# MTH535A: An Introduction To Bayesian Analysis

## Course Description

In this course, the students will be introduced to the Bayesian approach to statistics. We will start with some preliminary concepts about probability theory, and then we will gradually move towards data analysis. Students will learn the main philosophical difference between the commonly-taught Frequentist approach and the Bayesian paradigm. They will see some benefits of using a Bayesian approach, in terms of better uncertainty quantification and obtaining results that are more intuitive and more interpretable.

## Course Content

• Basics of Bayesian paradigm, with some examples
• A brief review of probability theory
• Bayes’ theorem and its implications, some examples
• Introduction to Bayesian inference
• Summarizing a posterior
• Conjugate priors
• Objective Bayes
• Bayesian computing: deterministic methods
• Markov chain Monte Carlo
• Diagnosing and improving convergence
• Bayesian linear models
• Bayesian Model comparisons
• Hierarchical models
• Case study demonstration
• Frequentist properties of Bayesian methods, decision theory

Course Materials

Lecture notes/slides will be provided. Additionally, students can read the following books:

1. Bayesian Statistical Methods by Brian J. Reich, Sujit K. Ghosh. Published in 2019, by Chapman and Hall/CRC (this book will be mainly followed).
2. Bayesian Essentials with R by Jean-Michel Marin and Christian P. Robert. Published in 2014, by Springer.
3. Bayesian Data Analysis by Andrew Gelman, John Carlin, Hal Stern, David Dunson, Aki Vehtari, and Donald Rubin. Published in 2013 by Chapman and Hall/CRC. For non-commercial purposes, a free electronic copy can be downloaded from http://www.stat.columbia.edu/~gelman/book/.
4. A First Course in Bayesian Statistical Methods by Peter D. Hoff. Published in 2009, by Springer.
5. An Introduction to Bayesian Analysis by Jayanta K. Ghosh, Mohan Delampady, Tapas Samanta. Published in 2006 by Springer (printed Indian edition available).

Quiz

There will be roughly eight/nine quizzes and the best of six grades will be considered for final evaluation. Quizzes will be of the system-graded type (multiple choices with single or multiple options correct, True/False).

Final project

A final group project report will be due by April 22, 2022. The group sizes will be four or five. Tentative plan: Students have to reproduce methods of some research papers (students can choose and can discuss with me) and have to apply them to new datasets.

Final Semester Score = (30Q + 20M + 20F + 30P)/100, where Q is the Quiz average, M is the midterm grade, F is the final exam grade, and P is the final project grade (all out of 100).

Course structure

The course structure is inspired by “A Bayesian Statistics Course for Undergraduates: Bayesian Thinking, Computing, and Research” by Jingchen Hu (2020), Journal of Statistics Education, 28:3, 229-235.

Some of the course contents are influenced by the courses Applied Bayesian Analysis (MS level) taught by Brian J. Reich and Bayesian Inference and Analysis (Ph.D. level) taught by Subhasish Ghosal, at North Carolina State University, United States.

## Course Audience

Prerequisites

1. Basic probability theory
2. Some algebra and calculus
3. Basic visualisation techniques
4. Statistical computing using R/JAGS

## Outcomes of this Course

After completing this course, students will learn the concepts of the Bayesian approach, the key differences between the Bayesian and Frequentist paradigms, different types of priors, and they will have the ability to do basic data analyses.