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List of Figures | |

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List of Tables | |

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Preface | |

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Acknowledgments | |

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Introduction | |

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Introducing Bayesian Analysis | |

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The foundations of Bayesian inference | |

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What is probability? | |

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Probability in classical statistics | |

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Subjective probability1 | |

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Subjective probability in Bayesian statistics | |

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Bayes theorem, discrete case | |

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Bayes theorem, continuous parameter | |

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Conjugate priors | |

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Bayesian updating with irregular priors | |

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Cromwell's Rule | |

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Bayesian updating as information accumulation | |

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Parameters as random variables, beliefs as distributions | |

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Communicating the results of a Bayesian analysis | |

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Bayesian point estimation | |

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Credible regions | |

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Asymptotic properties of posterior distributions | |

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Bayesian hypothesis testing | |

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Model choice | |

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Bayes factors | |

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From subjective beliefs to parameters and models | |

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Exchangeability | |

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Implications and extensions of de Finetti's Representation Theorem | |

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Finite exchangeability | |

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Exchangeability and prediction | |

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Conditional exchangeability and multiparameter models | |

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Exchangeability of parameters: hierarchical modeling | |

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Historical note | |

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Getting started: Bayesian analysis for simple models | |

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Learning about probabilities, rates and proportions | |

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Conjugate priors for probabilities, rates and proportions | |

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Bayes estimates as weighted averages of priors and data | |

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Parameterizations and priors | |

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The variance of the posterior density | |

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Associations between binary variables | |

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Learning from counts | |

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Predictive inference with count data | |

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Learning about a normal mean and variance | |

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Variance known | |

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Mean and variance unknown | |

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Conditionally conjugate prior | |

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An improper, reference prior | |

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Conflict between likelihood and prior | |

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Non-conjugate priors | |

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Regression models | |

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Bayesian regression analysis | |

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Likelihood function | |

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Conjugate prior | |

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Improper, reference prior | |

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Further reading | |

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Simulation Based Bayesian Analysis | |

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Monte Carlo methods | |

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Simulation consistency | |

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Inference for functions of parameters | |

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Marginalization via Monte Carlo integration | |

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Sampling algorithms | |

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Inverse-CDF method | |

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Importance sampling | |

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Accept-reject sampling | |

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Adaptive rejection sampling | |

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Further reading | |

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Markov chains | |

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Notation and definitions | |

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State space | |

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Transition kernel | |

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Properties of Markov chains | |

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Existence of a stationary distribution, discrete case | |

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Existence of a stationary distribution, continuous case | |

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Irreducibility | |

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Recurrence | |

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Invariant measure | |

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Reversibility | |

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Aperiodicity | |

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Convergence of Markov chains | |

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Speed of convergence | |

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Limit theorems for Markov chains | |

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Simulation inefficiency | |

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Central limit theorems for Markov chains | |

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Further reading | |

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Markov chain Monte Carlo | |

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Metropolis-Hastings algorithm | |

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Theory for the Metropolis-Hastings algorithm | |

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Choosing the proposal density | |

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Gibbs sampling | |

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Theory for the Gibbs sampler | |

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Connection to the Metropolis algorithm | |

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Deriving conditional densities for the Gibbs sampler: statistical models as conditional independence graphs | |

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Pathologies | |

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Data augmentation | |

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Missing data problems | |

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The slice sampler | |

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Implementing Markov chain Monte Carlo | |

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Software for Markov chain Monte Carlo | |

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Assessing convergence and run-length | |

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Working with BUGS/JAGS from R | |

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Tricks of the trade | |

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Thinning | |

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Blocking | |

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Reparameterization | |

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Other examples | |

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Further reading | |

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Advanced Applications in the Social Sciences | |

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Hierarchical Statistical Models | |

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Data and parameters that vary by groups: the case for hierarchical modeling | |

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Exchangeable parameters generate hierarchical models | |

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ï¿½Borrowing strengthï¿½ via exchangeability | |

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Hierarchical modeling as a 'semi-poolingï¿½ estimator | |

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Hierarchical modeling as a 'shrinkageï¿½ estimator | |

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Computation via Markov chain Monte Carlo | |

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ANOVA as a hierarchical model | |

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One-way analysis of variance | |

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Two-way ANOVA | |

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Hierarchical models for longitudinal data | |

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Hierarchical models for non-normal data | |

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Multi-level models | |

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Bayesian analysis of choice making | |

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Regression models for binary responses | |

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Probit model via data augmentation | |

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Probit model via marginal data augmentation | |

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Logit model | |

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Binomial model for grouped binary data | |

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Ordered outcomes | |

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Identification | |

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Multinomial outcomes | |

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Multinomial logit (MNL) | |

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Independence of irrelevant alternatives | |

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Multinomial probit | |

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Bayesian analysis via MCMC | |

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Bayesian approaches to measurement | |

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Bayesian inference for latent states | |

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A formal role for prior information | |

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Inference for many parameters | |

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Factor analysis | |

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Likelihood and prior densities | |

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Identification | |

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Posterior density | |

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Inference over rank orderings of the latent variable | |

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Incorporating additional information via hierarchical modeling | |

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Item-response models | |

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Dynamic measurement models | |

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State-space models for [pooling the polls] | |

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Bayesian inference | |

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Appendices | |

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Working with vectors and matrices | |

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Probability review | |

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Foundations of probability | |

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Probability densities and mass functions | |

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Probability mass functions for discrete random quantities | |

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Probability density functions for continuous random quantities | |

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Convergence of sequences of random variables | |

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Proofs of selected propositions | |

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Products of normal densities | |

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Conjugate analysis of normal data | |

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Asymptotic normality of the posterior density | |

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References | |

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Topic index | |

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Author index | |