288 Pages
    by Chapman & Hall

    288 Pages
    by Chapman & Hall

    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.



    Brian J. Reich, Associate Professor of Statistics at North Carolina State University, is currently the editor-in-chief of the Journal of Agricultural, Biological, and Environmental Statistics and was awarded the LeRoy & Elva Martin Teaching Award.



    Sujit K. Ghosh, Professor of Statistics at North Carolina State University, has over 22 years of research and teaching experience in conducting Bayesian analyses, received the Cavell Brownie mentoring award, and served as the Deputy Director at the Statistical and Applied Mathematical Sciences Institute.





     

    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

    Biography

    Brian J. Reich, Associate Professor of Statistics at North Carolina State University, is currently the editor-in-chief of the Journal of Agricultural, Biological, and Environmental Statistics and was awarded the LeRoy & Elva Martin Teaching Award.

    Sujit K. Ghosh, Professor of Statistics at North Carolina State University, has over 22 years of research and teaching experience in conducting Bayesian analyses, received the Cavell Brownie mentoring award, and served as the Deputy Director at the Statistical and Applied Mathematical Sciences Institute

    "Brian J. Reich and Sujit K. Ghosh make a valuable contribution to the growing canon of introductory texts on Bayesian statistics…The extensive data and problem sets provided are a major highlight of the work…Features that instructors will find quite appealing include the nice library of problem sets (with solutions to odd problems in chapters 1-5 online), the availability online of several nice worked data examples including code, and coverage of some topics not yet standard in introductory texts, including Bayesian computation with big data…A big plus is the recent addition of Python code (PyMC) online…Because several of the exercises are application based and incorporate data from a variety of disciplines, the book will surely capture the interest of its intended readership."
    ~Biometrics

    "A book that gives a comprehensive coverage of Bayesian inference for a diverse background of scientific practitioners is needed. The book Bayesian Statistical Methods seems to be a good candidate for this purpose, which aims at a balanced treatment between theory and computation. The authors are leading researchers and experts in Bayesian statistics. I believe this book is likely to be an excellent text book for an introductory course targeting at first-year graduate students or undergraduate statistics majors…This new book is more focused on the most fundamental components of Bayesian methods. Moreover, this book contains many simulated examples and real-data applications, with computer code provided to demonstrate the implementations."
    ~Qing Zhou, UCLA

    "The book gives an overview of Bayesian statistical modeling with a focus on the building blocks for fitting and analyzing hierarchical models. The book uses a number of interesting and realistic examples to illustrate the methods. The computational focus is in the use of JAGS, as a tool to perform Bayesian inference using Markov chain Monte Carlo methods…It can be targeted as a textbook for upper-division undergraduate students in statistics and some areas of science, engineering and social sciences with an interest in a reasonably formal development of data analytic methods and uncertainty quantification. It could also be used for a Master’s class in statistical modeling."
    ~Bruno Sansó, University of California Santa Cruz

    "The given manuscript sample is technically correct, clearly written, and at an appropriate level of difficulty… I enjoyed the real-life problems in the Chapter 1 exercises. I especially like the problem on the Federalist Papers, because the students can revisit this problem and perform more powerful inferences using the advanced Bayesian methods that they will learn later in the textbook… I would seriously consider adopting the book as a required textbook. This text provides more details, R codes, and illuminating visualizations compared to competing books, and more quickly introduces a broad scope of regression models that are important in practical applications."
    ~Arman Sabbaghi, Purdue University

    "The authors are leading researchers and experts in Bayesian statistics. I believe this book is likely to be an excellent textbook for an introductory course targeting at first-year graduate students or
    undergraduate statistics majors..."
    ~Qing Zhou, UCLA

    "I would seriously consider adopting the book as a required textbook. This text provides more details, R codes, and illuminating visualizations compared to competing books, and more quickly introduces a broad scope of regression models that are important in practical applications…"
    ~Arman Sabbaghi, Purdue University

    "The book gives an overview of Bayesian statistical modeling with a focus on the building blocks for fitting and analyzing hierarchical models. The book uses a number of interesting and realistic examples to illustrate the methods. The computational focus is in the use of JAGS, as a tool to perform Bayesian inference using Markov chain Monte Carlo methods…It can be targeted as a textbook for upper-division undergraduate students in statistics and some areas of science, engineering and social sciences with an interest in a reasonably formal development of data analytic methods and uncertainty quantification. It could also be used for a Master’s class in statistical modeling."
    ~Bruno Sansó, University of California Santa Cruz