Contents. Part I: Fundamentals of Bayesian Inference 1

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1 Contents Preface xiii Part I: Fundamentals of Bayesian Inference 1 1 Probability and inference The three steps of Bayesian data analysis General notation for statistical inference Bayesian inference Discrete probability examples: genetics and spell checking Probability as a measure of uncertainty Example of probability assignment: football point spreads Example: estimating the accuracy of record linkage Some useful results from probability theory Computation and software Bayesian inference in applied statistics Bibliographic note Exercises 27 2 Single-parameter models Estimating a probability from binomial data Posterior as compromise between data and prior information Summarizing posterior inference Informative prior distributions Estimating a normal mean with known variance Other standard single-parameter models Example: informative prior distribution for cancer rates Noninformative prior distributions Weakly informative prior distributions Bibliographic note Exercises 57 3 Introduction to multiparameter models Averaging over nuisance parameters Normal data with a noninformative prior distribution Normal data with a conjugate prior distribution Multinomial model for categorical data Multivariate normal model with known variance Multivariate normal with unknown mean and variance Example: analysis of a bioassay experiment Summary of elementary modeling and computation Bibliographic note Exercises 79 vii

2 viii CONTENTS 4 Asymptotics and connections to non-bayesian approaches Normal approximations to the posterior distribution Large-sample theory Counterexamples to the theorems Frequency evaluations of Bayesian inferences Bayesian interpretations of other statistical methods Bibliographic note Exercises 98 5 Hierarchical models Constructing a parameterized prior distribution Exchangeability and setting up hierarchical models Fully Bayesian analysis of conjugate hierarchical models Estimating exchangeable parameters from a normal model Example: parallel experiments in eight schools Hierarchical modeling applied to a meta-analysis Weakly informative priors for hierarchical variance parameters Bibliographic note Exercises 134 Part II: Fundamentals of Bayesian Data Analysis Model checking The place of model checking in applied Bayesian statistics Do the inferences from the model make sense? Posterior predictive checking Graphical posterior predictive checks Model checking for the educational testing example Bibliographic note Exercises Evaluating, comparing, and expanding models Measures of predictive accuracy Information criteria and cross-validation Model comparison based on predictive performance Model comparison using Bayes factors Continuous model expansion Implicit assumptions and model expansion: an example Bibliographic note Exercises Modeling accounting for data collection Bayesian inference requires a model for data collection Data-collection models and ignorability Sample surveys Designed experiments Sensitivity and the role of randomization Observational studies Censoring and truncation Discussion Bibliographic note Exercises 230

3 CONTENTS 9 Decision analysis Bayesian decision theory in different contexts Using regression predictions: incentives for telephone surveys Multistage decision making: medical screening Hierarchical decision analysis for radon measurement Personal vs. institutional decision analysis Bibliographic note Exercises 257 Part III: Advanced Computation Introduction to Bayesian computation Numerical integration Distributional approximations Direct simulation and rejection sampling Importance sampling How many simulation draws are needed? Computing environments Debugging Bayesian computing Bibliographic note Exercises Basics of Markov chain simulation Gibbs sampler Metropolis and Metropolis-Hastings algorithms Using Gibbs and Metropolis as building blocks Inference and assessing convergence Effective number of simulation draws Example: hierarchical normal model Bibliographic note Exercises Computationally efficient Markov chain simulation Efficient Gibbs samplers Efficient Metropolis jumping rules Further extensions to Gibbs and Metropolis Hamiltonian Monte Carlo Hamiltonian dynamics for a simple hierarchical model Stan: developing a computing environment Bibliographic note Exercises Modal and distributional approximations Finding posterior modes Boundary-avoiding priors for modal summaries Normal and related mixture approximations Finding marginal posterior modes using EM Approximating conditional and marginal posterior densities Example: hierarchical normal model (continued) Variational inference Expectation propagation Other approximations 343 ix

4 x CONTENTS Unknown normalizing factors Bibliographic note Exercises 349 Part IV: Regression Models Introduction to regression models Conditional modeling Bayesian analysis of the classical regression model Regression for causal inference: incumbency in congressional elections Goals of regression analysis Assembling the matrix of explanatory variables Regularization and dimension reduction for multiple predictors Unequal variances and correlations Including numerical prior information Bibliographic note Exercises Hierarchical linear models Regression coefficients exchangeable in batches Example: forecasting U.S. presidential elections Interpreting a normal prior distribution as additional data Varying intercepts and slopes Computation: batching and transformation Analysis of variance and the batching of coefficients Hierarchical models for batches of variance components Bibliographic note Exercises Generalized linear models Standard generalized linear model likelihoods Working with generalized linear models Weakly informative priors for logistic regression Example: hierarchical Poisson regression for police stops Example: hierarchical logistic regression for political opinions Models for multivariate and multinomial responses Loglinear models for multivariate discrete data Bibliographic note Exercises Models for robust inference Aspects of robustness Overdispersed versions of standard probability models Posterior inference and computation Robust inference and sensitivity analysis for the eight schools Robust regression using t-distributed errors Bibliographic note Exercises 446

5 CONTENTS 18 Models for missing data Notation Multiple imputation Missing data in the multivariate normal and t models Example: multiple imputation for a series of polls Missing values with counted data Example: an opinion poll in Slovenia Bibliographic note Exercises 467 Part V: Nonlinear and Nonparametric Models Parametric nonlinear models Example: serial dilution assay Example: population toxicokinetics Bibliographic note Exercises Basis function models Splines and weighted sums of basis functions Basis selection and shrinkage of coefficients Non-normal models and multivariate regression surfaces Bibliographic note Exercises Gaussian process models Gaussian process regression Example: birthdays and birthdates Latent Gaussian process models Functional data analysis Density estimation and regression Bibliographic note Exercises Finite mixture models Setting up and interpreting mixture models Example: reaction times and schizophrenia Label switching and posterior computation Unspecified number of mixture components Mixture models for classification and regression Bibliographic note Exercises Dirichlet process models Bayesian histograms Dirichlet process prior distributions Dirichlet process mixtures Beyond density estimation Hierarchical dependence Density regression Bibliographic note Exercises 573 xi

6 xii CONTENTS A Standard probability distributions 575 A.1 Continuous distributions 575 A.2 Discrete distributions 583 A.3 Bibliographic note 584 B Outline of proofs of limit theorems 585 B.1 Bibliographic note 588 C Computation in R and Stan 589 C.1 Getting started with R and Stan 589 C.2 Fitting a hierarchical model in Stan 589 C.3 Direct simulation, Gibbs, and Metropolis in R 594 C.4 Programming Hamiltonian Monte Carlo in R 601 C.5 Further comments on computation 605 C.6 Bibliographic note 606 References 607 Author Index 641 Subject Index 649

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