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The blog for the course will be found here. Please look at it regularly. Use it to ask questions or post comments. If you know the answer to a question, use it to answer. You will need a google account to post comments.

For completeness, the old blog will be found here, in case you need to look up something from the first part of the course.

Here is Jeff's handout, "R Basics"

0. Preliminaries

1. Introduction

2. Simple Examples

3. Bayes' Theorem

4. Interpretation

5. Single and Multiparameter Models

6. Bayesian Computation

7. Doing the Integrals

8. Normal Linear Models

9. Hierarchical Bayes

10. Priors

11. Bayesian Hypothesis Testing

12. Model Selection/Averaging

1. (Due 1/22/09) Read Chapter 1 of Albert's book; Do problems 4 and 5 on pp. 16-17. Remember, we prefer that you do this in small groups of two or three.

2. (Due 2/5/09) The problem set is on a pdf file that you can download by clicking here. Please read the blog entry for 1/27/09 for formatting and other issues.

3. (Due 2/12/09) The pdf file for the third problem set is here.

4. (Due 2/19/09) The pdf file for the fourth problem set is here.5. (Due 2/26/09) The pdf file for the fifth problem set is here.

6. (Due 3/5/09) The pdf file for the sixth problem set is here.

7. (Due 3/19/09) The pdf file for the seventh problem set is here.

8. (Due 4/2/09) The pdf file for the eighth problem set is here.

8. (Due 4/9/09) The pdf file for the ninth problem set is here.

(Due 4/16/09) This one is on a web page, and can be found here and printed out.

5. This is the multiparameter normal example from class 2/10/09.

6. Here is the betabinomial.r code we discussed in class on 2/19/09.

7. Here is the discrete Gibbs sampler code we discussed in class on 2/26/09.

7. Here is code for the homework set #6

7. Here is the code we ran in class on 3/5/09.

8. Here are the two code files we ran in class on 3/26/09. One is the R file, and one is the WinBUGS file.

8. Here is the combined R/WinBUGS file from class on 3/31/09.

8. Here is the combined R/WinBUGS file from class on 4/2/09

9. Here are the R/WinBUGS file and the data file for class on 4/9/09. Please look them over before class on Thursday.

Bayesian Inference (Stat 295)

This is a course in Bayesian statistics. Bayesian inference is a powerful and increasingly popular statistical approach, which allows one to deal with complex problems in a conceptually simple and unified way. The recent introduction of Markov Chain Monte Carlo (MCMC) simulation methods has made possible the solution of large problems in Bayesian inference that were formerly intractable. This course will introduce the student to the basic methods and techniques of modern Bayesian inference, including parameter estimation, MCMC simulation, hypothesis testing, and model selection/model averaging in the context of practical problems.

Bayesian Computation with R (Jim Albert, Springer Verlag) Please download R and the packages from this book and install them on your computer, as we will be using both in this course.

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Optional, but valuable:

Bayesian Data Analysis, Second Edition (Andrew Gelman, John B. Carlin, Hal S. Stein and Donald B. Rubin. London: Chapman and Hall)

Introduction to Statistical Thought (Michael Lavine), available here as a free web download.

Review of probability calculus. Interpretations of probability (e.g., frequency, degree-of-belief). Coherence. Bayes's Theorem. Joint, conditional, and marginal distribution. Independence. Prior distribution, likelihood, and posterior distribution. Bayesian estimation and inference on discrete state spaces. Likelihoods, odds and Bayes factors. Simple and composite alternatives.

Markov Chain Monte Carlo (MCMC) simulation as a method for practical calculation of Bayesian results. The Gibbs sampler. Metropolis-Hastings sampling. Metropolis-within-Gibbs sampling. Computer tools, e.g., BUGS,S+, R.

Bayesian point and interval parameter estimation. Bayesian credible intervals. Comparison with frequentist parameter estimation and confidence intervals. Bayesian inference on Gaussian distributions. Maximum Likelihood estimation as a Bayesian approximation. Laplace's approximation. Bayesian inference in non-Gaussian cases, e.g., Poisson, Cauchy, and arbitrary distributions. Linear and nonlinear models. Errors-in-variables models. Selection models. Hierarchical models

Prior selection. Subjective and objective priors. Priors as a way to encode actual prior knowledge. Sensitivity of the posterior distribution to the prior. Priors for hierarchical models.

Bayesian hypothesis testing. Comparison with frequentist hypothesis testing. Model selection and model averaging. Reversible jump MCMC for models of variable size. Approximations, e.g., AIC, BIC. Philosophical issues, likelihood principle, and the Bayesian Ockham's Razor.

The course grade will be based 80% on the assignments and 20% on class participation. By class participation, we mean that we will often leave unanswered questions in the notes that will be found on the web. You should read the notes in advance and attempt to answer these questions for yourselves. We will ask students for their answers to these questions in class. Also, we will sometimes ask for students' ideas about how they solved the assignments.

In general, we encourage students to work on the assignments in small groups of two or three (maximum). Statistics is by nature a cooperative enterprise. Statisticians act as experts in that field and advise clients (who are experts in their fields) on how to apply statistics to their problem. By working in groups, we hope to foster this sort of cooperative attitude between students in the class. If a group works on an assignment, we would like one paper turned in for the group, with everyone's name at the top. It goes without saying that we expect that everyone who works in a group will contribute roughly equally to the final result. For example, in a programming assignment, each member of the group should attempt to program the problem, and the group should then try to work out differences (e.g., if different students in the group arrive at different results, the group should try to figure out why this is so, to locate the sources of the discrepancies and fix them; if no resolution can be found, then the students should turn in a paper that displays the several different attempts with a discussion explaining the group's best understanding of the reasons for the discrepancy). Similarly, if a problem is worked and different members of the group obtain different answers, a similar resolution should be attempted, and if no agreement is obtained, the group should present a discussion. Our role will be to examine what each group presents and comment on them, as well as to provide a grade.

Jeff Buzas' information: Office hours Mondays from 10-11 and Wednesdays from 2-3 in room 105 Mansfield House, or by appointment. Phone number 656-2971, Email buzas@cems.uvm.edu .

Bill Jefferys' information: Office hours TTh 11:30-12:30. Office location is 107 Lord House. Phone number N/A. Email bill@bayesrules.net

A very basic discussion of the intuitive basis of Bayesian reasoning can be found at http://yudkowsky.net/bayes/bayes.html. This contains some javascript calculators to try out simple calculations.

Tom Loredo's Bayesian Inference in the Physical Sciences (BIPS) website has a lot of useful information about Bayesian inference. Note particularly the first five items in his Bayesian Reprints page, which are very nice tutorials on practical application of Bayesian inference. He also has extensive pointers to other websites including software, reprint archives, etc.

The book by E. T. Jaynes can be found in a preliminary form here.

First Bayes is a software package that is intended to help students with the first steps in understanding Bayesian inference. It runs under Windows. It concentrates on simple, closed-form examples but may be helpful to you.

The International Society for Bayesian Analysis (ISBA) is the international Bayesian organization. It sponsors meetings and publishes a newsletter. Dues are not expensive, and for students are set at a reduced rate of $10/year.

Bayesians Worldwide contains links to the home pages of a large number of Bayesians. Many of these individuals maintain collections of their reprints. Most of the prominent Bayesians are listed.

The Bayesian Songbook contains songs that have been presented at various Bayesian meetings over the years. Just for fun. There are also links to pictures of the infamous "Cabarets" at which these songs were sung.

Carnegie-Mellon University's statistics group has a library of many different statistics packages, including Bayesian packages. It can be accessed here.

Although CMU archives the R package, it's best to go to the R Project (CRAN) homepage, since you'll probably get the most recent version of it. Click here. R runs on Windows, Linux, UNIX and Macintosh (OS 9 or higher). The introductory tutorial for R can be found here. Many add-on packages for R are available at CRAN.

The BUGS project at the University of Cambridge offers the BUGS (Bayesian inference Using Gibbs Sampling) package. It does both Gibbs and Metropolis-Hastings sampling, and the software can be downloaded here. It runs on Windows and the "classic" version runs on UNIX. There is no Mac version of BUGS. However, if you purchase Virtual PC you can run it on a Mac (at reduced speed), but Virtual PC is not free. Virtual PC is sold by Microsoft. Another (cheap) system is sold by iEmulator. The new Macintosh computers (based on Intel chips) can run BUGS using Boot Camp and your own copy of Windows, and they will run it at full speed. There are other systems that will also run Windows on the new Macintoshes, for example, Parallels.

S Plus is not free, but there is a fairly inexpensive student package. It is sold by Mathsoft. Most of the functionality of S Plus can be found in the free R package (above) so unless you need something not available in R, you don't need to buy S Plus. Also, S Plus has had some memory management problems that cause problems in large simulations. R does not have this problem.

Matlab is another software package that has become popular for MCMC simulations. It is faster than R or S Plus. A student version is available. Matlab can be instructed to produce C or C++ code, which will run very fast.

Another software package that has been used successfully in MCMC simulations is Gauss, sold by Aptech.

Minitab is used by quite a few people.

SAS is extremely powerful. It is reputed to be the most difficult of the popular packages to learn. There are versions for Windows, and UNIX. The recent Intel Macintoshes can run it under Boot Camp.

Yet another popular package is SPSS. There are versions for Windows and Macintosh.

This page is under construction. Keep tuned for new material.

All materials at this website Copyright (C) 1994-2009 by William
H. Jefferys. All Rights Reserved.