Statistics 476: Monte Carlo Methods
Winter 2009

Instructor: Professor Steve Lalley
Office: 118 Eckhart Hall
Office Hours: Wednesday 2:00 - 3:00
Phone: 702-9890
E-mail: lalley "atsign" galton.uchicago.edu

This 5-week course will be a brief introduction to several useful techniques of simulation:

• importance sampling and sequential importance sampling
• MCMC (Markov chain Monte Carlo)
• Gibbs sampling
• perfect sampling

The utility of these methods will be illustrated by a number of substantial examples, including

• self-avoiding random walks
• enumeration of contingency tables with fixed margins
• simulation of Ising models
• code-breaking

Books:

• Jun Liu, Monte Carlo Strategies in Scientific Computing
• Casella & Robert, Monte Carlo Statistical Methods
• J. M. Hammersley & D. Handscomb, Monte Carlo Methods
• Markov Chains and Mixing Times by David A. Levin, Yuval Peres, Elizabeth L. Wilmer. See especially Chapters 13 and 14.

Articles:

• Radford Neal, Probabilistic Inference Using Markov Chain Monte Carlo Methods
• Chen et. al., Sequential Monte Carlo Methods for Statistical Analysis of Tables
• Madras & Sokal, The pivot algorithm: A highly efficient Monte Carlo method for the self-avoiding walk
• Alan Sokal, Monte Carlo Methods in Statistical Mechanics
• J. Propp and D. Wilson, Coupling from the past: a user's guide

Web Pages:

• David Wilson's Exact Sampling Web Page

Optional Exercises:

• HW 1: Max-Flow Min-Cut

Project 1:

Decode this coded text. The text is an encrypted English message. The encryption is done by a simple single-character substitution cipher, that is, a permutation of the characters of the original message. I suggest trying MCMC on the posterior distribution when the prior distribution is the uniform distribution on all permutations. You will have to develop methods for (a) computing a likelihood, and (b) running a suitable Markov chain on the space of permutations. You might find this web site useful. You might also find it helpful to try running your MCMC on chunks of the message of sizes 1000, 10000, etc., characters.

Project 2:

Here are two data sets, Ten Points and Twenty Points. Just for the heck of it, here's a third: Fifty Points. The points all lie inside the unit square. Use simulated annealing to find the shortest paths through the points in each of these data sets. Assume that there is a river that runs along the main diagonal x=y of the square, and that edges that cross this river count three times as much as edges that don't cross.

You should experiment with both the temperature schedule and the path mutation rules to see what works best, and you should do multiple runs to see how often you get trapped in local minima. Do most of your work first with the Ten Points data set, because the computation time for Twenty Points is likely to be much greater.