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Bayesian Statistical Methods provides data scientists with the foundational and computational tools needed to carry out a Bayesian analysis. This book focuses on Bayesian methods applied routinely in practice including multiple linear regression, mixed effects models and generalized linear models (GLM). The authors include many examples with complete R code and comparisons with analogous frequentist procedures.

In addition to the basic concepts of Bayesian inferential methods, the book covers many general topics:

Advice on selecting prior distributions
Computational methods including Markov chain Monte Carlo (MCMC)
Model-comparison and goodness-of-fit measures, including sensitivity to priors
Frequentist properties of Bayesian methods
Case studies covering advanced topics illustrate the flexibility of the Bayesian approach:

Semiparametric regression
Handling of missing data using predictive distributions
Priors for high-dimensional regression models
Computational techniques for large datasets
Spatial data analysis
The advanced topics are presented with sufficient conceptual depth that the reader will be able to carry out such analysis and argue the relative merits of Bayesian and classical methods. A repository of R code, motivating data sets, and complete data analyses are available on the book's website.



Table of Contents
1. Basics of Bayesian Inference

Probability background

Univariate distributions

Discrete distributions

Continuous distributions

Multivariate distributions

Marginal and conditional distributions

Bayes´ Rule

Discrete example of Bayes´ Rule

Continuous example of Bayes´ Rule

Introduction to Bayesian inference

Summarizing the posterior

Point estimation

Univariate posteriors

Multivariate posteriors

The posterior predictive distribution

Exercises

2. From Prior Information to Posterior Inference

Conjugate Priors

Beta-binomial model for a proportion

Poisson-gamma model for a rate

Normal-normal model for a mean

Normal-inverse gamma model for a variance

Natural conjugate priors

Normal-normal model for a mean vector

Normal-inverse Wishart model for a covariance matrix

Mixtures of conjugate priors

Improper Priors

Objective Priors

Jeffreys prior

Reference Priors

Maximum Entropy Priors

Empirical Bayes

Penalized complexity priors

Exercises

3. Computational approaches

Deterministic methods

Maximum a posteriori estimation

Numerical integration

Bayesian Central Limit Theorem (CLT)

Markov Chain Monte Carlo (MCMC) methods

Gibbs sampling

Metropolis-Hastings (MH) sampling

MCMC software options in R

Diagnosing and improving convergence

Selecting initial values

Convergence diagnostics

Improving convergence

Dealing with large datasets

Exercises

4. Linear models

Analysis of normal means

One-sample/paired analysis

Comparison of two normal means

Linear regression

Jeffreys prior

Gaussian prior

Continuous shrinkage priors

Predictions

Example: Factors that affect a home´s microbiome

Generalized linear models

Binary data

Count data

Example: Logistic regression for NBA clutch free throws

Example: Beta regression for microbiome data

Random effects

Flexible linear models

Nonparametric regression

Heteroskedastic models

Non-Gaussian error models

Linear models with correlated data

Exercises

5. Model selection and diagnostics

Cross validation

Hypothesis testing and Bayes factors

Stochastic search variable selection

Bayesian model averaging

Model selection criteria

Goodness-of-fit checks

Exercises

6. Case studies using hierarchical modeling

Overview of hierarchical modeling

Case study: Species distribution mapping via data fusion

Case study: Tyrannosaurid growth curves

Case study: Marathon analysis with missing data

7. Statistical properties of Bayesian methods

Decision theory

Frequentist properties

Bias-variance tradeoff

Asymptotics

Simulation studies

Exercises

Appendices

Probability distributions

Univariate discrete

Multivariate discrete

Univariate continuous

Multivariate continuous

List of conjugacy pairs

Derivations

Normal-normal model for a mean

Normal-normal model for a mean vector

Normal-inverse Wishart model for a covariance matrix

Jeffreys´ prior for a normal model

Jeffreys´ prior for multiple linear regression

Convergence of the Gibbs sampler

Marginal distribution of a normal mean under Jeffreys' prior

Marginal posterior of the regression coefficients under Jeffreys prior

Proof of posterior consistency

Computational algorithms

Integrated nested Laplace approximation (INLA)

Metropolis-adjusted Langevin algorithm

Hamiltonian Monte Carlo (HMC)

Delayed Rejection and Adaptive Metropolis

Slice sampling

Software comparison

Example - Simple linear regression

Example - Random slopes model