Course Description
Printable syllabus
Contents
Overview of the Course
Basic Info:
Professor
Jonathan GilliganOffice: SC 5735 (7th floor of Stevenson 5)
Email: jonathan.gilligan@vanderbilt.edu
Office Hours: Monday 2:00–3:00 pm, Tuesday 4:00–5:00, or by appointment.
Schedule
Class meetings: TR 11:00–12:15, SC 6740
Catalog Description
EES 589103 Bayesian Statistical Methods The class will begin with an introduction to Bayesian statistics and then focus on practical application of regression methods to data. We will use R together with the Stan software package for Hamiltonian Monte Carlo methods and the RINLA software package for Integrated Nested Laplace Approximation (INLA) analysis (https://www.rinla.org/) . The course will combine practical applications of Bayesian methods to real (often messy) data with more philosophical discussions of Bayesian approaches to statistics and how to interpret results of statistical analyses. We will focus on regression methods, including hierarchical or multilevel regression modeling methods, which can be very powerful when you have data that has a nested structure (e.g., cities and counties within states or species within genera). Students will do projects applying Bayesian methods to their own data sets.
Prerequisites
You should be comfortable with differential and integral calculus and have some previous experience with standard statistics.
This course will be very mathematical and will make extensive use of the R software system, but I do not assume that you already know R or advanced mathematics beyond calculus.
Narrative Description
Bayesian statistics is a branch of statistics that has been around for almost 300 years, but for most of that time, it was very difficult to apply to practical problems because the mathematical equations were too difficult to solve. In the last 30 years, as computers have become much faster and more powerful, new computational methods have emerged that make Bayesian statistics practical for research and applications.
Bayesian analysis is widely used across a wide variety of research as well as practical applications. It is used to analyze results from highenergy particle physics experiments to discover new subatomic particles. There are many other applications in a wide variety of domains. It’s used by geologists to improve estimates of mineral distributions and radon hazards. It’s used by biologists to identify and categorize variations in the genomes of humans and other species. It’s used extensively in medicine to analyze the results of clinical trials, to determine the pharmacokinetics of drug metabolism, and to assess the predictive value of tests for diseases such as cancer or COVID infection. It’s used in political science and sociology to improve the accuracy of public opinion surveys and to understand patterns of voting. It’s widely used in marketing to identify consumer preferences and improve the effectiveness of advertising. If you use Google, Amazon, Netflix, Stitchfix, or practically any large online platform for shopping or entertainment, advanced Bayesian methods form the basis of their recommendations. Bayesian analysis has also been applied effectively to law and criminology to assess the value of evidence in proving guilt or innocence. It has been applied to public health to estimate the prevalence of dieseases and tomake more effective treatment decisions when medical tests are uncertain. It is widely used in meteorology to make weather forecasts and in climate science to combine data from many different sources and come up with quantitative predictions and detailed understanding of their associated uncertainties. Bayesian methods are also widely used in computational applications, such as image analysis and reconstruction, computational text analysis, and natural language processing. One of the earliest practical applications of Bayesian textual analysis, in 1964, identified the anonymous authors of the Federalist Papers. More recent applications of Bayesian textual analysis are used to separate desired email from spam.
Bayesian statistical methods are valuable because they provide a systematic way to combine what you already know about a problem with new data from experiments or observations, and the results of Bayesian analyses are more straightforward to interpret than conventional statistics.
This course will provide a general introduction to Bayesian statistics and will combine practical instruction in how to do Bayesian data analysis and philosophical discussions about how to think about the assumptions that go into a Bayesian analysis and how to interpret the results that it produces.
You do not need to have any prior knowledge of computer programming, but I do expect that you are familiar with basic statistics and calculus (both derivatives and integrals).
Goals for the Course
By the end of the semester, you will:
 Understand Bayes’s theorem and how to apply it.
 Understand problems with the traditional statistical emphasis on nullhypothesis significance testing (NHST), why Bayesian approaches to NHST don’t solve these problems, and how Bayeian statistics offers superior alternatives to NHST.
 Understand how think about statistical models, how to choose an appropriate model for your problems, and understand the tradeoffs between different kinds of models.
 Be able to design and conduct a comprehensive Bayesian analysis of data from start to finish.
 Understand how to choose appropriate priors for your Bayesian analyses and how to test whether your choice of priors is sound.
 Understand how to set up, perform, assess the validity of, and interpret the results of Bayesian regression analysis.
 Understand why Markov Chain Monte Carlo (MCMC) sampling is used in Bayesian analysis, what the limits of MCMC are, and how to test your MCMC analyses for validity.
 Understand and be able to perform analyses using more complex statistical models, such as interaction models, generalized linear models, models of discrete (categorical and count) data.
 Understand what multilevel or hierarchical models are, when to use them, and how to interpret the results of a multilevel analysis.
 Understand the Integrated Nested Laplace Approximation (INLA), why you might use INLA instead of MCMC analysis, and what the limits of INLA analysis are.
 Understand several types of Bayesian geospatial analysis, including Matern covariance models and conditional autoregressive (CAR) models.
Structure of the Course:
I divide the semester into three parts:

Introduction to Bayes’s Theorem and its Applications: The first part of the course introduces the basic concepts of Bayesian statistics, using simplified approximations to calculate difficult equations. This section will focus on linear regression methods.

Monte Carlo Methods: Next, we study Monte Carlo methods, which help us solve more difficult problems that our earlier approximations are not powerful enough for. This section will introduce statistical models of discrete data (counts, categories, etc.), and generalized linear models. It will conclude with multilevel statistical models, which can be very powerful methods for working with large and complex data sets.

Geospatial Modeling: Finally, we will learn a different approach, called the Integrated Nested Laplace Approximation (INLA), which is very well suited for analyzing geospatial data that may be too difficult to analyze uding Monte Carlo methods.
Reading Material
There are two required textbooks and two optional textbooks:
Textbooks

Richard McElreath, <em>Statistical Rethinking: A Bayesian Course with Examples in R and Stan</em> , 2^{nd} ed. (CRC Press, 2020). ISBN: 9780367139919
This will be the main textbook for most of the semester.
There is a companion web site to the book, and the author has posted videos of his lectures on YouTube.
McElreath uses basic R, which is fine, but many people have learned to use a more modern dialect of R called the “tidyverse,” which is described at length in our companion book, <em>R for Data Science</em> . For people who are used to the Tidyverse, there is an online ebook
 A. Solomon Kurz, <em>Statistical rethinking with brms, ggplot2, and the tidyverse:</em> Second edition ,
that translates McElreath’s R code into the tidyverse, dialect. If you want to use this, you will read the text in McElreath’s book, but use the code from Kurz’s ebook.

Virgilio GomezRubio, <em>Bayeian Inference with INLA</em> , (CRC Press, 2021) ISBN: 9781032174532
We will only use this book for a few weeks in the third part of the semester, when we are studying geospatial methods.
The book is fairly expensive, but there is a free online ebook version of this book that you can use, if you don’t want to buy it.
We will use this book in the third part of the semester, when we study Bayesian geospatial methods. This book covers a method called the Integrated Nested Laplace Approximation, which is very powerful for statistical models that are too computationally intensive for Monte Carlo methods.
Optional Textbooks
There are two optional textbooks that you don’t need to buy, but which may be very useful as companions to the assigned textbooks.

John Kruschke, <em>Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan</em> , 2^{nd} ed. (Academic Press, 2014). ISBN: 9780124058880
This is an excellent introduction to Bayesian data analysis for beginners. It is gentler than Statistical Rethinking, and would be better suited for an undergraduate course, but I decided not to use it as the main textbook for this class because it focuses more on the statistical methods and does not give as much application of them to real scientific problems.
The author writes very clearly and this book may be helpful if you find some of the material in Statistical Rethinking confusing. I have asked the Science and Engineering Library to put a copy on reserve so you will be able to access it without buying a copy.

Garett Grolemund and Hadley Wickham, <em>R for Data Science</em> , (O’Reilly 2017) ISBN: 9781491910399
This book is the best practical introduction I have found for getting started in R and getting things done in data analysis. The author is the chief data scientist as the RStudio company and wrote a huge number of widely used free packages to extend and enrich R. This book follows his philosophy of how to organize data sensibly for analyzing and presenting it.
Additional Resources
This course only scratches the surface of what is possible with Bayesian statistics. I have prepared a handout with a lot of additional recommendations for reading about these powerful methods and how to use them.
Class Web Site
In addition to Brightspace , I have set up a companion web site for this course at https://ees5891.jgilligan.org , where I post the reading and homework assignments, my slides from class, and other useful material. That web site will be the central place to keep up with material for the course during the semester. This web site will direct you to Brightspace if there is anything you need to find there.
Computer Software
For this class, we will work in R, and I strongly recommend that you install the free version of RStudio Desktop for working with R. All the software we will use this semester is free and can be downloaded and installed on Windows, Mac, and Linux systems. You can find details at the tools page of the ees5891.jgilligan.org
Assignments
Overview of reading assignments
I will give out detailed reading that give specific pages to read for each class and notes on important things you should understand. I expect you to complete the reading before you come to class on the day for which the reading is assigned, so you can participate in discussions of the assigned material and ask questions if there are things you don’t understand.
Graded Work
Homework
Homework must be turned in at the beginning of class on the day it’s due.
Project
In the second half of the semester, you will do a research project, in which you will choose a data set that’s interesting to you and apply Bayesian methods to analyze it. You will present the results of your project in class during the last week of the semester and turn in a written report about your project.
Tests and Examinations
There will not be any tests or examinations in this course. Your grade will be based on class participation, homework, modeling projects, and inclass presentations.
Basis for Grading
Class participation  5%  
Homework  45%  
Research Project  50% 
Final Note:
I have made every effort to plan a busy, exciting, and instructive semester. I may find during the term that I need to revise the syllabus to give more time to some subjects or to pass more quickly over others rather than covering them in depth. Thus, while I will attempt to follow this syllabus as closely as I can, you should realize that it is subject to change during the semester.