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Preface | |

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Fundamentals of Bayesian Inference | |

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Probability and inference | |

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The three steps of Bayesian data analysis | |

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General notation for statistical inference | |

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Bayesian inference | |

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Discrete probability examples: genetics and spell checking | |

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Probability as a measure of uncertainty | |

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Example of probability assignment: football point spreads | |

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Example: estimating the accuracy of record linkage | |

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Some useful results from probability theory | |

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Computation and software | |

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Bayesian inference in applied statistics | |

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Bibliographic note | |

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Exercises | |

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Single-parameter models | |

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Estimating a probability from binomial data | |

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Posterior as compromise between data and prior information | |

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Summarizing posterior inference | |

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Informative prior distributions | |

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Estimating a normal mean with known variance | |

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Other standard single-parameter models | |

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Example: informative prior distribution for cancer rates | |

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Noninformative prior distributions | |

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Weakly informative prior distributions | |

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Bibliographic note | |

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Exercises | |

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Introduction to multiparameter models | |

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Averaging over 'nuisance parameters' | |

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Normal data with a noninformative prior distribution | |

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Normal data with a conjugate prior distribution | |

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Multinomial model for categorical data | |

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Multivariate normal model with known variance | |

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Multivariate normal with unknown mean and variance | |

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Example: analysis of a bioassay experiment | |

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Summary of elementary modeling and computation | |

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Bibliographic note | |

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Exercises | |

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Asymptotics and connections to non-Bayesian approaches | |

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Normal approximations to the posterior distribution | |

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Large-sample theory | |

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Counterexamples to the theorems | |

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Frequency evaluations of Bayesian inferences | |

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Bayesian interpretations of other statistical methods | |

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Bibliographic note | |

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Exercises | |

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Hierarchical models | |

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Constructing a parameterized prior distribution | |

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Exchangeability and setting up hierarchical models | |

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Fully Bayesian analysis of conjugate hierarchical models | |

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Estimating exchangeable parameters from a normal model | |

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Example: parallel experiments in eight schools | |

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Hierarchical modeling applied to a meta-analysis | |

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Weakly informative priors for hierarchical variance parameters | |

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Bibliographic note | |

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Exercises | |

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Fundamentals of Bayesian Data Analysis | |

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Model checking | |

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The place of model checking in applied Bayesian statistics | |

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Do the inferences from the model make sense? | |

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Posterior predictive checking | |

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Graphical posterior predictive checks | |

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Model checking for the educational testing example | |

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Bibliographic note | |

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Exercises | |

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Evaluating, comparing, and expanding models | |

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Measures of predictive accuracy | |

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Information criteria and cross-validation | |

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Model comparison based on predictive performance | |

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Model comparison using Bayes factors | |

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Continuous model expansion | |

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Implicit assumptions and model expansion: an example | |

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Bibliographic note | |

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Exercises | |

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Modeling accounting for data collection | |

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Bayesian inference requires a model for data collection | |

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Data-collection models and ignoreability | |

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Sample surveys | |

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Designed experiments | |

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Sensitivity and the role of randomization | |

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Observational studies | |

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Censoring and truncation | |

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Discussion | |

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Bibliographic note | |

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Exercises | |

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Decision analysis | |

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Bayesian decision theory in different contexts | |

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Using regression predictions: incentives for telephone surveys | |

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Multistage decision making: medical screening | |

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Hierarchical decision analysis for radon measurement | |

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Personal vs. institutional decision analysis | |

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Bibliographic note | |

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Exercises | |

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Advanced Computation | |

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Introduction to Bayesian computation | |

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Numerical integration | |

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Distributional approximations | |

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Direct simulation and rejection sampling | |

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Importance sampling | |

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How many simulation draws are needed? | |

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Computing environments | |

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Debugging Bayesian computing | |

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Bibliographic note | |

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Exercises | |

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Basics of Markov chain simulation | |

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Gibbs sampler | |

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Metropolis and Metropolis-Hastings algorithms | |

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Using Gibbs and Metropolis as building blocks | |

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Inference and assessing convergence | |

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Effective number of simulation draws | |

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Example: hierarchical normal model | |

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Bibliographic note | |

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Exercises | |

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Computationally efficient Markov chain simulation | |

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Efficient Gibbs samplers | |

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Efficient Metropolis jumping rules | |

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Further extensions to Gibbs and Metropolis | |

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Hamiltonian Monte Carlo | |

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Hamiltonian dynamics for a simple hierarchical model | |

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Stan: developing a computing environment | |

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Bibliographic note | |

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Exercises | |

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Modal and distributional approximations | |

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Finding posterior modes | |

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Boundary-avoiding priors for modal summaries | |

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Normal and related mixture approximations | |

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Finding marginal posterior modes using EM | |

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Approximating conditional and marginal posterior densities | |

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Example: hierarchical normal model (continued) | |

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Variational inference | |

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Expectation propagation | |

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Other approximations | |

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Unknown normalizing factors | |

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Bibliographic note | |

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Exercises | |

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Regression Models | |

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Introduction to regression models | |

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Conditional modeling | |

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Bayesian analysis of the classical regression model | |

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Regression for causal inference: incumbency in congressional elections | |

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Goals of regression analysis | |

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Assembling the matrix of explanatory variables | |

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Regularization and dimension reduction for multiple predictors | |

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Unequal variances and correlations | |

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Including numerical prior information | |

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Bibliographic note | |

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Exercises | |

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Hierarchical linear models | |

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Regression coefficients exchangeable in batches | |

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Example: forecasting U.S. presidential elections | |

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Interpreting a normal prior distribution as additional data | |

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Varying intercepts and slopes | |

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Computation: batching and transformation | |

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Analysis of variance and the batching of coefficients | |

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Hierarchical models for batches of variance components | |

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Bibliographic note | |

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Exercises | |

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Generalized linear models | |

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Standard generalized linear model likelihoods | |

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Working with generalized linear models | |

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Weakly informative priors for logistic regression | |

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Example: hierarchical Poisson regression for police stops | |

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Example: hierarchical logistic regression for political opinions | |

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Models for multivariate and multinomial responses | |

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Loglinear models for multivariate discrete data | |

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Bibliographic note | |

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Exercises | |

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Models for robust inference | |

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Aspects of robustness | |

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Overdispersed versions of standard probability models | |

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Posterior inference and computation | |

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Robust inference and sensitivity analysis for the eight schools | |

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Robust regression using t-distributed errors | |

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Bibliographic note | |

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Exercises | |

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Models for missing data | |

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Notation | |

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Multiple imputation | |

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Missing data in the multivariate normal and t models | |

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Example: multiple imputation for a series of polls | |

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Missing values with counted data | |

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Example: an opinion poll in Slovenia | |

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Bibliographic note | |

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Exercises | |

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Nonlinear and Nonparametric Models | |

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Parametric nonlinear models | |

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Example: serial dilution assay | |

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Example: population toxicokinetics | |

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Bibliographic note | |

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Exercises | |

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Basis function models | |

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Splines and weighted sums of basis functions | |

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Basis selection and shrinkage of coefficients | |

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Non-normal models and multivariate regression surfaces | |

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Bibliographic note | |

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Exercises | |

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Gaussian process models | |

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Gaussian process regression | |

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Example: birthdays and birthdates | |

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Latent Gaussian process models | |

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Functional data analysis | |

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Density estimation and regression | |

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Bibliographic note | |

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Exercises | |

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Finite mixture models | |

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Setting up and interpreting mixture models | |

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Example: reaction times and schizophrenia | |

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Label switching and posterior computation | |

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Unspecified number of mixture components | |

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Mixture models for classification and regression | |

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Bibliographic note | |

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Exercises | |

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Dirichlet process models | |

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Bayesian histograms | |

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Dirichlet process prior distributions | |

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Dirichlet process mixtures | |

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Beyond density estimation | |

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Hierarchical dependence | |

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Density regression | |

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Bibliographic note | |

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Exercises | |

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Standard probability distributions | |

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Continuous distributions | |

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Discrete distributions | |

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Bibliographic note | |

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Outline of proofs of limit theorems | |

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Bibliographic note | |

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Computation in R and Stan | |

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Getting started with R and Stan | |

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Fitting a hierarchical model in Stan | |

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Direct simulation, Gibbs, and Metropolis in R | |

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Programming Hamiltonian Monte Carlo in R | |

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Further comments on computation | |

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Bibliographic note | |

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References | |

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Author Index | |

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Subject Index | |