Albert Bayesian Computation with R

2. Auflage 2009
ISBN: 978-0-387-92298-0
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 300 Seiten

Reihe: Use R!

There has been dramatic growth in the development and application of Bayesian inference in statistics. Berger (2000) documents the increase in Bayesian activity by the number of published research articles, the number of books,andtheextensivenumberofapplicationsofBayesianarticlesinapplied disciplines such as science and engineering. One reason for the dramatic growth in Bayesian modeling is the availab- ity of computational algorithms to compute the range of integrals that are necessary in a Bayesian posterior analysis. Due to the speed of modern c- puters, it is now possible to use the Bayesian paradigm to ?t very complex models that cannot be ?t by alternative frequentist methods. To ?t Bayesian models, one needs a statistical computing environment. This environment should be such that one can: write short scripts to de?ne a Bayesian model use or write functions to summarize a posterior distribution use functions to simulate from the posterior distribution construct graphs to illustrate the posterior inference An environment that meets these requirements is the R system. R provides a wide range of functions for data manipulation, calculation, and graphical d- plays. Moreover, it includes a well-developed, simple programming language that users can extend by adding new functions. Many such extensions of the language in the form of packages are easily downloadable from the Comp- hensive R Archive Network (CRAN).

Autoren/Hrsg.

Albert, Jim

Inhaltsverzeichnis

1;Preface;5
2;Contents;8
3;An Introduction to R;12
3.1;Overview;12
3.2;Exploring a Student Dataset;12
3.2.1;Introduction to the Dataset;12
3.2.2;Reading the Data into R;13
3.2.3;R Commands to Summarize and Grapha Single Batch;13
3.2.4;R Commands to Compare Batches;16
3.2.5;R Commands for Studying Relationships;17
3.3;Exploring the Robustness of the t Statistic;19
3.3.1;Introduction;19
3.3.2;Writing a Function to Compute the t Statistic;20
3.3.3;Programming a Monte Carlo Simulation;21
3.3.4;The Behavior of the True Significance Level Under Different Assumptions;22
3.4;Further Reading;24
3.5;Summary of R Functions;25
3.6;Exercises;26
4;Introduction to Bayesian Thinking;29
4.1;Introduction;29
4.2;Learning About the Proportion of Heavy Sleepers;29
4.3;Using a Discrete Prior;30
4.4;Using a Beta Prior;32
4.5;Using a Histogram Prior;36
4.6;Prediction;38
4.7;Further Reading;44
4.8;Summary of R Functions;44
4.9;Exercises;45
5;Single-Parameter Models;48
5.1;Introduction;48
5.2;Normal Distribution with Known Mean but Unknown Variance;48
5.3;Estimating a Heart Transplant Mortality Rate;50
5.4;An Illustration of Bayesian Robustness;53
5.5;Mixtures of Conjugate Priors;58
5.6;A Bayesian Test of the Fairness of a Coin;61
5.7;Further Reading;66
5.8;Summary of R Functions;66
5.9;Exercises;67
6;Multiparameter Models;71
6.1;Introduction;71
6.2;Normal Data with Both Parameters Unknown;71
6.3;A Multinomial Model;74
6.4;A Bioassay Experiment;77
6.5;Comparing Two Proportions;83
6.6;Further Reading;88
6.7;Summary of R Functions;88
6.8;Exercises;89
7;Introduction to Bayesian Computation;95
7.1;Introduction;95
7.2;Computing Integrals;96
7.3;Setting Up a Problem in R;97
7.4;A Beta-Binomial Model for Overdispersion;98
7.5;Approximations Based on Posterior Modes;102
7.6;The Example;103
7.7;Monte Carlo Method for Computing Integrals;105
7.8;Rejection Sampling;106
7.9;Importance Sampling;109
7.9.1;Introduction;109
7.9.2;Using a Multivariate t as a Proposal Density;111
7.10;Sampling Importance Resampling;113
7.11;Further Reading;116
7.12;Summary of R Functions;117
7.13;Exercises;118
8;Markov Chain Monte Carlo Methods;124
8.1;Introduction;124
8.2;Introduction to Discrete Markov Chains;124
8.3;Metropolis-Hastings Algorithms;127
8.4;Gibbs Sampling;129
8.5;MCMC Output Analysis;129
8.6;A Strategy in Bayesian Computing;131
8.7;Learning About a Normal Population from Grouped Data;131
8.8;Example of Output Analysis;136
8.9;Modeling Data with Cauchy Errors;138
8.10;Analysis of the Stanford Heart Transplant Data;147
8.11;Further Reading;152
8.12;Summary of R Functions;153
8.13;Exercises;154
9;Hierarchical Modeling;160
9.1;Introduction;160
9.2;Three Examples;160
9.3;Individual and Combined Estimates;162
9.4;Equal Mortality Rates?;164
9.5;Modeling a Prior Belief of Exchangeability;168
9.6;Posterior Distribution;170
9.7;Simulating from the Posterior;170
9.8;Posterior Inferences;175
9.8.1;Shrinkage;175
9.8.2;Comparing Hospitals;176
9.9;Bayesian Sensitivity Analysis;178
9.10;Posterior Predictive Model Checking;180
9.11;Further Reading;182
9.12;Summary of R Functions;182
9.13;Exercises;183
10;Model Comparison;187
10.1;Introduction;187
10.2;Comparison of Hypotheses;187
10.3;A One-Sided Test of a Normal Mean;188
10.4;A Two-Sided Test of a Normal Mean;191
10.5;Comparing Two Models;192
10.6;Models for Soccer Goals;193
10.7;Is a Baseball Hitter Really Streaky?;196
10.8;A Test of Independence in a Two-Way Contingency Table;200
10.9;Further Reading;205
10.10;Summary of R Functions;205
10.11;Exercises;207
11;Regression Models;211
11.1;Introduction;211
11.2;Normal Linear Regression;211
11.2.1;The Model;211
11.2.2;The Posterior Distribution;212
11.2.3;Prediction of Future Observations;212
11.2.4;Computation;213
11.2.5;Model Checking;213
11.2.6;An Example;214
11.3;Model Selection Using Zellner's g Prior;223
11.4;Survival Modeling;228
11.5;Further Reading;233
11.6;Summary of R Functions;233
11.7;Exercises;235
12;Gibbs Sampling;241
12.1;Introduction;241
12.2;Robust Modeling;242
12.3;Binary Response Regression with a Probit Link;246
12.3.1;Missing Data and Gibbs Sampling;246
12.3.2;Proper Priors and Model Selection;249
12.4;Estimating a Table of Means;254
12.4.1;Introduction;254
12.4.2;A Flat Prior Over the Restricted Space;256
12.4.3;A Hierarchical Regression Prior;260
12.4.4;Predicting the Success of Future Students;265
12.5;Further Reading;266
12.6;Summary of R Functions;266
12.7;Exercises;267
13;Using R to Interface with WinBUGS;271
13.1;Introduction to WinBUGS;271
13.2;An R Interface to WinBUGS;272
13.3;MCMC Diagnostics Using the coda Package;273
13.4;A Change-Point Model;274
13.5;A Robust Regression Model;278
13.6;Estimating Career Trajectories;282
13.7;Further Reading;287
13.8;Exercises;288
14;References;293
15;Index;298