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This book examines advanced Bayesian computational methods. It presents methods for sampling from posterior distributions and discusses how to compute posterior quantities of interest using Markov chain Monte Carlo (MCMC) samples. This book examines each of these issues in detail and heavily focuses on computing various posterior quantities of interest from a given MCMC sample. Several topics are addressed, including techniques for MCMC sampling, Monte Carlo methods for estimation of posterior quantities, improving simulation accuracy, marginal posterior density estimation, estimation of normalizing constants, constrained parameter problems, highest posterior density interval calculations, computation of posterior modes, and posterior computations for proportional hazards models and Dirichlet process models. The authors also discuss computions involving model comparisons, including both nested and non-nested models, marginal likelihood methods, ratios of normalizing constants, Bayes factors, the Savage-Dickey density ratio, Stochastic Search Variable Selection, Bayesian Model Averaging, the reverse jump algorithm, and model adequacy using predictive and latent residual approaches. The book presents an equal mixture of theory and applications involving real data. The book is intended as a graduate textbook or a reference book for a one semester course at the advanced masters or Ph.D. level. It would also serve as a useful reference book for applied or theoretical researchers as well as practitioners. Ming-Hui Chen is Associate Professor of Mathematical Sciences at Worcester Polytechnic Institute, Qu-Man Shao is Assistant Professor of Mathematics at the University of Oregon. Joseph G. Ibrahim is Associate Professor of Biostatistics at the Harvard School of Public Health and Dana-Farber Cancer Institute.
Show moreThis book examines advanced Bayesian computational methods. It presents methods for sampling from posterior distributions and discusses how to compute posterior quantities of interest using Markov chain Monte Carlo (MCMC) samples. This book examines each of these issues in detail and heavily focuses on computing various posterior quantities of interest from a given MCMC sample. Several topics are addressed, including techniques for MCMC sampling, Monte Carlo methods for estimation of posterior quantities, improving simulation accuracy, marginal posterior density estimation, estimation of normalizing constants, constrained parameter problems, highest posterior density interval calculations, computation of posterior modes, and posterior computations for proportional hazards models and Dirichlet process models. The authors also discuss computions involving model comparisons, including both nested and non-nested models, marginal likelihood methods, ratios of normalizing constants, Bayes factors, the Savage-Dickey density ratio, Stochastic Search Variable Selection, Bayesian Model Averaging, the reverse jump algorithm, and model adequacy using predictive and latent residual approaches. The book presents an equal mixture of theory and applications involving real data. The book is intended as a graduate textbook or a reference book for a one semester course at the advanced masters or Ph.D. level. It would also serve as a useful reference book for applied or theoretical researchers as well as practitioners. Ming-Hui Chen is Associate Professor of Mathematical Sciences at Worcester Polytechnic Institute, Qu-Man Shao is Assistant Professor of Mathematics at the University of Oregon. Joseph G. Ibrahim is Associate Professor of Biostatistics at the Harvard School of Public Health and Dana-Farber Cancer Institute.
Show more1 Introduction.- 1.1 Aims.- 1.2 Outline.- 1.3 Motivating Examples.- 1.4 The Bayesian Paradigm.- Exercises.- 2 Markov Chain Monte Carlo Sampling.- 2.1 Gibbs Sampler.- 2.2 Metropolis-Hastings Algorithm.- 2.3 Hit-and-Run Algorithm.- 2.4 Multiple-Try Metropolis Algorithm.- 2.5 Grouping, Collapsing, and Reparameterizations.- 2.6 Acceleration Algorithms for MCMC Sampling.- 2.7 Dynamic Weighting Algorithm.- 2.8 Toward “Black-Box” Sampling.- 2.9 Convergence Diagnostics.- Exercises.- 3 Basic Monte Carlo Methods for Estimating Posterior Quantities.- 3.1 Posterior Quantities.- 3.2 Basic Monte Carlo Methods.- 3.3 Simulation Standard Error Estimation.- 3.4 Improving Monte Carlo Estimates.- 3.5 Controlling Simulation Errors.- Exercises.- 4 Estimating Marginal Posterior Densities.- 4.1 Marginal Posterior Densities.- 4.2 Kernel Methods.- 4.3 IWMDE Methods.- 4.4 Illustrative Examples.- 4.5 Performance Study Using the Kullback-Leibler Divergence.- Exercises.- 5 Estimating Ratios of Normalizing Constants.- 5.1 Introduction.- 5.2 Importance Sampling.- 5.3 Bridge Sampling.- 5.4 Path Sampling.- 5.5 Ratio Importance Sampling.- 5.6 A Theoretical Illustration.- 5.7 Computing Simulation Standard Errors.- 5.8 Extensions to Densities with Different Dimensions.- 5.9 Estimation of Normalizing Constants After Transformation.- 5.10 Other Methods.- 5.11 An Application of Weighted Monte Carlo Estimators.- 5.12 Discussion.- Exercises.- 6 Monte Carlo Methods for Constrained Parameter Problems.- 6.1 Constrained Parameter Problems.- 6.2 Posterior Moments and Marginal Posterior Densities.- 6.3 Computing Normalizing Constants for Bayesian Estimation.- 6.4 Applications.- 6.5 Discussion.- Exercises.- 7 Computing Bayesian Credible and HPD Intervals.- 7.1 Bayesian Credible and HPD Intervals.- 7.2 EstimatingBayesian Credible Intervals.- 7.3 Estimating Bayesian HPD Intervals.- 7.4 Extension to the Constrained Parameter Problems.- 7.5 Numerical Illustration.- 7.6 Discussion.- Exercises.- 8 Bayesian Approaches for Comparing Nonnested Models.- 8.1 Marginal Likelihood Approaches.- 8.2 Scale Mixtures of Multivariate Normal Link Models.- 8.3 “Super-Model” or “Sub-Model” Approaches.- 8.4 Criterion-Based Methods.- 9 Bayesian Variable Selection.- 9.1 Variable Selection for Logistic Regression Models.- 9.2 Variable Selection for Time Series Count Data Models.- 9.3 Stochastic Search Variable Selection.- 9.4 Bayesian Model Averaging.- 9.5 Reversible Jump MCMC Algorithm for Variable Selection.- Exercises.- 10 Other Topics.- 10.1 Bayesian Model Adequacy.- 10.2 Computing Posterior Modes.- 10.3 Bayesian Computation for Proportional Hazards Models.- 10.4 Posterior Sampling for Mixture of Dirichlet Process Models.- Exercises.- References.- Author Index.
Springer Book Archives
"This book combines the theory topics with good computer and
application examples from the field of food science, agriculture,
cancer and others. The volume will provide an excellent research
resource for statisticians with an interest in computer intensive
methods for modelling with different sorts of prior
information."
A.V. Tsukanov in "Short Book Reviews", Vol. 20/3, December 2000
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