Instructor: Stephen Kissler
Class times: TBD
Class location: TBD
Office hours: TBD
Motivation and learning objectives
Bayesian inference is the state of the art when it comes to conducting rigorous, principled statistics on scientific quantities. Bayesian statistical theory itself is beautiful in how it parallels the way we naturally think: we engage with the world with sets of expectations that are adjusted iteratively through experience. Also, in the Bayesian framework, answers take the form of distributions rather than specific values, which I think is the most honest way of capturing the limits of our knowledge: no matter how much we’ve learned, we’ll always be just a little bit uncertain about things, and that’s ok.
Unlike much of frequentist statistics (think of your grandmother’s hypothesis testing), Bayesian inference usually makes heavy use of computers. The downside of this is that Bayesian inference can take time, and the algorithms can be fiddly to code well. The upside of this is that, with the help of computers, we can build unbelievably complex, layered models that allow us to make remarkable insights about the systems we study, going well past the bounds of traditional statistics.
By completing this course, you will be able to:
- Identify problems that would benefit from Bayesian thinking
- Explain the intuition behind Bayesian inference
- Develop hierarchical models to describe scientific phenomena
- Write computer code to fit these models to data
- Diagnose good vs. bad fits and understand how to fix the bad ones
- Communicate the results of a Bayesian analysis in a clear, principled, and rigorous manner
Ready? Let’s dive in.
Students should have completed the Calculus I/II sequence and Introductory Probability. Students should also have basic experience coding in R, Python, MATLAB, and/or Julia. We will extensively use the Stan platform for Bayesian computation, though no experience with that is necessary. Examples will be conducted in R.
The flipped classroom
This course uses a flipped classrom approach. Lectures are split into two parts of roughly 25 minutes each. The first part covers the motivation and theory for the day’s topic, while the second covers practical worked examples.
We’ll use class time to discuss key topics and to work together through additional examples, mainly drawn from the assigned problem sets. These sessions will be more than just a study hall: I will guide the sessions from the back of the classroom, and students will work through the problems on the board. It’s a bit like an oral exam, except I’ll be feeding you a lot more information and I won’t be evaluating you on what you do or don’t know, only whether or not you participate. To get the most from class time, it’s important to watch the lectures and have taken a crack at the problem set before each class!
For the in-class examples, I’ll randomly call on students using a random name generator (code here) to present concepts and work through problems at the board. I recognize there will be days when you’re away or just not on your game, so the board-work will be on an opt-in basis. Opting in will count for part of your grade (see Evaluation); to get full points, you’ll need to opt in to at least 30 of the 39 class sessions, so that we should have a decently large pool of students opting in each day.
Every other week, I’ll assign a problem set. The problem set will be released on the course webpage on Mondays at 9am and will be due on Fridays (12 days later) at 5pm, with adjustments made for holidays (e.g., if there’s a Monday holiday, I’ll release the problem set on Tuesday morning). Feel free to work with other students on the problem sets, but the write-ups should obviously reflect your own work. Problem sets will involve a mix of derivation-type problems and computational problems.
The end-of-term project makes up a major part of the course grade. The project is intended to involve a variety of the topics we’ve discusse during the course. A project proposal is due at the end of Week 9, leaving six weeks for completion.
You can work on teams of up to three; pairs and solo-work is fine, too, and I’ll adjust my expectations accordingly (a group of three should be able to accomplish somewhat more than a single person!).
Textbook and materials
This course will be based principally on Bayesian Data Analysis, 3rd Edition by Gelman, Carlin, Stern, Dunson, Vehtari, and Rubin.
A full PDF of the textbook is availble here along with additional materials.
We’ll also dip briefly into Rasmussen and Williams’ Gaussian Processes for Machine Learning, available here
30% of the course grade will be based on the final project. 10% will be based on in-class participation. The remaining 60% will be evenly split between weekly homework assignments. There are no exams.
Week 1: Probability and inference (BDA 1)
- Class 1: Basics of Bayesian inference (BDA 1.1-1.4)
- Class 2: Probability as a measure of uncertainty (BDA 1.5-1.8)
- Class 3: Computation and software (BDA 1.9-1.10)
Week 2: Single-parameter models (BDA 2)
- Class 1: Posteriors (BDA 2.1-2.3)
- Class 2: Priors 1 (BDA 2.4-2.7)
- Class 3: Priors 2 (BDA 2.8-2.9)
Week 3: Multi-parameter models (BDA 3)
- Class 1: Normal data (BDA 3.1-3.3)
- Class 2: Categorical data (BDA 3.4)
- Class 3: Multivariate normal data (BDA 3.5-3.8)
Week 4: Hierarchical models (BDA 5)
- Class 1: Constructing priors (BDA 5.1)
- Class 2: Constructing models (BDA 5.3-5.5)
- Class 3: Meta-analysis (BDA 5.6-5.7)
Week 5: Model evaluation (BDA 6-7)
- Class 1: Posterior predictive checking (BDA 6.1-6.4)
- Class 2: Cross-validation (BDA 7.1-7.2)
- Class 3: Model comparison (BDA 7.3-7.4)
Week 6: Modeling accounting for data collection (BDA 8)
- Class 1: Data collection models (BDA 8.1-8.3)
- Class 2: Designed experiments (BDA 8.4-8.5)
- Class 3: Observational studies (BDA 8.6-8.7)
Week 7: Introduction to Bayesian computation (BDA 10)
- Class 1: Numerical integration (BDA 10.1-10.2)
- Class 2: Sampling (BDA 10.3-10.5)
- Class 3: Computing environments (BDA 10.6-10.7)
Week 8: Markov chain simultion (BDA 11-12)
- Class 1: Gibbs, Metropolis, and Metropolis-Hastings (BDA 11.1-11.3)
- Class 2: Inference and assessing convergence (BDA 11.4-11.5)
- Class 3: Efficient sampling (BDA 12.1-12.5)
Week 9: Introduction to regression models (BDA 14)
- Class 1: A Bayesian approach to regression (BDA 14.1-14.4)
- Class 2: Assembling a regression model (BDA 14.5-14.6)
- Class 3: Dealing with unequal variances and correlations (BDA 14.7-14.8)
Week 10: Hierarchical linear models (BDA 15)
- Class 1: Batching regression coefficients (BDA 15.1-15.2)
- Class 2: Varying intercepts and slopes (BDA 15.4-15.5)
- Class 3: Analaysis of variance (BDA 15.6-15.7)
Week 11: Generalized linear models (BDA 16)
- Class 1: Working with GLMs (BDA 16.1-16.2)
- Class 2: Single-parameter GLMs (BDA 16.3-16.5)
- Class 3: Multivariate GLMs (BDA 16.6-16.7)
Week 12: Finite mixture models (BDA 22)
- Class 1: Setting up mixture models (BDA 22.1-22.2)
- Class 2: Label switching and specifying mixture components (BDA 22.3-22.4)
- Class 3: Mixture models for classification and regression (BDA 22.5)
Week 13: Gaussian process models (BDA 21)
- Class 1: Gaussian process regression (BDA 21.1-22.2)
- Class 2: Latent Gaussian process models (BDA 21.3-21.4)
- Class 3: Density estimation and regression (BDA 21.5)
Week 14: Project wrap-up
Week 15: Project presentations
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