Introductory material

0. Approach to teaching Bayes

1. Introduction to Bayes

2. Probabilities: a quick example on death penalty

3. Probability rules

4. Bayes theorem

Submarine Activity: click here

5. Quick overview of PMFs and PDFs

6. The likelihood function

• Likelihood for simple models
• Likelihood for regression models
• Notes on model notation and subscripts

7. Maximum likelihood estimation (MLE)

• Basic ideas

Logistic + Poisson regressions activity: click here

Biomass recovery activity: click here

Basics of Bayes

1. Conjugate likelihood-prior pairs:

• Basketball example I: Binomial-beta pair
• Basketball example II: Informative priors
• Basketball example III: Final remarks

Cancer risk activity: click here

2. An example for the normal-normal conjugate pair

3. Monte Carlo integration

• MC1
• MC2

4. Gibbs sampling

• Full conditional distributions
• Customized Gibbs sampler in R

Climate change activity: click here

5. Generative models and inverse modeling

6. JAGS

• Linear models: 2-sample t-test

ANOVA activity: click here

ANCOVA activity: click here

7. Convergence

• MCMC convergence
• MCMC lack of convergence

Mixture model activity: click here

Population size activity: description and activity

Educational data activity: click here

Occupancy model activity: click here

Sensitive question activity: click here

8. Metropolis-Hastings algorithm

• Motivation
• Algorithm
• Intuition

Poisson regression activity: click here

9. Mixed models

• Introduction to mixed models
• Shrinkage
• Correlation induced by random effects

Radon activity: click here

10. Sources of uncertainty

• Example of river pollutant
• Example of river pollutant across a range of covariate values
• Model fit and predictive distribution

River pollutant activity: click here

Less common but interesting models

1. Robust regression

• Introduction to robust regression
• Integrating \(\tau_i\) out from the robust regression model

Robust regression activity: click here

2. Models with latent continuous responses

• Introduction to models with latent continuous responses
• Modeling censored data

Censored data activity: click here

Ordinal data activity: theory and activity

3. Spatial and temporal correlation

• Introduction
• Simulating data and fitting an AR1 model
• Comparing results from AR1 model and simple regression

Geospatial model assignment: click here

Model notation assignment: click here

Spatial capture-recapture (SCR) model:

• Introduction
• Model 1
• Model 2
• Model 3
• Model 4

Troubleshooting algorithms and models

1. Improving MCMC algorithms by centering covariates

• Improving MCMC results
• Diagnosing the problem
• Centering covariates

2. Common problems in JAGS (and other MCMC algorithms)

• Bad data
• Bad initial values
• Unidentifiable parameters
• Unidentifiable parameters when covariates sum to one
• Speeding up MCMC algorithms

Debugging activity: click here

Identifiability activity: click here

There are many many other types of errors that JAGS can provide. Brian Reich describes a subset of them in his website https://www4.stat.ncsu.edu/~bjreich/BSMdata/errors.html.

Exploring other tools to fit Bayesian models

1. NIMBLE

Miscellanea

0. Learning how to program

1. Big sum and big product notation

2. “Proportional to” notation

3. Difference between “dnorm” and “rnorm”

4. Basic math facts

5. Derivation of normal-normal conjugate pair

6. Why can we deduce the posterior distribution just by knowing \(p(\pi|X)\) up to a proportionality constant?

7. Inner product to help write long expression in a regression model with many covariates

8. Additional modeling resources