Statistics 692: Introduction to Bayesian Analysis, Professor Michael Newton, Fall 2006.

Lectures: Tuesdays/Thursdays 9:30 - 10:45 in rm 133 SMI
Office hours: Tuesdays 11-12, Wednesdays 1-2 in rm 1245A MSC

Overview: Broadly speaking, statistical analysis aims to extract useful information from data in a way that accounts for underlying sources of variation. Bayesian statistical analysis is a major branch of statistics that achieves this information extraction by an explicit development of probability concerning what is unknown conditional on what is observed. This is an inversion of sampling theoretic probability, which characterizes the probability of data in terms of unknown parameters, and thus it relies on the famous mathematical result for such inversions called Bayes's rule.

In this course we will investigate many elements of Bayesian Statistical Analysis as outlined in the syllabus below. Main topics include a study of probability structures for multiple random variables, inference derived from such probability structures, and computational approaches to evaluate these inferences. We will explore statistical inference using case studies from biomedical sciences, social sciences, agriculture, business, and elsewhere. This course is aimed at advanced undergraduates in Statistics or graduate students with interest in the area.

Probability
1. Basic elements: events, random variables, urn models, exchangeability, independence, conditional independence, mixing, Bayes rule
2. Structure of joint distributions: directed acyclic graphs, conditional independence graphs, marginal and conditional distributions
3. Simulation techniques: direct sampling, Gibbs sampling

Inference

1. de Finetti's theorem
2. One-parameter models: likelihood, prior, posterior, prior predictive; inference via estimation, testing, confidence sets, likelihood principle
3. Multiparameter non-hierarchical models: marginal inference, normal models, multinomial-dirichlet
4. Hierarchical models: priors, shrinkage, empirical Bayes
5. Topics: model choice, model averaging, model checking, Markov chain computation, asymptotic approximations, decision theory

Examples

1. image analysis
2. aligning DNA or protein sequences
3. estimating population sizes
4. ranking varieties in agricultural field trials
5. ...and more...