Introduction to Applied Bayesian Statistics and Estimation for Social Scientists (eBook)

Preface 7
Acknowledgements 11
Contents 13
List of Figures 18
List of Tables 25
1 Introduction 27
1.1 Outline 29
1.2 A note on programming 31
1.3 Symbols used throughout the book 32
2 Probability Theory and Classical Statistics 34
2.1 Rules of probability 34
2.2 Probability distributions in general 37
2.3 Some important distributions in social science 50
2.4 Classical statistics in social science 58
2.5 Maximum likelihood estimation 60
2.6 Conclusions 69
2.7 Exercises 69
3 Basics of Bayesian Statistics 71
3.1 Bayes’ Theorem for point probabilities 71
3.2 Bayes’ Theorem applied to probability distributions 74
3.3 Bayes’ Theorem with distributions: A voting example 77
3.4 A normal prior–normal likelihood example with s2 known 86
3.5 Some useful prior distributions 92
3.6 Criticism against Bayesian statistics 94
3.7 Conclusions 97
3.8 Exercises 98
4 Modern Model Estimation Part 1: Gibbs Sampling 100
4.1 What Bayesians want and why 100
4.2 The logic of sampling from posterior densities 101
4.3 Two basic sampling methods 103
4.4 Introduction to MCMC sampling 111
4.5 Conclusions 126
4.6 Exercises 128
5 Modern Model Estimation Part 2: Metroplis– Hastings Sampling 129
5.1 A generic MH algorithm 130
5.2 Example: MH sampling when conditional densities are difficult to derive 137
5.3 Example: MH sampling for a conditional density with an unknown form 140
5.4 Extending the bivariate normal example: The full multiparameter model 143
5.5 Conclusions 150
5.6 Exercises 151
6 Evaluating Markov Chain Monte Carlo ( MCMC) Algorithms and Model Fit 153
6.1 Why evaluate MCMC algorithm performance? 154
6.2 Some common problems and solutions 154
6.3 Recognizing poor performance 157
6.4 Evaluating model fit 175
6.5 Formal comparison and combining models 181
6.6 Conclusions 185
6.7 Exercises 185
7 The Linear Regression Model 187
7.1 Development of the linear regression model 187
7.2 Sampling from the posterior distribution for the model parameters 190
7.3 Example: Are people in the South "nicer” than others? 196
7.4 Incorporating missing data 204
7.5 Conclusions 213
7.6 Exercises 214
8 Generalized Linear Models 215
8.1 The dichotomous probit model 217
8.2 The ordinal probit model 239
8.3 Conclusions 250
8.4 Exercises 251
9 Introduction to Hierarchical Models 253
9.1 Hierarchical models in general 254
9.2 Hierarchical linear regression models 262
9.3 A note on fixed versus random effects models and other terminology 286
9.4 Conclusions 290
9.5 Exercises 291
10 Introduction to Multivariate Regression Models 292
10.1 Multivariate linear regression 292
10.2 Multivariate probit models 298
10.3 A multivariate probit model for generating distributions of multistate life tables 324
10.4 Conclusions 336
10.5 Exercises 338
11 Conclusion 339
A Background Mathematics 343
A.1 Summary of calculus 343
A.2 Summary of matrix algebra 350
A.3 Exercises 355
B The Central Limit Theorem, Confidence Intervals, and Hypothesis Tests 357
B.1 A simulation study 357
B.2 Classical inference 358
References 365
Index 372