Bayesian Statistics
This course will cover Bayesian approaches
to modern statistics.
The Bayesian approach has emerged over recent
years as a key element in our toolbox for
handling large complex systems coherently.
Our emphasis will be on computing examples in R
(and hopefully some Python, and maybe some C++).
Course Materials will be available at
http://www.rob-mcculloch.org/.
Grades will be based on weekly assignments (50%)
and a final project (50%).
Both of these may be done in groups.
For the final project, you should do a real
data analysis using a Bayesian approach.
Do whatever you want as long as it is fun.
I will not collect many homeworks and I will
not check them carefully so really, the course
is about the project.
While there is no required textbook,
books that closely follow the topics we will
cover are:
``A First Course in Bayesian Statistical Methods''
by Peter D. Hoff
``Bayesian Ideas and Data Analysis,
An Introduction for Scientists''
by Christensen, Johnson, Branscum and Hanson
Topics are below.
Notes are on the webpage, but we may change
things as we go along.
Topics:
Bayes Theorem
The IID Bernoulli Model
conditional probability
The Normal Mean (standard deviation known)
The Normal Standard Deviation (mean known)
The Normal Mean with the constant
Testing equality of normal means
Bayesian Model Selection
multivariate change of variable
The Dirichlet Distribution and the multinomial
More on the Multivariate Normal
Markov Chains
Gibbs Sampling and Hierarchical Means
Regression
Stochastic Search Variable Selection
Probit (Albert and Chib) and Mixture of Normals
Metropolis Algorithm
AR(p)
Kalman Filter (intro to state space models)
Which roughly conforms to the book by Hoff:
1. Introduction and examples
2. One-parameter models
3. Multinomial data
4. Monte Carlo approximation
5. The Normal model
6. Posterior approximation with the Gibbs sampler
7. The multivariate normal model
8. Group comparisons and hierarchical modeling
9. Linear regression
10. Nonconjugate priors and the Metropolis-Hastings algorithm
11. Linear and generalized linear mixed effects models
12. Latent variable methods for ordinal data.
Other topics, time permitting:
Dirichlet Process priors.
Foundations: coherence, complete-class theorem, the Likelihood principal.