Newton: 1-19-99

STAT 775: Bayesian Analysis

Time:
Tuesdays,Thursdays, 9:30-10:45
Place:
2345 Engineering Hall
Professor:
• Michael Newton
• Office: 4369 CSS
• Phone: 262-0086 or 263-0357
• e-mail: newton@stat.wisc.edu
Office Hours:
Drop in usually ok, or by appointment.
Main Text:
• A. Gelman, J.B. Carlin, H.S. Stern, and D.B. Rubin, 1995 Bayesian Data Analysis Chapman & Hall. (Ordered by UBS; Expected end September)
On Reserve at Wendt:
• G.E.P. Box and G.C. Tiao, 1973. Bayesian Inference in Statistical Analysis. Wiley Classics.
• J.M. Berardo and A.F.M. Smith, 1994. Bayesian Theory. Wiley.
• M. Schervish, 1995. Theory of Statistics. Springer-Verlag.
• J.O. Berger, 1985. Statistical Decision Theory and Bayesian Analysis 2nd edition. Springer-Verlag.
• C.P. Robert, 1994. The Bayesian Choice: A Decision-Theoretic Motivation . Springer-Verlag.
Student work:
• Regular Homework
• Midterm (date TBA); Exam.

Course Description

---

This course introduces graduate students to Bayesian statistical analysis. There are no graduate-level prerequisites, although students are expected to be familiar with essential features of probability and statistical inference as usually covered in an intermediate undergraduate course. A basic premise in Bayesian analysis is that probability measures a degree of belief in any uncertain event, and thus is personalistic. The implication for scientific applications is that all inference proceeds by manipulating joint probability distributions. The first part of the course studies such probability distributions and conditional independence concepts in more detail. Monte Carlo methods arise naturally from these discussions, and we will consider static and dynamic methods by taking different graphical summaries of the dependence structure of a joint distribution. We take a predictive approach to Bayesian analysis, following de Finetti, and motivate the development of statistical modeling via exchangeability and the de Finetti theorem. From here, we study the key components of Bayesian analysis in {\em one-layer} problems, including prior, posterior, and predictive distributions. We introduce some basic elements of statistical decision theory at this stage, and discuss the Stein effect before moving on to more advanced modeling and inference problems. The theory and methods are illustrated with examples from a wide range of current research topics, and calculations are done in the Splus/R computer language.

---

1. Scope of Bayesian Analysis: accounting for uncertainty; large parameter spaces; combining information; knowledge representation; probability as degree of belief.
2. Structure of Joint Distributions:
   1. directed acyclic graphs; conditional independence; example structures, independence, Markov chain, hidden Markov model; direct simulation.
   2. undirected graphs; cliques; Markov random field; Gibbs distribution; Gibbs sampler; Hammersley Clifford Theorem.
3. Statistical Inference I:
   1. de Finetti's theorem: exchangeability, Polya urn.
   2. one-layer problems: prior, likelihood, posterior, prior predictive, posterior predictive. Bayes rule. Some exponential family examples including normal, dirichlet-multinomial.
   3. Priors: conjugate, non-informative, Jeffreys
4. Decision Theory (a sampling):
   1. risk, loss, Bayes risk, admissibility, Bayes estimate.
   2. classification
   3. Stein effect, shrinkage.
   4. hypothesis testing: prior/posterior odds; Bayes factor; connection to p-values.
5. Statistical Inference II: Advanced Topics: further examples such as ridge regression, splines, state-space models; hierarchical modeling; model checking/selection/averaging; large-sample theory; Schwartz criterion; robustness; nonparametric Bayes, as time permits.

---