W.s. Kendall, F. Liang, J.-s. Wang. Includes Bibliographical References And Index.