STA 7828

Description and Goals

Monte Carlo methods are increasingly used in many scientific areas, including statistical physics (where they originated), Bayesian and frequentist statistical inference, and image reconstruction. The basic idea is to carry out a simulation to estimate quantities of interest that cannot be computed analytically. This course will begin with a brief discussion of some standard Monte Carlo schemes, before moving to Monte Carlo methods based on Markov chains. Consider the situation where there is a distribution π on some space, and we are interested in estimating π or R f dπ where f is some function, but π is analytically intractable. Markov chain Monte Carlo proceeds as follows. We set up a Markov chain with the property that its transition function has π as its stationary distribution. Then we run a chain X1, X2, . . . with this transition function. If the Markov chain converges to its stationary distribution (i.e. for large n, the distribution of Xn is approximately π), then by running the chain long enough, we can obtain a sample from π. This sample can be used to estimate π or some feature of it such as R f dπ. In this course I will explain the method in detail, describe the main implementations, and discuss some classes of problems in statistics, primarily in Bayesian inference, where it has had success. The method is not fool-proof. I will talk about some of the mathematical results pertaining to convergence issues, and also discuss some practical convergence diagnostics.

Final Grades

Exam 1: Wednesday October 2, 8:20 pm, room TBA. Note the evening time slot. 25%
Exam 2: Wednesday November 6, 8:20 pm, room TBA. Note the evening time slot. 25%
Final: Thursday December 12, 12:30pm–2:30pm. Comprehensive, but with
emphasis on material covered after Exam 2. 30%
HW: There will be about 8 homeworks assigned during the semester. 20%

Attendance and Late Policy

Homework must be turned in at the beginning of the lecture on the due date.
Late homework will not be accepted. The solutions to the homework assignments must be
entirely your own (this applies also to R code).

View the full course syllabus here (PDF).