Efnisyfirlit

• Cover
• Half-Title Page
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Part I Fundamentals of Bayesian Inference
• Chapter 1 Probability and inference
• 1.1 The three steps of Bayesian data analysis
• 1.2 General notation for statistical inference
• Parameters, data, and predictions
• Observational units and variables
• Exchangeability
• Explanatory variables
• Hierarchical modeling
• 1.3 Bayesian inference
• Bayes’ rule
• Prediction
• Likelihood
• Likelihood and odds ratios
• 1.4 Discrete examples: genetics and spell checking
• Inference about a genetic status
• Spelling correction
• 1.5 Probability as a measure of uncertainty
• Subjectivity and objectivity
• 1.6 Example: probabilities from football point spreads
• Football point spreads and game outcomes
• Assigning probabilities based on observed frequencies
• A parametric model for the difference between outcome and point spread
• Assigning probabilities using the parametric model
• 1.7 Example: calibration for record linkage
• Existing methods for assigning scores to potential matches
• Estimating match probabilities empirically
• External validation of the probabilities using test data
• 1.8 Some useful results from probability theory
• Modeling using conditional probability
• Means and variances of conditional distributions
• Transformation of variables
• 1.9 Computation and software
• Summarizing inferences by simulation
• Sampling using the inverse cumulative distribution function
• Simulation of posterior and posterior predictive quantities
• 1.10 Bayesian inference in applied statistics
• 1.11 Bibliographic note
• 1.12 Exercises
• Chapter 2 Single-parameter models
• 2.1 Estimating a probability from binomial data
• Prediction
• 2.2 Posterior as compromise between data and prior information
• 2.3 Summarizing posterior inference
• Posterior quantiles and intervals
• 2.4 Informative prior distributions
• Binomial example with different prior distributions
• Conjugate prior distributions
• Nonconjugate prior distributions
• Conjugate prior distributions, exponential families, and sufficient statistics
• 2.5 Normal distribution with known variance
• Likelihood of one data point
• Conjugate prior and posterior distributions
• Posterior predictive distribution
• Normal model with multiple observations
• 2.6 Other standard single-parameter models
• Normal distribution with known mean but unknown variance
• Poisson model
• Poisson model parameterized in terms of rate and exposure
• Exponential model
• 2.7 Example: informative prior distribution for cancer rates
• A puzzling pattern in a map
• Bayesian inference for the cancer death rates
• Relative importance of the local data and the prior distribution
• Constructing a prior distribution
• 2.8 Noninformative prior distributions
• Proper and improper prior distributions
• Improper prior distributions can lead to proper posterior distributions
• Jeffreys’ invariance principle
• Various noninformative prior distributions for the binomial parameter
• Pivotal quantities
• Difficulties with noninformative prior distributions
• 2.9 Weakly informative prior distributions
• Constructing a weakly informative prior distribution
• 2.10 Bibliographic note
• 2.11 Exercises
• Chapter 3 Introduction to multiparameter models
• 3.1 Averaging over ‘nuisance parameters’
• 3.2 Normal data with a noninformative prior distribution
• A noninformative prior distribution
• The joint posterior distribution, p
• The conditional posterior distribution, p
• The marginal posterior distribution, p
• Sampling from the joint posterior distribution
• Analytic form of the marginal posterior distribution of µ
• Posterior predictive distribution for a future observation
• 3.3 Normal data with a conjugate prior distribution
• A family of conjugate prior distributions
• The joint posterior distribution, p
• The conditional posterior distribution, p
• The marginal posterior distribution, p
• Sampling from the joint posterior distribution
• Analytic form of the marginal posterior distribution
• 3.4 Multinomial model for categorical data
• 3.5 Multivariate normal model with known variance
• Multivariate normal likelihood
• Conjugate analysis
• 3.6 Multivariate normal with unknown mean and variance
• Conjugate inverse-Wishart family of prior distributions
• Different noninformative prior distributions
• Scaled inverse-Wishart model
• 3.7 Example: analysis of a bioassay experiment
• The scientific problem and the data
• Modeling the dose-response relation
• The likelihood
• The prior distribution
• A rough estimate of the parameters
• Obtaining a contour plot of the joint posterior density
• Sampling from the joint posterior distribution
• The posterior distribution of the LD50
• 3.8 Summary of elementary modeling and computation
• 3.9 Bibliographic note
• 3.10 Exercises
• Chapter 4 Asymptotics and connections to non-Bayesian approaches
• 4.1 Normal approximations to the posterior distribution
• Normal approximation to the joint posterior distribution
• Interpretation of the posterior density function relative to its maximum
• Summarizing posterior distributions by point estimates and standard errors
• Data reduction and summary statistics
• Lower-dimensional normal approximations
• 4.2 Large-sample theory
• Notation and mathematical setup
• Asymptotic normality and consistency
• Likelihood dominating the prior distribution
• 4.3 Counterexamples to the theorems
• 4.4 Frequency evaluations of Bayesian inferences
• Large-sample correspondence
• Point estimation, consistency, and efficiency
• Confidence coverage
• 4.5 Bayesian interpretations of other statistical methods
• Maximum likelihood and other point estimates
• Unbiased estimates
• Confidence intervals
• Hypothesis testing
• Multiple comparisons and multilevel modeling
• Nonparametric methods, permutation tests, jackknife, bootstrap
• 4.6 Bibliographic note
• 4.7 Exercises
• Chapter 5 Hierarchical models
• 5.1 Constructing a parameterized prior distribution
• Analyzing a single experiment in the context of historical data
• Logic of combining information
• 5.2 Exchangeability and hierarchical models
• Exchangeability
• Exchangeability when additional information is available on the units
• Objections to exchangeable models
• The full Bayesian treatment of the hierarchical model
• The hyperprior distribution
• Posterior predictive distributions
• 5.3 Bayesian analysis of conjugate hierarchical models
• Analytic derivation of conditional and marginal distributions
• Drawing simulations from the posterior distribution
• Application to the model for rat tumors
• 5.4 Normal model with exchangeable parameters
• The data structure
• Constructing a prior distribution from pragmatic considerations
• The hierarchical model
• The joint posterior distribution
• The conditional posterior distribution of the normal means, given the hyperparameters
• The marginal posterior distribution of the hyperparameters
• Computation
• Posterior predictive distributions
• Difficulty with a natural non-Bayesian estimate of the hyperparameters
• 5.5 Example: parallel experiments in eight schools
• Inferences based on nonhierarchical models and their problems
• Posterior simulation under the hierarchical model
• Results
• Discussion
• 5.6 Hierarchical modeling applied to a meta-analysis
• Defining a parameter for each study
• A normal approximation to the likelihood
• Goals of inference in meta-analysis
• What if exchangeability is inappropriate?
• A hierarchical normal model
• Results of the analysis and comparison to simpler methods
• 5.7 Weakly informative priors for variance parameters
• Concepts relating to the choice of prior distribution
• Classes of noninformative and weakly informative prior distributions for hierarchical variance parameters
• Application to the 8-schools example
• Weakly informative prior distribution for the 3-schools problem
• 5.8 Bibliographic note
• 5.9 Exercises
• Part II Fundamentals of Bayesian Data Analysis
• Chapter 6 Model checking
• 6.1 The place of model checking in applied Bayesian statistics
• Sensitivity analysis and model improvement
• Judging model flaws by their practical implications
• 6.2 Do the inferences from the model make sense?
• External validation
• Choices in defining the predictive quantities
• 6.3 Posterior predictive checking
• Notation for replications
• Test quantities
• Tail-area probabilities
• Choosing test quantities
• Multiple comparisons
• Interpreting posterior predictive p-values
• Limitations of posterior tests
• P-values and u-values
• Model checking and the likelihood principle
• Marginal predictive checks
• 6.4 Graphical posterior predictive checks
• Direct data display
• Displaying summary statistics or inferences
• Residual plots and binned residual plots
• General interpretation of graphs as model checks
• 6.5 Model checking for the educational testing example
• Assumptions of the model
• Comparing posterior inferences to substantive knowledge
• Posterior predictive checking
• Sensitivity analysis
• 6.6 Bibliographic note
• 6.7 Exercises
• Chapter 7 Evaluating, comparing, and expanding models
• 7.1 Measures of predictive accuracy
• Predictive accuracy for a single data point
• Averaging over the distribution of future data
• Evaluating predictive accuracy for a fitted model
• Choices in defining the likelihood and predictive quantities
• 7.2 Information criteria and cross-validation
• Estimating out-of-sample predictive accuracy using available data
• Log predictive density asymptotically, or for normal linear models
• Akaike information criterion (AIC)
• Deviance information criterion (DIC) and effective number of parameters
• Watanabe-Akaike or widely applicable information criterion (WAIC)
• Effective number of parameters as a random variable
• Bayesian’ information criterion (BIC)
• Leave-one-out cross-validation
• Comparing different estimates of out-of-sample prediction accuracy
• 7.3 Model comparison based on predictive performance
• Evaluating predictive error comparisons
• Bias induced by model selection
• Challenges
• 7.4 Model comparison using Bayes factors
• 7.5 Continuous model expansion
• Sensitivity analysis
• Adding parameters to a model
• Accounting for model choice in data analysis
• Selection of predictors and combining information
• Alternative model formulations
• Practical advice for model checking and expansion
• 7.6 Implicit assumptions and model expansion: an example
• 7.7 Bibliographic note
• 7.8 Exercises
• Chapter 8 Modeling accounting for data collection
• 8.1 Bayesian inference requires a model for data collection
• Generality of the observed- and missing-data paradigm
• 8.2 Data-collection models and ignorability
• Notation for observed and missing data
• Stability assumption
• Fully observed covariates
• Data model, inclusion model, and complete and observed data likelihood
• Joint posterior distribution of parameters θ from the sampling model and I from the missing-data model
• Finite-population and superpopulation inference
• Ignorability
• ‘Missing at random’ and ‘distinct parameters’
• Ignorability and Bayesian inference under different data-collection schemes
• Propensity scores
• Unintentional missing data
• 8.3 Sample surveys
• Simple random sampling of a finite population
• Stratified sampling
• Cluster sampling
• Unequal probabilities of selection
• 8.4 Designed experiments
• Completely randomized experiments
• Randomized blocks, Latin squares, etc.
• Sequential designs
• Including additional predictors beyond the minimally adequate summary
• 8.5 Sensitivity and the role of randomization
• Complete randomization
• Randomization given covariates
• Designs that ‘cheat’
• Bayesian analysis of nonrandomized studies
• 8.6 Observational studies
• Comparison to experiments
• Bayesian inference for observational studies
• Causal inference and principal stratification
• Complier average causal effects and instrumental variables
• Bayesian causal inference with noncompliance
• 8.7 Censoring and truncation
• 1. Data missing completely at random
• 2. Data missing completely at random with unknown probability of missingness
• 3. Censored data
• 4. Censored data with unknown censoring point
• 5. Truncated data
• 6. Truncated data with unknown truncation point
• More complicated patterns of missing data
• 8.8 Discussion
• 8.9 Bibliographic note
• 8.10 Exercises
• Chapter 9 Decision analysis
• 9.1 Bayesian decision theory in different contexts
• Bayesian inference and decision trees
• Summarizing inference and model selection
• 9.2 Using regression predictions: survey incentives
• Background on survey incentives
• Data from 39 experiments
• Setting up a Bayesian meta-analysis
• Inferences from the model
• Inferences about costs and response rates for the Social Indicators Survey
• Loose ends
• 9.3 Multistage decision making: medical screening
• Example with a single decision point
• Adding a second decision point
• 9.4 Hierarchical decision analysis for home radon
• Background
• The individual decision problem
• Decision-making under certainty
• Bayesian inference for county radon levels
• Hierarchical model.
• Inferences.
• Bayesian inference for the radon level in an individual house
• Decision analysis for individual homeowners
• Deciding whether to remediate given a measurement.
• Aggregate consequences of individual decisions
• Applying the recommended decision strategy to the entire country.
• Evaluation of different decision strategies.
• 9.5 Personal vs. institutional decision analysis
• 9.6 Bibliographic note
• 9.7 Exercises
• Part III Advanced Computation
• Chapter 10 Introduction to Bayesian computation
• Normalized and unnormalized densities
• Log densities
• 10.1 Numerical integration
• Simulation methods
• Deterministic methods
• 10.2 Distributional approximations
• Crude estimation by ignoring some information
• 10.3 Direct simulation and rejection sampling
• Direct approximation by calculating at a grid of points
• Simulating from predictive distributions
• Rejection sampling
• 10.4 Importance sampling
• Accuracy and efficiency of importance sampling estimates
• Importance resampling
• Uses ofimportance sampling in Bayesian computation
• 10.5 How many simulation draws are needed?
• 10.6 Computing environments
• The Bugs family of programs
• Stan
• Other Bayesian software
• 10.7 Debugging Bayesian computing
• Debugging using fake data
• Model checking and convergence checking as debugging
• 10.8 Bibliographic note
• 10.9 Exercises
• Chapter 11 Basics of Markov chain simulation
• 11.1 Gibbs sampler
• 11.2 Metropolis and Metropolis-Hastings algorithms
• The Metropolis algorithm
• Relation to optimization
• Why does the Metropolis algorithm work?
• The Metropolis-Hastings algorithm
• Relation between the jumping rule and efficiency of simulations
• 11.3 Using Gibbs and Metropolis as building blocks
• Interpretation of the Gibbs sampler as a special case of the Metropolis-Hastings algorithm
• Gibbs sampler with approximations
• 11.4 Inference and assessing convergence
• Difficulties of inference from iterative simulation
• Discarding early iterations of the simulation runs
• Dependence of the iterations in each sequence
• Multiple sequences with overdispersed starting points
• Monitoring scalar estimands
• Challenges ofmonitoring convergence: mixing and stationarity
• Splitting each saved sequence into two parts
• Assessing mixing using between- and within-sequence variances
• 11.5 Effective number of simulation draws
• Bounded or long-tailed distributions
• Stopping the simulations
• 11.6 Example: hierarchical normal model
• Data from a small experiment
• The model
• Starting points
• Gibbs sampler
• Numerical results with the coagulation data
• The Metropolis algorithm
• Metropolis results with the coagulation data
• 11.7 Bibliographic note
• 11.8 Exercises
• Chapter 12 Computationally efficient Markov chain simulation
• 12.1 Efficient Gibbs samplers
• Transformations and reparameterization
• Auxiliary variables
• Parameter expansion
• 12.2 Efficient Metropolis jumping rules
• Adaptive algorithms
• 12.3 Further extensions to Gibbs and Metropolis
• Slice sampling
• Reversible jump sampling for moving between spaces of differing dimensions
• Simulated tempering and parallel tempering
• Particle filtering, weighting, and genetic algorithms
• 12.4 Hamiltonian Monte Carlo
• The momentum distribution, p(Ï•)
• The three steps of an HMC iteration
• Restricted parameters and areas of zero posterior density
• Setting the tuning parameters
• Varying the tuning parameters during the run
• Locally adaptive HMC
• Combining HMC with Gibbs sampling
• 12.5 Hamiltonian Monte Carlo for a hierarchical model
• Transforming to log Ï„
• 12.6 Stan: developing a computing environment
• Entering the data and model
• Setting tuning parameters in the warm-up phase
• No-U-turn sampler
• Inferences and postprocessing
• 12.7 Bibliographic note
• 12.8 Exercises
• Chapter 13 Modal and distributional approximations
• 13.1 Finding posterior modes
• Conditional maximization
• Newton’s method
• Quasi-Newton and conjugate gradient methods
• Numerical computation of derivatives
• 13.2 Boundary-avoiding priors for modal summaries
• Posterior modes on the boundary of parameter space
• Zero-avoiding prior distribution for a group-level variance parameter
• Boundary-avoiding prior distribution for a correlation parameter
• Degeneracy-avoiding prior distribution for a covariance matrix
• 13.3 Normal and related mixture approximations
• Fitting multivariate normal densities based on the curvature at the modes
• Laplace’s method for analytic approximation of integrals
• Mixture approximation for multimodal densities
• Multivariate t approximation instead of the normal
• Sampling from the approximate posterior distributions
• 13.4 Finding marginal posterior modes using EM
• Derivation of the EM and generalized EM algorithms
• Implementation of the EM algorithm
• Example. Normal distribution with unknown mean and variance and partially conjugate prior distribution
• Extensions of the EM algorithm
• Supplemented EM and ECM algorithms
• Parameter-expanded EM (PX-EM)
• 13.5 Conditional and marginal posterior approximations
• Approximating the conditional posterior density, p(γ|Ï•, y)
• Approximating the marginal posterior density, p(Ï•|y), using an analytic approximation to p(γ|Ï•, y)
• 13.6 Example: hierarchical normal model (continued)
• Crude initial parameter estimates
• Conditional maximization to find the joint mode of p(θ, μ, log σ, log Ï„|y)
• Factoring into conditional and marginal posterior densities
• Finding the marginal posterior mode of p(μ, log σ, log Ï„|y) using EM
• Constructing an approximation to the joint posterior distribution
• Comparison to other computations
• 13.7 Variational inference
• Minimization of Kullback-Leibler divergence
• The class of approximate distributions
• The variational Bayes algorithm
• Example. Educational testing experiments
• Proof that each step of variational Bayes decreases the Kullback-Leibler divergence
• Model checking
• Variational Bayes followed by importance sampling or particle filtering
• EM as a special case of variational Bayes
• More general forms of variational Bayes
• 13.8 Expectation propagation
• Expectation propagation for logistic regression
• Extensions of expectation propagation
• 13.9 Other approximations
• Integrated nested Laplace approximation (INLA)
• Central composite design integration (CCD)
• Approximate Bayesian computation (ABC)
• 13.10 Unknown normalizing factors
• Posterior computations involving an unknown normalizing factor
• Bridge and path sampling
• 13.11 Bibliographic note
• 13.12 Exercises
• Part IV: Regression Models
• Chapter 14 Introduction to regression models
• 14.1 Conditional modeling
• Notation
• Formal Bayesian justification of conditional modeling
• 14.2 Bayesian analysis of classical regression
• Notation and basic model
• The standard noninformative prior distribution
• The posterior distribution
• Sampling from the posterior distribution
• The posterior predictive distribution for new data
• Model checking and robustness
• 14.3 Regression for causal inference: incumbency and voting
• Units of analysis, outcome, and treatment variables
• Setting up control variables so that data collection is approximately ignorable
• Implicit ignorability assumption
• Transformations
• Posterior inference
• Model checking and sensitivity analysis
• 14.4 Goals of regression analysis
• Predicting y from x for new observations
• Causal inference
• Do not control for post-treatment variables when estimating the causal effect.
• 14.5 Assembling the matrix of explanatory variables
• Identifiability and collinearity
• Nonlinear relations
• Indicator variables
• Categorical and continuous variables
• Interactions
• Controlling for irrelevant variables
• Selecting the explanatory variables
• 14.6 Regularization and dimension reduction
• Lasso
• 14.7 Unequal variances and correlations
• Modeling unequal variances and correlated errors
• Bayesian regression with a known covariance matrix
• Bayesian regression with unknown covariance matrix
• Variance matrix known up to a scalar factor
• Weighted linear regression
• Parametric models for unequal variances
• Estimating several unknown variance parameters
• General models for unequal variances
• 14.8 Including numerical prior information
• Coding prior information on a regression parameter as an extra ‘data point’
• Interpreting prior information on several coefficients as several additional ‘data points’
• Prior information about variance parameters
• Prior information in the form of inequality constraints on parameters
• 14.9 Bibliographic note
• 14.10 Exercises
• chapter 15 Hierarchical linear models
• 15.1 Regression coefficients exchangeable in batches
• Simple varying-coefficients model
• Intraclass correlation
• Mixed-effects model
• Several sets of varying coefficients
• Exchangeability
• 15.2 Example: forecasting U.S. presidential elections
• Unit of analysis and outcome variable
• Preliminary graphical analysis
• Fitting a preliminary, nonhierarchical, regression model
• Checking the preliminary regression model
• Extending to a varying-coefficients model
• Forecasting
• Posterior inference
• Reasons for using a hierarchical model
• 15.3 Interpreting a normal prior distribution as extra data
• Interpretation as a single linear regression
• More than one way to set up a model
• 15.4 Varying intercepts and slopes
• Inverse-Wishart model
• Scaled inverse-Wishart model
• Predicting business school grades for different groups of students
• 15.5 Computation: batching and transformation
• Gibbs sampler, one batch at a time
• All-at-once Gibbs sampler
• Parameter expansion
• Transformations for HMC
• 15.6 Analysis of variance and the batching of coefficients
• Notation and model
• Computation
• Finite-population and superpopulation standard deviations
• 15.7 Hierarchical models for batches of variance components
• Superpopulation and finite-population standard deviations
• 15.8 Bibliographic note
• 15.9 Exercises
• Chapter 16 Generalized linear models
• 16.1 Standard generalized linear model likelihoods
• Continuous data
• Poisson
• Binomial
• Overdispersed models
• 16.2 Working with generalized linear models
• Canonical link functions
• Offsets
• Interpreting the model parameters
• Understanding discrete-data models in terms of latent continuous data
• Bayesian nonhierarchical and hierarchical generalized linear models
• Noninformative prior distributions on β
• Conjugate prior distributions
• Nonconjugate prior distributions
• Hierarchical models
• Normal approximation to the likelihood
• Approximate normal posterior distribution
• More advanced computational methods
• 16.3 Weakly informative priors for logistic regression
• The problem of separation
• Computation with a specified normal prior distribution
• Approximate EM algorithm with a t prior distribution
• Default prior distribution for logistic regression coefficients
• Other models
• Bioassay example
• Weakly informative default prior compared to actual prior information
• 16.4 Overdispersed Poisson regression for police stops
• Aggregate data
• Regression analysis to control for precincts
• 16.5 State-level opinons from national polls
• 16.6 Models for multivariate and multinomial responses
• Multivariate outcomes
• Extension of the logistic link
• Special methods for ordered categories
• Using the Poisson model for multinomial responses
• 16.7 Loglinear models for multivariate discrete data
• The Poisson or multinomial likelihood
• Setting up the matrix of explanatory variables
• Prior distributions
• Computation
• 16.8 Bibliographic note
• 16.9 Exercises
• Chapter 17 Models for robust inference
• 17.1 Aspects of robustness
• Robustness of inferences to outliers
• Sensitivity analysis
• 17.2 Overdispersed versions of standard models
• The t distribution in place of the normal
• Negative binomial alternative to Poisson
• Beta-binomial alternative to binomial
• The t distribution alternative to logistic and probit regression
• Why ever use a nonrobust model?
• 17.3 Posterior inference and computation
• Notation for robust model as expansion of a simpler model
• Gibbs sampling using the mixture formulation
• Sampling from the posterior predictive distribution for new data
• Computing the marginal posterior distribution of the hyperparameters by importance weighting
• Approximating the robust posterior distributions by importance resampling
• 17.4 Robust inference for the eight schools
• Robust inference based on a t4 population distribution
• Sensitivity analysis based on tν distributions with varying values of ν
• Treating ν as an unknown parameter
• Discussion
• 17.5 Robust regression using t-distributed errors
• Iterative weighted linear regression and the EM algorithm
• Gibbs sampler and Metropolis algorithm
• 17.6 Bibliographic note
• 17.7 Exercises
• Chapter 18 Models for missing data
• 18.1 Notation
• 18.2 Multiple imputation
• Computation using EM and data augmentation
• Inference with multiple imputations
• 18.3 Missing data in the multivariate normal and t models
• Finding posterior modes using EM
• Drawing samples from the posterior distribution of the model parameters
• Extending the normal model using the t distribution
• Nonignorable models
• 18.4 Example: multiple imputation for a series of polls
• Background
• Multivariate missing-data framework
• A hierarchical model for multiple surveys
• Use of the continuous model for discrete responses
• Computation
• Accounting for survey design and weights
• Results
• 18.5 Missing values with counted data
• 18.6 Example: an opinion poll in Slovenia
• Crude estimates
• The likelihood and prior distribution
• The model for the ‘missing data’
• Using the EM algorithm to find the posterior mode of θ
• Using SEM to estimate the posterior variance matrix and obtain a normal approximation
• Multiple imputation using data augmentation
• Posterior inference for the estimand of interest
• 18.7 Bibliographic note
• 18.8 Exercises
• Part V: Nonlinear and Nonparametric Models
• Chapter 19 Parametric nonlinear models
• 19.1 Example: serial dilution assay
• Laboratory data
• The model
• Inference
• Comparison to existing estimates
• 19.2 Example: population toxicokinetics
• Background
• Toxicokinetic model
• Difficulties in estimation and the role of prior information
• Measurement model
• Population model for parameters
• Prior information
• Joint posterior distribution for the hierarchical model
• Computation
• Inference for quantities of interest
• Evaluating the fit of the model
• Use of a complex model with an informative prior distribution
• 19.3 Bibliographic note
• 19.4 Exercises
• Chapter 20 Basis function models
• 20.1 Splines and weighted sums of basis functions
• 20.2 Basis selection and shrinkage of coefficients
• Shrinkage priors
• 20.3 Non-normal models and regression surfaces
• Other error distributions
• Multivariate regression surfaces
• 20.4 Bibliographic note
• 20.5 Exercises
• Chapter 21 Gaussian process models
• 21.1 Gaussian process regression
• Covariance functions
• Inference
• Covariance function approximations
• Marginal likelihood and posterior
• 21.2 Example: birthdays and birthdates
• Decomposing the time series as a sum of Gaussian processes
• An improved model
• 21.3 Latent Gaussian process models
• 21.4 Functional data analysis
• 21.5 Density estimation and regression
• Density estimation
• Density regression
• Latent-variable regression
• 21.6 Bibliographic note
• 21.7 Exercises
• Chapter 22 Finite mixture models
• 22.1 Setting up and interpreting mixture models
• Finite mixtures
• Continuous mixtures
• Identifiability of the mixture likelihood
• Prior distribution
• Ensuring a proper posterior distribution
• Number of mixture components
• More general formulation
• Mixtures as true models or approximating distributions
• Basics of computation for mixture models
• Crude estimates
• Posterior modes and marginal approximations using EM and variational Bayes
• Posterior simulation using the Gibbs sampler
• Posterior inference
• 22.2 Example: reaction times and schizophrenia
• Initial statistical model
• Crude estimate of the parameters
• Finding the modes of the posterior distribution using ECM
• Normal and t approximations at the major mode
• Simulation using the Gibbs sampler
• Possible difficulties at a degenerate point
• Inference from the iterative simulations
• Posterior predictive distributions
• Checking the model
• Expanding the model
• Checking the new model
• 22.3 Label switching and posterior computation
• 22.4 Unspecified number of mixture components
• 22.5 Mixture models for classification and regression
• Classification
• Regression
• 22.6 Bibliographic note
• 22.7 Exercises
• Chapter 23 Dirichlet process models
• 23.1 Bayesian histograms
• 23.2 Dirichlet process prior distributions
• Definition and basic properties
• Stick-breaking construction
• 23.3 Dirichlet process mixtures
• Specification and Polya urns
• Blocked Gibbs sampler
• Hyperprior distribution
• 23.4 Beyond density estimation
• Nonparametric residual distributions
• Nonparametric models for parameters that vary by group
• Functional data analysis
• 23.5 Hierarchical dependence
• Dependent Dirichlet processes
• Hierarchical Dirichlet processes
• Nested Dirichlet processes
• Convex mixtures
• 23.6 Density regression
• Dependent stick-breaking processes
• 23.7 Bibliographic note
• 23.8 Exercises
• Appendixes
• Appendix A Standard probability distributions
• A.1 Continuous distributions
• Uniform
• Univariate normal
• Lognormal
• Multivariate normal
• Gamma
• Inverse-gamma
• Chi-square
• Inverse chi-square
• Exponential
• Weibull
• Wishart
• Inverse-Wishart
• LKJ correlation
• t
• Beta
• Dirichlet
• Constrained distributions
• A.2 Discrete distributions
• Poisson
• Binomial
• Multinomial
• Negative binomial
• Beta-binomial
• A.3 Bibliographic note
• Appendix B Outline of proofs of limit theorems
• Mathematical framework
• Convergence of the posterior distribution for a discrete parameter space
• Convergence of the posterior distribution for a continuous parameter space
• Convergence of the posterior distribution to normality
• Multivariate form
• B.1 Bibliographic note
• Appendix C Computation in R and Stan
• C.1 Getting started with R and Stan
• C.2 Fitting a hierarchical model in Stan
• Stan program
• R script for data input, starting values, and running Stan
• Accessing the posterior simulations in R
• Posterior predictive simulations and graphs in R
• Alternative prior distributions
• Using the t model
• C.3 Direct simulation, Gibbs, and Metropolis in R
• Marginal and conditional simulation for the normal model
• Gibbs sampler for the normal model
• Gibbs sampling for the t model with fixed degrees of freedom
• Gibbs-Metropolis sampling for the t model with unknown degrees of freedom
• Parameter expansion for the t model
• C.4 Programming Hamiltonian Monte Carlo in R
• C.5 Further comments on computation
• C.6 Bibliographic note
• References
• Author Index
• Subject Index