Bayesian Biostatistics -- Contents -- Preface -- Notation, terminology and some guidance for reading the book -- Part I BASIC CONCEPTS IN BAYESIAN METHODS -- 1 Modes of statistical inference -- 1.1 The frequentist approach: A critical reflection -- 1.1.1 The classical statistical approach -- 1.1.2 The P-value as a measure of evidence -- 1.1.3 The confidence interval as a measure of evidence -- 1.1.4 An historical note on the two frequentist paradigms -- 1.2 Statistical inference based on the likelihood function -- 1.2.1 The likelihood function -- 1.2.2 The likelihood principles -- 1.3 The Bayesian approach: Some basic ideas -- 1.3.1 Introduction -- 1.3.2 Bayes theorem - discrete version for simple events -- 1.4 Outlook -- Exercises -- 2 Bayes theorem: Computing the posterior distribution -- 2.1 Introduction -- 2.2 Bayes theorem - the binary version -- 2.3 Probability in a Bayesian context -- 2.4 Bayes theorem - the categorical version -- 2.5 Bayes theorem - the continuous version -- 2.6 The binomial case -- 2.7 The Gaussian case -- 2.8 The Poisson case -- 2.9 The prior and posterior distribution of h(θ) -- 2.10 Bayesian versus likelihood approach -- 2.11 Bayesian versus frequentist approach -- 2.12 The different modes of the Bayesian approach -- 2.13 An historical note on the Bayesian approach -- 2.14 Closing remarks -- Exercises -- 3 Introduction to Bayesian inference -- 3.1 Introduction -- 3.2 Summarizing the posterior by probabilities -- 3.3 Posterior summary measures -- 3.3.1 Characterizing the location and variability of the posterior distribution -- 3.3.2 Posterior interval estimation -- 3.4 Predictive distributions -- 3.4.1 The frequentist approach to prediction -- 3.4.2 The Bayesian approach to prediction -- 3.4.3 Applications -- 3.5 Exchangeability -- 3.6 A normal approximation to the posterior.

3.6.1 A Bayesian analysis based on a normal approximation to the likelihood -- 3.6.2 Asymptotic properties of the posterior distribution -- 3.7 Numerical techniques to determine the posterior -- 3.7.1 Numerical integration -- 3.7.2 Sampling from the posterior -- 3.7.3 Choice of posterior summary measures -- 3.8 Bayesian hypothesis testing -- 3.8.1 Inference based on credible intervals -- 3.8.2 The Bayes factor -- 3.8.3 Bayesian versus frequentist hypothesis testing -- 3.9 Closing remarks -- Exercises -- 4 More than one parameter -- 4.1 Introduction -- 4.2 Joint versus marginal posterior inference -- 4.3 The normal distribution with μ and σ2 unknown -- 4.3.1 No prior knowledge on μ and σ2 is available -- 4.3.2 An historical study is available -- 4.3.3 Expert knowledge is available -- 4.4 Multivariate distributions -- 4.4.1 The multivariate normal and related distributions -- 4.4.2 The multinomial distribution -- 4.5 Frequentist properties of Bayesian inference -- 4.6 Sampling from the posterior distribution: The Method of Composition -- 4.7 Bayesian linear regression models -- 4.7.1 The frequentist approach to linear regression -- 4.7.2 A noninformative Bayesian linear regression model -- 4.7.3 Posterior summary measures for the linear regression model -- 4.7.4 Sampling from the posterior distribution -- 4.7.5 An informative Bayesian linear regression model -- 4.8 Bayesian generalized linear models -- 4.9 More complex regression models -- 4.10 Closing remarks -- Exercises -- 5 Choosing the prior distribution -- 5.1 Introduction -- 5.2 The sequential use of Bayes theorem -- 5.3 Conjugate prior distributions -- 5.3.1 Univariate data distributions -- 5.3.2 Normal distribution - mean and variance unknown -- 5.3.3 Multivariate data distributions -- 5.3.4 Conditional conjugate and semiconjugate distributions -- 5.3.5 Hyperpriors.

5.4 Noninformative prior distributions -- 5.4.1 Introduction -- 5.4.2 Expressing ignorance -- 5.4.3 General principles to choose noninformative priors -- 5.4.4 Improper prior distributions -- 5.4.5 Weak/vague priors -- 5.5 Informative prior distributions -- 5.5.1 Introduction -- 5.5.2 Data-based prior distributions -- 5.5.3 Elicitation of prior knowledge -- 5.5.4 Archetypal prior distributions -- 5.6 Prior distributions for regression models -- 5.6.1 Normal linear regression -- 5.6.2 Generalized linear models -- 5.6.3 Specification of priors in Bayesian software -- 5.7 Modeling priors -- 5.8 Other regression models -- 5.9 Closing remarks -- Exercises -- 6 Markov chain Monte Carlo sampling -- 6.1 Introduction -- 6.2 The Gibbs sampler -- 6.2.1 The bivariate Gibbs sampler -- 6.2.2 The general Gibbs sampler -- 6.2.3 Remarks -- 6.2.4 Review of Gibbs sampling approaches -- 6.2.5 The Slice sampler -- 6.3 The Metropolis(-Hastings) algorithm -- 6.3.1 The Metropolis algorithm -- 6.3.2 The Metropolis-Hastings algorithm -- 6.3.3 Remarks -- 6.3.4 Review of Metropolis(-Hastings) approaches -- 6.4 Justification of the MCMC approaches -- 6.4.1 Properties of the MH algorithm -- 6.4.2 Properties of the Gibbs sampler -- 6.5 Choice of the sampler -- 6.6 The Reversible Jump MCMC algorithm -- 6.7 Closing remarks -- Exercises -- 7 Assessing and improving convergence of the Markov chain -- 7.1 Introduction -- 7.2 Assessing convergence of a Markov chain -- 7.2.1 Definition of convergence for a Markov chain -- 7.2.2 Checking convergence of the Markov chain -- 7.2.3 Graphical approaches to assess convergence -- 7.2.4 Formal diagnostic tests -- 7.2.5 Computing the Monte Carlo standard error -- 7.2.6 Practical experience with the formal diagnostic procedures -- 7.3 Accelerating convergence -- 7.3.1 Introduction -- 7.3.2 Acceleration techniques.

7.4 Practical guidelines for assessing and accelerating convergence -- 7.5 Data augmentation -- 7.6 Closing remarks -- Exercises -- 8 Software -- 8.1 WinBUGS and related software -- 8.1.1 A first analysis -- 8.1.2 Information on samplers -- 8.1.3 Assessing and accelerating convergence -- 8.1.4 Vector and matrix manipulations -- 8.1.5 Working in batch mode -- 8.1.6 Troubleshooting -- 8.1.7 Directed acyclic graphs -- 8.1.8 Add-on modules: GeoBUGS and PKBUGS -- 8.1.9 Related software -- 8.2 Bayesian analysis using SAS -- 8.2.1 Analysis using procedure GENMOD -- 8.2.2 Analysis using procedure MCMC -- 8.2.3 Other Bayesian programs -- 8.3 Additional Bayesian software and comparisons -- 8.3.1 Additional Bayesian software -- 8.3.2 Comparison of Bayesian software -- 8.4 Closing remarks -- Exercises -- Part II BAYESIAN TOOLS FOR STATISTICAL MODELING -- 9 Hierarchical models -- 9.1 Introduction -- 9.2 The Poisson-gamma hierarchical model -- 9.2.1 Introduction -- 9.2.2 Model specification -- 9.2.3 Posterior distributions -- 9.2.4 Estimating the parameters -- 9.2.5 Posterior predictive distributions -- 9.3 Full versus empirical Bayesian approach -- 9.4 Gaussian hierarchical models -- 9.4.1 Introduction -- 9.4.2 The Gaussian hierarchical model -- 9.4.3 Estimating the parameters -- 9.4.4 Posterior predictive distributions -- 9.4.5 Comparison of FB and EB approach -- 9.5 Mixed models -- 9.5.1 Introduction -- 9.5.2 The linear mixed model -- 9.5.3 The generalized linear mixed model -- 9.5.4 Nonlinear mixed models -- 9.5.5 Some further extensions -- 9.5.6 Estimation of the random effects and posterior predictive distributions -- 9.5.7 Choice of the level-2 variance prior -- 9.6 Propriety of the posterior -- 9.7 Assessing and accelerating convergence -- 9.8 Comparison of Bayesian and frequentist hierarchical models -- 9.8.1 Estimating the level-2 variance.

9.8.2 ML and REML estimates compared with Bayesian estimates -- 9.9 Closing remarks -- Exercises -- 10 Model building and assessment -- 10.1 Introduction -- 10.2 Measures for model selection -- 10.2.1 The Bayes factor -- 10.2.2 Information theoretic measures for model selection -- 10.2.3 Model selection based on predictive loss functions -- 10.3 Model checking -- 10.3.1 Introduction -- 10.3.2 Model-checking procedures -- 10.3.3 Sensitivity analysis -- 10.3.4 Posterior predictive checks -- 10.3.5 Model expansion -- 10.4 Closing remarks -- Exercises -- 11 Variable selection -- 11.1 Introduction -- 11.2 Classical variable selection -- 11.2.1 Variable selection techniques -- 11.2.2 Frequentist regularization -- 11.3 Bayesian variable selection: Concepts and questions -- 11.4 Introduction to Bayesian variable selection -- 11.4.1 Variable selection for K small -- 11.4.2 Variable selection for K large -- 11.5 Variable selection based on Zellner's g-prior -- 11.6 Variable selection based on Reversible Jump Markov chain Monte Carlo -- 11.7 Spike and slab priors -- 11.7.1 Stochastic Search Variable Selection -- 11.7.2 Gibbs Variable Selection -- 11.7.3 Dependent variable selection using SSVS -- 11.8 Bayesian regularization -- 11.8.1 Bayesian LASSO regression -- 11.8.2 Elastic Net and further extensions of the Bayesian LASSO -- 11.9 The many regressors case -- 11.10 Bayesian model selection -- 11.11 Bayesian model averaging -- 11.12 Closing remarks -- Exercises -- Part III BAYESIAN METHODS IN PRACTICAL APPLICATIONS -- 12 Bioassay -- 12.1 Bioassay essentials -- 12.1.1 Cell assays -- 12.1.2 Animal assays -- 12.2 A generic in vitro example -- 12.3 Ames/Salmonella mutagenic assay -- 12.4 Mouse lymphoma assay (L5178Y TK+/-) -- 12.5 Closing remarks -- 13 Measurement error -- 13.1 Continuous measurement error -- 13.1.1 Measurement error in a variable.

13.1.2 Two types of measurement error on the predictor in linear and nonlinear models.