Introduction to Bayesian Data Analysis

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

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.

• 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.

• 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: