Modules

Week 1

Week Learning Objectives

By the end of this module, you will be able to

• Navigate the course website and Blackboard site
• Identify the Slack channels relevant for the course
• Describe the historical origin of Bayesian statistics
• Identify components in research papers involving Bayesian analyses
• Render a simple Quarto (.qmd) file

Task List

1. Review the syllabus
2. Review the resources (slides and note)
3. Install/Update R and RStudio on your computer
4. Attend the Tuesday and Thursday class meetings
5. Complete the assigned readings
   • McElreath ch. 1
   • Supplemental (i.e., optional) reading: Gigerenzer (2004)
   • Markdown Basics
6. Introduce yourself on the #introduction Slack channel (as part of HW 1)
7. Complete Homework 1 (see instruction on Brightspace)

Slides

Link to HTML slides

Week 2

Week Learning Objectives

By the end of this module, you will be able to

• Describe the subjectivist interpretation of probability, and contrast it with the frequentist interpretation
• Compute probability density using simulations
• Compute joint, marginal, and conditional probabilities with two variables
• Apply Bayes’ rule to obtain posterior from prior and data
• Explain what data-order invariance and exchangeability are
• Use grid approximation to obtain the posterior for a Bernoulli model

Task List

1. Review the resources (slides and notes)
2. Attend the Tuesday and Thursday class meetings
3. Complete the assigned readings
   • McElreath ch. 2, 3
   • James ch. 1
4. Complete Homework 2 (due in two weeks; see instruction on Brightspace)

Slides

Link to HTML slides

Week 3

Week Learning Objectives

By the end of this module, you will be able to

• Apply Bayesian workflow to analyze real data with a Bernoulli model
• Explain the idea of a conjugate prior
• Summarize the posterior distribution using simulations
• Apply Bayesian terminology in summarizing the posterior
• Use R to perform prior and posterior predictive checks

Task List

1. Review the resources (slides and notes)
2. Attend the Tuesday and Thursday class meetings
3. Complete the assigned readings
   • Johnson ch. 3
   • Getting started with CmdStanR
4. Complete Homework 2 (see instruction on Brightspace)

Lecture Videos

Recap of the Beta-Bernoulli Model

Check your learning

If we do not use a conjugate prior for the Bernoulli model, what is the distribution of the posterior \(\theta\)?

Normal distribution
Beta distribution
Something not from a known family of distributions

Summarizing the Posterior

Check your learning

In the posterior distribution above, the interval bounding the shaded area is

an equal-tailed credible interval
a highest density interval
a one-sided credible interval

Posterior Predictive Check

Check your learning

A researcher asks a participant 10 true/false questions to assess their statistics knowledge. A posterior predictive distribution in this case would be

a probability distribution of the knowledge of the participant, given their answers
a probability distribution of future true/false responses from the participant, based on the posterior distribution of their statistics knowledge
a probability distribution of the observed responses from the participant, based on the prior distribution of their statistics knowledge
a probability distribution of the knowledge of the participant, before incorporating the observed responses from them

Stan (Part 1)

Stan (Part 2)

Check your learning

Which of the following is the correct way to declare a data variable of responses on a 5-point scale from N participants?

array[N] int<lower=1, upper=5> y
array[N] int<lower=1, upper=5> y;
vector[N] y
vector[N] y;

Stan (Part 3)

Slides

Link to HTML slides

Week 4

Week Learning Objectives

By the end of this module, you will be able to

• Explain the logic of a hierarchical model
• Apply the binomial distribution to describe the sum of multiple Bernoulli trials
• Program a hierarchical binomial model in Stan
• Analyze secondary data using a hierarchical normal model (i.e., random-effect meta-analysis)

Task List

1. Review the resources (slides and notes)
2. Watch the lecture videos below (to be posted)
3. Complete the assigned readings
   • McElreath ch. 13.1, 13.2
   • Gabry et al. (2019)
   • Gelman et al. (2020)
4. Start working on Homework 3 (see instruction on Brightspace)

Lecture Videos

Hierarchical Models

Check your learning

In terms of statistical inference, a binomial variable \(Z\) with N = 10 and theta (\(\theta\)) is equivalent to

10 Bernoulli variables that are dependent, and each has the same theta
10 Bernoulli variables that are exchangeable, and each has a different theta
10 binary variables that are dependent, and each has a different theta
10 binary variables that are exchangeable, and each has the same theta

Partial Pooling

Check your learning

Based on the notation in the slides, a Beta2(0.5, 6) distribution is the same as

A Beta distribution with a = 2, b = 6
A Beta distribution with a = 3, b = 3
A Beta distribution with a = 4, b = 2

Check your learning

A Gamma distribution is handy as a prior for kappa (\(\kappa\)) because

A Gamma distribution is defined in the range 0 to infinity, which matches the range of kappa
A Gamma distribution only has one parameter, which is easy to work with
A Gamma distribution is a conjugate prior to the Beta model

Hierarchical Binomial

The following videos are from an older class, so the slides may look slightly different.

For the first video, please skip to 04:12.

Check your learning

With shrinkage, the posterior distribution of individual-specific \(\theta_j\) is

pulled farther away from the center of the common distribution of theta
typically more concentrated due to borrowing information from other clusters
A Gamma distribution is a conjugate prior to the Beta model

Hierarchical Normal

The following videos are from an older class, so the slides may look slightly different.

Check your learning

In the hierarchical normal distribution discussed in the lecture, the treatment effect estimate in each study

has the same mean with the hierarchical model
has a different mean but the same standard deviation
has a different mean and a difference standard deviation

Slides

Link to HTML slides

Week 5

Week Learning Objectives

By the end of this module, you will be able to

• Interpret the coefficients in a linear regression model
• Obtain posterior predictive distributions and checks
• Explain how the assumptions of regression are coded in the model equations
• Perform Bayesian regression with the R package brms
• Interpret results from an interaction model using plots and posterior predictions

Task List

1. Review the resources (slides and notes)
2. Attend the Tuesday and Thursday class meetings
3. Complete the assigned readings
   • McElreath ch. 4, 5, 7, 8
4. Complete Homework 3 (see instruction on Brightspace)

Slides

Link to HTML slides

Week 6

Week Learning Objectives

By the end of this module, you will be able to

• Explain how information criteria approximates out-of-sample divergence from the “true” model
• Use WAIC and LOO-IC to compare models

Task List

1. Review the resources (slides and notes)
2. Attend the Tuesday and Thursday class meetings
3. Complete Homework 4 (see instruction on Brightspace)

Slides

Link to HTML slides

Week 9

Week Learning Objectives

By the end of this module, you will be able to

• Draw a directed acyclic graph (DAG) to represent causal assumptions
• Use a DAG to guide analyses for obtaining causal effects
• Describe how randomization can remove potential confounders
• Explain how the back-door criterion can be used to identify a set of adjusted variables with nonexperimental data
• Perform a mediation analysis and interpret the results

Task List

1. Complete the assigned readings
   • McElreath ch 6
2. Review the resources (slides and notes)
3. Attend the Tuesday and Thursday class meetings
4. Complete Project Prospectus (see instruction on Brightspace)
5. Schedule a meeting with the instructor for your prospectus (sign-up link will be posted on Slack)

Slides

Link to HTML slides

Week 10

Week Learning Objectives

By the end of this module, you will be able to

• Explain what is unique for samples using Markov Chain Monte Carlo (MCMC)
• Explain why we need MCMC to approximate the posterior
• Describe when MCMC samples are representative and accurate for approximating the posterior
• Use R to perform convergence diagnostics for MCMC samples

Task List

1. Complete the assigned readings
   • McElreath ch 9
2. Review the resources (slides and notes)
3. Attend the Tuesday and Thursday class meetings
4. Complete Homework 6 (see instruction on Brightspace)

Slides

Link to HTML slides

Week 12

Week Learning Objectives

By the end of this module, you will be able to

• Describe the three components of the generalized linear model (GLM)
• Name examples of the GLM (e.g., linear regression, Poisson regression)
• Obtain posterior predictive distributions and checks

Task List

1. Complete the assigned readings
   • McElreath ch 10, 11
2. Review the resources (slides and notes)
3. Attend the Tuesday and Thursday class meetings
4. Complete Homework 7 (see instruction on Brightspace)

Slides

Link to HTML slides

Week 13

Week Learning Objectives

By the end of this module, you will be able to

• Provide examples of clustered data
• Fit Bayesian multilevel models (MLMs)
• Name advantages of MLM
• Interpret coefficients in MLM

Task List

1. Complete the assigned readings
   • McElreath ch 13, 14.1, 14.2
2. Review the resources (slides and notes)
3. Attend the Tuesday class meetings
4. Continue to work on final project

Slides

Link to HTML slides

P.S.: If you’d like to print the slides to PDF, follow https://quarto.org/docs/presentations/revealjs/presenting.html#print-to-pdf