
Introduction to the study of random processes, including Markov chains, Markov random fields, martingales, random walks, Brownian motion and diffusions. Techniques in probability, such as coupling and large deviations. Applications to image reconstruction, Bayesian statistics, finance, and probabilistic analysis of algorithms. After Statistics 241a or equivalent.

Instructor:

David Pollard

When: 
Mon, Wed 1:00  2:15 
Where: 
WLH 119 
Office hours: 
Tues 4:00 ?? (might be changed later in the semester), and after each class 
TA: 
Zhao Ren 
Problem session: 
to be arranged if needed 
Webpage: 
http://www.stat.yale.edu/~pollard/Courses/251.spring2013

Intended audience: 
Students (both graduate and
undergraduate) who are comfortable with introductory
probability (as covered in Stat 241/541).

Text: 
The course will be based on a book manuscript
written by Joe Chang. I will take some material from each chapter, posting supplementary notes if I deviate too far from Joe's treatment of any topic.
Table of contents for Chang manuscript
Topics to be covered (tentative):

Basic theory for Markov Chains with finite or countable state spaces: transition probabilities; periodicity; transience and recurrence; stationary distributions; convergence to stationary distributions. [Chang Chapters 1 and 2]

Markov random fields [Chang Chapter 3]

Martingales [Chang Chapter 4]

Brownian Motion and Diffusion [Chang Chapters 5 and 6]

Grades: 

The final grade will be based on the weekly homework (60%) plus a
final project (40%).
Early in the semester, I am also open to suggestions for alternative ways to assign final
grades.

Homework due each Wednesday. All help received for the homework must be explicitly acknowledged.

Students who wish to work in teams (no more than 2 to a team)
should submit a single a solution set. All members of a team will be
expected to understand the team's solutions sufficiently well to
explain the reasoning at the blackboard. Occasional meetings with DP
will be arranged.

Other: 
 Handouts.

Class materials for an introductory probability course
(Stat
241/541,
Fall 2011), containing more extensive elementary discussion of
probabilistic ideas.
See, in particular, the first two chapters, on (conditional) probability and (conditional) expectations.

The course would be more fun for me (and probably for you too) if I could assume that students were willing to engage in some matrix computation. I use R, which is available for free from
CRAN. It is not so hard to get R to carry out basic computations. Significant work is required to become a grand master.
