Bayesian Analysis for the Social Sciences

by Jackman, Simon

• ISBN13: 9780470011546
• ISBN10: 0470011548
• Edition: 1st
• Format: Hardcover
• Copyright: 2009-12-02
• Publisher: Wiley

Summary

Bayesian methods are increasingly being used in the social sciences, as the problems encountered lend themselves so naturally to the subjective qualities of Bayesian methodology. This book provides an accessible introduction to Bayesian methods, tailored specifically for social science students. It contains lots of real examples from political science, psychology, sociology, and economics, exercises in all chapters, and detailed descriptions of all the key concepts, without assuming any background in statistics beyond a first course. It features examples of how to implement the methods using WinBUGS - the most-widely used Bayesian analysis software in the world - and R - an open-source statistical software. The book is supported by a Website featuring WinBUGS and R code, data sets, and solutions to exercises.

Author Biography

Simon Jackman is a political scientist by trade but has a tremendous amount of experience in using Bayesian methods for solving problems in the social and political sciences, and teaching Bayesian methods to social science students.

Table of Contents

List of Figures	p. xiii
List of Tables	p. xix
Preface	p. xxi
Acknowledgments	p. xxv
Introduction	p. xxvii
Introducing Bayesian Analysis	p. 1
The foundations of Bayesian inference	p. 3
What is probability?	p. 3
Probability in classical statistics	p. 4
Subjective probability1	p. 5
Subjective probability in Bayesian statistics	p. 7
Bayes theorem, discrete case	p. 8
Bayes theorem, continuous parameter	p. 13
Conjugate priors	p. 15
Bayesian updating with irregular priors	p. 16
Cromwell's Rule	p. 18
Bayesian updating as information accumulation	p. 19
Parameters as random variables, beliefs as distributions	p. 21
Communicating the results of a Bayesian analysis	p. 22
Bayesian point estimation	p. 23
Credible regions	p. 26
Asymptotic properties of posterior distributions	p. 29
Bayesian hypothesis testing	p. 31
Model choice	p. 36
Bayes factors	p. 37
From subjective beliefs to parameters and models	p. 38
Exchangeability	p. 39
Implications and extensions of de Finetti's Representation Theorem	p. 42
Finite exchangeability	p. 43
Exchangeability and prediction	p. 43
Conditional exchangeability and multiparameter models	p. 44
Exchangeability of parameters: hierarchical modeling	p. 45
Historical note	p. 46
Getting started: Bayesian analysis for simple models	p. 49
Learning about probabilities, rates and proportions	p. 49
Conjugate priors for probabilities, rates and proportions	p. 51
Bayes estimates as weighted averages of priors and data	p. 58
Parameterizations and priors	p. 61
The variance of the posterior density	p. 64
Associations between binary variables	p. 67
Learning from counts	p. 73
Predictive inference with count data	p. 78
Learning about a normal mean and variance	p. 80
Variance known	p. 80
Mean and variance unknown	p. 83
Conditionally conjugate prior	p. 92
An improper, reference prior	p. 93
Conflict between likelihood and prior	p. 98
Non-conjugate priors	p. 98
Regression models	p. 99
Bayesian regression analysis	p. 102
Likelihood function	p. 103
Conjugate prior	p. 104
Improper, reference prior	p. 107
Further reading	p. 124
Simulation Based Bayesian Analysis	p. 129
Monte Carlo methods	p. 133
Simulation consistency	p. 134
Inference for functions of parameters	p. 140
Marginalization via Monte Carlo integration	p. 142
Sampling algorithms	p. 153
Inverse-CDF method	p. 153
Importance sampling	p. 156
Accept-reject sampling	p. 159
Adaptive rejection sampling	p. 163
Further reading	p. 167
Markov chains	p. 171
Notation and definitions	p. 172
State space	p. 173
Transition kernel	p. 173
Properties of Markov chains	p. 176
Existence of a stationary distribution, discrete case	p. 177
Existence of a stationary distribution, continuous case	p. 178
Irreducibility	p. 179
Recurrence	p. 182
Invariant measure	p. 184
Reversibility	p. 185
Aperiodicity	p. 186
Convergence of Markov chains	p. 187
Speed of convergence	p. 189
Limit theorems for Markov chains	p. 191
Simulation inefficiency	p. 191
Central limit theorems for Markov chains	p. 195
Further reading	p. 196
Markov chain Monte Carlo	p. 201
Metropolis-Hastings algorithm	p. 201
Theory for the Metropolis-Hastings algorithm	p. 202
Choosing the proposal density	p. 204
Gibbs sampling	p. 214
Theory for the Gibbs sampler	p. 218
Connection to the Metropolis algorithm	p. 221
Deriving conditional densities for the Gibbs sampler: statistical models as conditional independence graphs	p. 225
Pathologies	p. 229
Data augmentation	p. 236
Missing data problems	p. 237
The slice sampler	p. 244
Implementing Markov chain Monte Carlo	p. 251
Software for Markov chain Monte Carlo	p. 251
Assessing convergence and run-length	p. 252
Working with BUGS/JAGS from R	p. 256
Tricks of the trade	p. 261
Thinning	p. 261
Blocking	p. 264
Reparameterization	p. 270
Other examples	p. 272
Further reading	p. 292
Advanced Applications in the Social Sciences	p. 299
Hierarchical Statistical Models	p. 301
Data and parameters that vary by groups: the case for hierarchical modeling	p. 301
Exchangeable parameters generate hierarchical models	p. 305
ÆBorrowing strengthÆ via exchangeability	p. 307
Hierarchical modeling as a 'semi-poolingÆ estimator	p. 307
Hierarchical modeling as a 'shrinkageÆ estimator	p. 308
Computation via Markov chain Monte Carlo	p. 310
ANOVA as a hierarchical model	p. 317
One-way analysis of variance	p. 317
Two-way ANOVA	p. 329
Hierarchical models for longitudinal data	p. 345
Hierarchical models for non-normal data	p. 354
Multi-level models	p. 362
Bayesian analysis of choice making	p. 379
Regression models for binary responses	p. 379
Probit model via data augmentation	p. 380
Probit model via marginal data augmentation	p. 389
Logit model	p. 393
Binomial model for grouped binary data	p. 395
Ordered outcomes	p. 397
Identification	p. 399
Multinomial outcomes	p. 415
Multinomial logit (MNL)	p. 415
Independence of irrelevant alternatives	p. 423
Multinomial probit	p. 424
Bayesian analysis via MCMC	p. 426
Bayesian approaches to measurement	p. 435
Bayesian inference for latent states	p. 435
A formal role for prior information	p. 436
Inference for many parameters	p. 436
Factor analysis	p. 438
Likelihood and prior densities	p. 439
Identification	p. 440
Posterior density	p. 442
Inference over rank orderings of the latent variable	p. 448
Incorporating additional information via hierarchical modeling	p. 449
Item-response models	p. 454
Dynamic measurement models	p. 471
State-space models for [pooling the polls]	p. 473
Bayesian inference	p. 474
Appendices	p. 489
Working with vectors and matrices	p. 491
Probability review	p. 497
Foundations of probability	p. 497
Probability densities and mass functions	p. 498
Probability mass functions for discrete random quantities	p. 501
Probability density functions for continuous random quantities	p. 503
Convergence of sequences of random variables	p. 511
Proofs of selected propositions	p. 513
Products of normal densities	p. 513
Conjugate analysis of normal data	p. 516
Asymptotic normality of the posterior density	p. 533
References	p. 535
Topic index	p. 553
Author index	p. 559
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