The BUGS Book: A Practical Introduction to Bayesian Analysis

by Lunn; David

• ISBN13: 9781584888499
• ISBN10: 1584888490
• Format: Nonspecific Binding
• Copyright: 2012-10-02
• Publisher: Chapman & Hall/

Summary

In recent years, Bayesian methods have become the most widely used statistical methods for data analysis and modeling. The BUGS software has become the most popular software for Bayesian analysis worldwide. Authored by the team that originally developed this software, Bayesian Analysis using BUGS provides a practical introduction to this program and its use. The text presents complete coverage of all the functionalities of BUGS, including prediction, missing data, model criticism, and prior sensitivity. It also features a large number of worked examples, a wide range of applications from various disciplines, and numerous detailed exercises in every chapter.

Table of Contents

Preface	p. xiii
Introduction: Probability and parameters	p. 1
Probability	p. 1
Probability distributions	p. 5
Calculating properties of probability distributions	p. 7
Monte Carlo integration	p. 8
Monte Carlo simulations using BUGS	p. 13
Introduction to BUGS	p. 13
Background	p. 13
Directed graphical models	p. 13
The BUGS language	p. 15
Running BUGS models	p. 16
Running WinBUGS for a simple example	p. 17
DoodleBUGS	p. 21
Using BUGS to simulate from distributions	p. 22
Transformations of random variables	p. 24
Complex calculations using Monte Carlo	p. 26
Multivariate Monte Carlo analysis	p. 27
Predictions with unknown parameters	p. 29
Introduction to Bayesian inference	p. 33
Bayesian learning	p. 33
Bayes' theorem for observable quantities	p. 33
Bayesian inference for parameters	p. 34
Posterior predictive distributions	p. 36
Conjugate Bayesian inference	p. 36
Binomial data	p. 37
Normal data with unknown mean, known variance	p. 41
Inference about a discrete parameter	p. 45
Combinations of conjugate analyses	p. 49
Bayesian and classical methods	p. 51
Likelihood-based inference	p. 52
Exchangeability	p. 52
Long-run properties of Bayesian methods	p. 53
Model-based vs procedural methods	p. 54
The "likelihood principle"	p. 55
Introduction to Markov chain Monte Carlo methods	p. 57
Bayesian computation	p. 57
Single-parameter models	p. 57
Multi-parameter models	p. 59
Monte Carlo integration for evaluating posterior integrals	p. 61
Markov chain Monte Carlo methods	p. 62
Gibbs sampling	p. 63
Gibbs sampling and directed graphical models	p. 64
Derivation of full conditional distributions in BUGS	p. 68
Other MCMC methods	p. 68
Initial values	p. 70
Convergence	p. 71
Detecting convergence/stationarity by eye	p. 72
Formal detection of convergence/stationarity	p. 73
Efficiency and accuracy	p. 77
Monte Carlo standard error of the posterior mean	p. 77
Accuracy of the whole posterior	p. 78
Beyond MCMC	p. 79
Prior distributions	p. 81
Different purpose of priors	p. 81
Vague, "objective," and "reference" priors	p. 82
Introduction	p. 82
Discrete uniform distributions	p. 83
Continuous uniform distributions and Jeffreys prior	p. 83
Location parameters	p. 84
Proportions	p. 84
Counts and rates	p. 85
Scale parameters	p. 87
Distributions on the positive integers	p. 88
More complex situations	p. 89
Representation of informative priors	p. 89
Elicitation of pure judgement	p. 90
Discounting previous data	p. 93
Mixture of prior distributions	p. 95
Sensitivity analysis	p. 97
Regression models	p. 103
Linear regression with normal errors	p. 103
Linear regression with non-normal errors	p. 107
Non-linear regression with normal errors	p. 109
Multivariate responses	p. 112
Generalised linear regression models	p. 114
Inference on functions of parameters	p. 118
Further reading	p. 119
Categorical data	p. 121
2 X 2 tables	p. 121
Tables with one margin fixed	p. 122
Case-control studies	p. 125
Tables with both margins fixed	p. 126
Multinomial models	p. 126
Conjugate analysis	p. 126
Non-conjugate analysis-parameter constraints	p. 128
Categorical data with covariates	p. 129
Multinomial and Poisson regression equivalence	p. 131
Contingency tables	p. 132
Ordinal regression	p. 132
Further reading	p. 134
Model checking and comparison	p. 137
Introduction	p. 137
Deviance	p. 138
Residuals	p. 140
Standardised Pearson residuals	p. 140
Multivariate residuals	p. 142
Observed p-values for distributional shape	p. 143
Deviance residuals and tests of fit	p. 145
Predictive checks and Bayesian p-values	p. 147
Interpreting discrepancy statistics - how big is big?	p. 147
Out-of-sample prediction	p. 148
Checking functions based on data alone	p. 148
Checking functions based on data and parameters	p. 152
Goodness of fit for grouped data	p. 155
Model assessment by embedding in larger models	p. 157
Model comparison using deviances	p. 159
pD: The effective number of parameters	p. 159
Issues with pD	p. 161
Alternative measures of the effective number of parameters	p. 164
DIC for model comparison	p. 165
How and why does WinBUGS partition DIC and pD?	p. 167
Alternatives to DIC	p. 168
Bayes factors	p. 169
Lindley-Bartlett paradox in model selection	p. 171
Computing marginal likelihoods	p. 172
Model uncertainty	p. 173
Bayesian model averaging	p. 173
MCMC sampling over a space of models	p. 173
Model averaging when all models are wrong	p. 175
Model expansion	p. 176
Discussion on model comparison	p. 177
Prior-data conflict	p. 178
Identification of prior-data conflict	p. 179
Accommodation of prior-data conflict	p. 180
Issues in Modelling	p. 185
Missing data	p. 185
Missing response data	p. 186
Missing covariate data	p. 189
Prediction	p. 193
Measurement error	p. 195
Cutting feedback	p. 201
New distributions	p. 204
Specifying a new sampling distribution	p. 204
Specifying a new prior distribution	p. 205
Censored, truncated, and grouped observations	p. 206
Censored observations	p. 206
Truncated sampling distributions	p. 208
Grouped, rounded, or interval-censored data	p. 209
Constrained parameters	p. 211
Univariate fully specified prior distributions	p. 211
Multivariate fully specified prior distributions	p. 211
Prior distributions with unknown parameters	p. 214
Bootstrapping	p. 214
Ranking	p. 215
Hierarchical models	p. 219
Exchangeability	p. 219
Priors	p. 223
Unit-specific parameters	p. 223
Parameter constraints	p. 223
Priors for variance components	p. 225
Hierarchical regression models	p. 227
Data formatting	p. 230
Hierarchical models for variances	p. 237
Redundant parameterisations	p. 240
More general formulations	p. 242
Checking of hierarchical models	p. 242
Comparison of hierarchical models	p. 249
"Focus": The crucial element of model comparison in hierarchical models	p. 250
Further resources	p. 252
Specialised models	p. 253
Time-to-event data	p. 253
Parametric survival regression	p. 254
Time series models	p. 257
Spatial models	p. 262
Intrinsic conditionally autoregressive (CAR) models	p. 263
Supplying map polygon data to WinBUGS and creating adjacency matrices	p. 264
Multivariate CAR models	p. 268
Proper CAR model	p. 269
Poisson-gamma moving average models	p. 269
Geostatistical models	p. 270
Evidence synthesis	p. 273
Meta-analysis	p. 273
Generalised evidence synthesis	p. 274
Differential equation and pharmacokinetic models	p. 278
Finite mixture and latent class models	p. 280
Mixture models using an explicit likelihood	p. 283
Piecewise parametric models	p. 286
Change-point models	p. 286
Splines	p. 288
Semiparametric survival models	p. 288
Bayesian nonparametric models	p. 291
Dirichlet process mixtures	p. 293
Stick-breaking implementation	p. 293
Different implementations of BUGS	p. 297
Introduction-BUGS engines and interfaces	p. 297
Expert systems and MCMC methods	p. 298
Classic BUGS	p. 299
WinBUGS	p. 300
Using WinBUGS: compound documents	p. 301
Formatting data	p. 301
Using the WinBUGS graphical interface	p. 304
Doodles	p. 308
Scripting	p. 308
Interfaces with other software	p. 310
R2WinBUGS	p. 311
WBDev	p. 313
OpenBUGS	p. 315
Differences from WinBUGs	p. 317
OpenBUGS on Linux	p. 317
BRugs	p. 318
Parallel computation	p. 319
JAGS	p. 320
Extensibility: modules	p. 321
Language differences	p. 321
Other differences from WinBUGS	p. 324
Running JAGS from the command line	p. 325
Running JAGS from R	p. 326
BUGS language syntax	p. 329
Introduction	p. 329
Distributions	p. 329
Standard distributions	p. 329
Censoring and truncation	p. 330
Non-standard distributions	p. 331
Deterministic functions	p. 331
Standard functions	p. 331
Special functions	p. 331
Add-on functions	p. 332
Repetition	p. 332
Multivariate quantities	p. 333
Indexing	p. 334
Functions as indices	p. 334
Implicit indexing	p. 334
Nested indexing	p. 334
Data transformations	p. 335
Commenting	p. 335
Functions in BUGS	p. 337
Standard functions	p. 337
Trigonometric functions	p. 337
Matrix algebra	p. 337
Distribution utilities and model checking	p. 340
Functionals and differential equations	p. 341
Miscellaneous	p. 342
Distributions in BUGS	p. 343
Continuous univariate, unrestricted range	p. 343
Continuous univariate, restricted to be positive	p. 345
Continuous univariate, restricted to a finite interval	p. 349
Continuous multivariate distributions	p. 350
Discrete univariate distributions	p. 351
Discrete multivariate distributions	p. 354
Bibliography	p. 357
Index	p. 373
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