Bayesian data analysis

Preface
Part I£º Fundamentals of Bayesian Inference
1 Probability and inference
I£®I The three steps of Bayesian data analysis
1£®2 General notation for statistical inference
1£®3 Bayesian inference
1£®4 Discrete examples£º genetics and spell checking
1£®5 Probability as a measure of uncertainty
1£®6 Example£º probabilities from football point spreads
1£®7 Example£º calibration for record linkage
1£®8 Some useful results from probability theory
1£®9 Computation and software
I£®I0 Bayesian inference in applied statistics
i£®Ii Bibliographic note
1£®12 Exercises
2 Single-parameter models
2£®1 Estimating a probability from binomial data
2£®2 Posterior as compromise between data and prior information
2£®3 Summarizing posterior inference
2£®4 Informative prior distributions
2£®5 Normal distribution with known variance
2£®6 Other standard single-parameter models
2£®7 Example£º informative prior distribution for cancer rates
2£®8 Noninformative prior distributions
2£®9 Weakly informative prior distributions
2£®10 Bibliographic note
2£®11 Exercises
3 Introduction to multiparameter models
3£®1 Averaging over 'nuisance parameters'
3£®2 Normal data with a noninformative prior distribution
3£®3 Normal data with a conjugate prior distribution
3£®4 Multinomial model for categorical data
3£®5 Multivariate normal model with known variance
3£®6 Multivariate normal with unknown mean and variance
3£®7 Example£º analysis of a bioassay experiment
3£®8 Summary of elementary modeling and computation
3£®9 Bibliographic note
3£®10 Exercises
4 Asymptotics and connections to non-Bayesian approaches
4£®1 Normal approximations to the posterior distribution
4£®2 Large-sample theory
4£®3 Counterexamples to the theorems
4£®4 Frequency evaluations of Bayesian inferences
4£®5 Bayesian interpretations of other statistical methods
4£®6 Bibliographic note
4£®7 Exercises
5 Hierarchical models
5£®1 Constructing a parameterized prior distribution
5£®2 Exchangeability and hierarchical models
5£®3 Bayesian analysis of conjugate hierarchical models
5£®4 Normal model with exchangeable parameters
5£®5 Example£º parallel experiments in eight schools
5£®6 Hierarchical modeling applied to a meta-analysis
5£®7 Weakly informative priors for variance parameters
5£®8 Bibliographic note
5£®9 Exercises

Part II£º Fundamentals of Bayesian Data Analysis
6 Model checking
6£®1 The place of model checking in applied Bayesian statistics
6£®2 Do the inferences from the model make sense?
6£®3 Posterior predictive checking
6£®4 Graphical posterior predictive checks
6£®5 Model checking for the educational testing example
6£®6 Bibliographic note
6£®7 Exercises
? Evaluating£¬ comparing£¬ and expanding models
7£®1 Measures of predictive accuracy
7£®2 Information criteria and cross-validation
7£®3 Model comparison based on predictive performance
7£®4 Model comparison using Bayes factors
7£®5 Continuous model expansion
7£®6 Implicit assumptions and model expansion£º an example
7£®7 Bibliographic note
7£®8 Exercises
8 Modeling accounting for data collection
8£®1 Bayesian inference requires a model for data collection
8£®2 Data-collection models and ignorability
8£®3 Sample surveys
8£®4 Designed experiments
8£®5 Sensitivity and the role of randomization
8£®6 Observational studies
8£®7 Censoring and truncation
8£®8 Discussion
8£®9 Bibliographic note
8£®10 Exercises
9 Decision analysis
9£®1 Bayesian decision theory in different contexts
9£®2 Using regression predictions£º survey incentives
9£®3 Multistage decision making£º medical screening
9£®4 Hierarchical decision analysis for home radon
9£®5 Personal vs£® institutional decision analysis
9£®6 Bibliographic note
9£®7 Exercises

Part III£º Advanced Computation
10 Introduction to Bayesian computation
10£®1 Numerical integration
10£®2 Distributional approximations
10£®3 Direct simulation and rejection sampling
10£®4 Importance sampling
10£®5 How many simulation draws are needed?
10£®6 Computing environments
10£®7 Debugging Bayesian computing
10£®8 Bibliographic note
10£®9 Exercises
11 Basics of Markov chain simulation
11£®1 Gibbs sampler
11£®2 Metropolis and Metropolis-Hastings algorithms
11£®3 Using Gibbs and Metropolis as building blocks
11£®4 Inference and assessing convergence
11£®5 Effective number of simulation draws
11£®6 Example£º hierarchical normal model
11£®7 Bibliographic note
11£®8 Exercises
12 Computationally efficient Markov chain simulation
12£®1 Efficient Gibbs samplers
12£®2 Efficient Metropolis jumping rules
12£®3 Further extensions to Gibbs and Metropolis
12£®4 Hamiltonian Monte Carlo
12£®5 Hamiltonian Monte Carlo for a hierarchical model
12£®6 Stan£º developing a computing environment
12£®7 Bibliographic note
12£®8 Exercises
13 Modal and distributional approximations
13£®1 Finding posterior modes
13£®2 Boundary-avoiding priors for modal summaries
13£®3 Normal and related mixture approximations
13£®4 Finding marginal posterior modes using EM
13£®5 Conditional and marginal posterior approximations
13£®6 Example£º hierarchical normal model £¨continued£©
13£®7 Variational inference
13£®8 Expectation propagation
13£®9 Other approximations
13£®10 Unknown normalizing factors
13£®11 Bibliographic note
13£®12 Exercises

Part IV£º Regression Models
14 Introduction to regression models
14£®1 Conditional modeling
14£®2 Bayesian analysis of classical regression
14£®3 Regression for causal inference£º incumbency and voting
14£®4 Goals of regression analysis
14£®5 Assembling the matrix of explanatory variables
14£®6 Regularization and dimension reduction
14£®7 Unequal variances and correlations
14£®8 Including numerical prior information
14£®9 Bibliographic note
14£®10 Exercises
15 Hierarchical linear models
15£®1 Regression coefficients exchangeable in batches
15£®2 Example£º forecasting U£®S£® presidential elections
15£®3 Interpreting a normal prior distribution as extra data
15£®4 Varying intercepts and slopes
15£®5 Computation£º batching and transformation
15£®6 Analysis of variance and the batching of coefficients
15£®7 Hierarchical models for batches of variance components
15£®8 Bibliographic note
15£®9 Exercises
16 Generalized linear models
16£®1 Standard generalized linear model likelihoods
16£®2 Working with generalized linear models
16£®3 Weakly informative priors for logistic regression
16£®4 Overdispersed Poisson regression for police stops
16£®5 State-level opinons from national polls
16£®6 Models for multivariate and multinomial responses
16£®7 Loglinear models for multivariate discrete data
16£®8 Bibliographic note
16£®9 Exercises
17 Models for robust inference
17£®1 Aspects of robustness
17£®2 Overdispersed versions of standard models
17£®3 Posterior inference and computation
17£®4 Robust inference for the eight schools
17£®5 Robust regression using t-distributed errors
17£®6 Bibliographic note
17£®7 Exercises
18 Models for missing data
18£®1 Notation
18£®2 Multiple imputation
18£®3 Missing data in the multivariate normal and t models
18£®4 Example£º multiple imputation for a series of polls
18£®5 Missing values with counted data
18£®6 Example£º an opinion poll in Slovenia
18£®7 Bibliographic note
18£®8 Exercises

Part V£º Nonlinear and Nonparametric Models
19 Parametric nonlinear models
19£®1 Example£º serial dilution assay
19£®2 Example£º population toxicokinetics
19£®3 Bibliographic note
19£®4 Exercises
20 Basis function models
20£®1 Splines and weighted sums of basis functions
20£®2 Basis selection and shrinkage of coefficients
20£®3 Non-normal models and regression surfaces
20£®4 Bibliographic note
20£®5 Exercises
21 Gaussian process models
21£®1 Gaussian process regression
21£®2 Example£º birthdays and birthdates
21£®3 Latent Gaussian process models
21£®4 Functional data analysis
21£®5 Density estimation and regression
21£®6 Bibliographic note
21£®7 Exercises
22 Finite mixture models
22£®1 Setting up and interpreting mixture models
22£®2 Example£º reaction times and schizophrenia
22£®3 Label switching and posterior computation
22£®4 Unspecified number of mixture components
22£®5 Mixture models for classification and regression
22£®6 Bibliographic note
22£®7 Exercises
23 Dirichlet process models
23£®1 Bayesian histograms
23£®2 Dirichlet process prior distributions
23£®3 Dirichlet process mixtures
23£®4 Beyond density estimation
23£®5 Hierarchical dependence
23£®6 Density regression
23£®7 Bibliographic note
23£®8 Exercises

Appendixes
A Standard probability distributions
A£®1 Continuous distributions
A£®2 Discrete distributions
A£®3 Bibliographic note
B Outline of proofs of limit theorems
B£®1 Bibliographic note
C Computation in R and Stan
C£®1 Getting started with R and Stan
C£®2 Fitting a hierarchical model in Stan
C£®3 Direct simulation£¬ Gibbs£¬ and Metropolis in R
C£®4 Programming Hamiltonian Monte Carlo in R
C£®5 Further comments on computation
C£®6 Bibliographic note
References
Author Index
Subject Index