Introduction to Bayesian Data Analysis (openHPI)

Introduction: Why are Bayesian methods important for data analysts?

Here are some of the advantages of Bayesian methods over the standard frequentist approach used in data analysis:

- Prior knowledge/expertise can be incorporated into the data analysis

- Models can be flexibly specified to reflect the assumed generative process

- The results of the analysis – the posterior distributions of the parameters of interest – have an intuitive interpretation

- Hypothesis testing can be carried out in a more meaningful manner than the standard used null hypothesis significance testing

Prerequisites: Who is this course for?

We assume the following in this course:

- Basic familiarity with the programming language R, openHPI offers a free R course for Beginners (in German)

- Experience with data analysis using linear models

- It is helpful (but not necessary) to have had some exposure to linear mixed models using the R library lme4

- High-school mathematics (pre-calculus)

- Some basic concepts from probability theory (sum and product rule, conditional probability)

This course is not appropriate for participants who don't know R programming and who have no experience at all with data analysis.

Course outcomes: What will you learn from this course?

- Some basic ideas relating to random variables

- Some fundamental properties of probability distributions

- Application of Bayes' rule in data analysis

- The concept of likelihood and its role in Bayesian statistical modeling

- Bayesian regression models using brms (a front-end for Stan)

- How to visualize and interpret prior and posterior distributions

- How to generate prior and posterior predictive distributions for evaluating models

- How to interpret the results of simple regression models

After completing this course, you will be in a good position to learn how to use more advanced Bayesian methods, such as hierarchical models, finite mixture models, multinomial processing tree models, measurement error models, etc.

What you'll learn

- Bayesian statistics

- Data analysis

- Bayesian regression models using brms

Course contents

Week 0 - Initial Setup:

Installing R and RStudio, rstan, brms, and other necessary packages in R; Setting up R markdown for reproducible data analyses.

Week 1 - Introduction:

Learn the foundational ideas about random variables and probability distributions; Reading: Chapter 1 of the textbook (excluding the section on bivariate distributions).

Week 2 - Bayesian data analysis:

Understand Bayes' rule, derive the posterior using Bayes' rule; visualize the prior, likelihood, and posterior; distinguish the relationship between the prior, likelihood, and posterior; incorporate prior knowledge into the analysis; Reading: Chapter 2.

Week 3 - Computational Bayesian data analysis:

Derive the posterior through sampling; perform simple regression modeling of a simple button-pressing task using Stan/brms; do prior predictive distributions, sensitivity analysis, and different classes of prior; do posterior predictive distributions; derive the log-normal likelihood; Reading: Chapter 3.

Week 4 - Bayesian regression and hierarchical models:

Perform simple linear regressions using the normal and binomial likelihoods to answer the following research questions: (i) Does attentional load affect pupil size? (ii) Does trial id affect response times? (iii) Does set size affect recall accuracy? Take a brief look-ahead at linear mixed models; Reading: Chapter 4 and up to section 5.3 of chapter 5.

Final Exam:

Final Exam