Monte Carlo Methods in Bayesian Computation

1 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.