Bayesian Statistics for Beginners

Bayesian statistics is currently undergoing something of a renaissance. At its heart is a method of statistical inference in which Bayes’ theorem is used to update the probability for a hypothesis as more evidence or information becomes available. It is an approach that is ideally suited to making initial assessments based on incomplete or imperfect information; as that information is gathered and disseminated, the Bayesian approach corrects or replaces the assumptions and alters its decision-making accordingly to generate a new set of probabilities. As new data/evidence becomes available the probability for a particular hypothesis can therefore be steadily refined and revised. It is very well-suited to the scientific method in general and is widely used across the social, biological, medical, and physical sciences. Key to this book’s novel and informal perspective is its unique pedagogy, a question and answer approach that utilizes accessible language, humor, plentiful illustrations, and frequent reference to on-line resources.

Bayesian Statistics for Beginners is an introductory textbook suitable for senior undergraduate and graduate students, professional researchers, and practitioners seeking to improve their understanding of the Bayesian statistical techniques they routinely use for data analysis in the life and medical sciences, psychology, public health, business, and other fields.

Table of Contents

SECTION 1 Basics of Probability

1 Introduction to Probability

2 Joint, Marginal, and Conditional Probability

SECTION 2 Bayes’ Theorem and Bayesian Inference

3 Bayes’ Theorem

4 Bayesian Inference

5 The Author Problem: Bayesian Inference with Two Hypotheses

6 The Birthday Problem: Bayesian Inference with Multiple Discrete Hypotheses

7 The Portrait Problem: Bayesian Inference with Joint Likelihood

SECTION 3 Probability Functions

8 Probability Mass Functions

9 Probability Density Functions

SECTION 4 Bayesian Conjugates

10 The White House Problem: The Beta-Binomial Conjugate

11 The Shark Attack Problem: The Gamma-Poisson Conjugate

12 The Maple Syrup Problem: The Normal-Normal Conjugate

SECTION 5 Markov Chain Monte Carlo

13 The Shark Attack Problem Revisited: MCMC with the Metropolis Algorithm

14 MCMC Diagnostic Approaches

15 The White House Problem Revisited: MCMC with the Metropolis-Hastings Algorithm

16 The Maple Syrup Problem Revisited: MCMC with Gibbs Sampling

SECTION 6 Applications

17 The Survivor Problem: Simple Linear Regression with MCMC

18 The Survivor Problem Continued: Introduction to Bayesian Model Selection

19 The Lorax Problem: Introduction to Bayesian Networks

20 The Once-ler Problem: Introduction to Decision Trees

APPENDIX 1 The Beta-Binomial Conjugate Solution

APPENDIX 2 The Gamma-Poisson Conjugate Solution

APPENDIX 3 The Normal-Normal Conjugate Solution

APPENDIX 4 Conjugate Solutions for Simple Linear Regression

APPENDIX 5 The Standardization of Regression Data

Review

“While reading this book, I joined the authors on a learning endeavor thanks to their honesty and intellectual vulnerability. Their lack of experience with Bayesian statistics helps them to be effective communicators . . . If you are interested in starting your Bayesian journey, then Bayesian Statistics for Beginners is an excellent place to begin.” — Taylor Saucier, Department of Wildlife, Fisheries, and Aquaculture, Mississippi State University, The Journal of Wildlife Management

About the Authors

Therese Donovan, Wildlife Biologist, U.S. Geological Survey, Vermont Cooperative Fish and Wildlife Research Unit, University of Vermont, USA,Ruth M. Mickey, Professor Emerita, Department of Mathematics and Statistics, University of Vermont, USA

Therese Donovan is a wildlife biologist with the U.S. Geological Survey, Vermont Cooperative Fish and Wildlife Research Unit. Based in the Rubenstein School of Environment and Natural Resources at the University of Vermont, Therese teaches graduate courses on ecological modeling and conservation biology. She works with a variety of student and professional collaborators on research problems focused on the conservation of vertebrates. Therese is the Director of the Vermont Cooperative Fish and Wildlife Unit Spreadsheet Project, a suite of on-line tutorials in Excel and R for modeling and analysis of wildlife populations. She lives in Vermont with her husband, Peter, and two children, Evan and Ana.

Ruth Mickey is a Professor Emerita of Statistics at the University of Vermont. Most of Ruth’s career was spent in the Department of Mathematics and Statistics, where she taught courses in Applied Multivariate Analysis, Categorical Data, Survey Sampling, Analysis of Variance and Regression, and Probability. She served as an advisor or committee member of numerous MS and PhD committees over a broad range of academic disciplines. She worked on the development of statistical methods and applications to advance public health and natural resources issues throughout her career.