Probability for Statistical Inference 2 (430-2-20)
Instructors
Miklos Zoltan Racz
Meeting Info
Technological Institute LG68: Tues, Thurs 11:00AM - 12:20PM
Overview of class
This course is the second in a two-quarter sequence of graduate probability theory with an eye towards statistical inference. The first course covers an introduction to mathematical probability theory including measure theory, law of large numbers, and the central limit theorem. In this course, after a brief review of the aforementioned topics, we cover Markov chains, conditional expectation, martingales, Poisson processes, and selected advanced topics, together with statistical applications.
A brief outline of topics covered is as follows:
Lectures 1 - 2: Introduction and overview. Review of measure theory, concentration inequalities, law of large numbers, central limit theorem.
Lectures 3 - 7: Markov chains: introduction, stationary distribution, convergence, ergodic theorem, recurrence and transience. Statistical applications.
Lectures 8 - 9: Conditional expectation.
Lecture 10: Midterm exam.
Lectures 11 - 14: Martingales: introduction, stopping times, optional stopping theorem, applications, martingale convergence, and more. Statistical applications.
Lectures 15 - 16: Poisson processes: introduction, superposition, thinning. Continuous time Markov chains. Statistical applications.
Lectures 17 - 18: Selected topics, time and interest permitting, which may include: Brownian motion, heavy tails, extreme value theory, Markov chain mixing times, large deviations, concentration of measure, random matrices, and more, with statistical applications.
Week 10: Review and final exam.
Registration Requirements
STAT 430-1 or permission of instructor. (If you would like to take the course and have not taken STAT 430-1, please email the instructor to discuss.)
Learning Objectives
The course is designed for PhD students whose ultimate research will involve rigorous mathematical probability, with a focus on statistical applications. The learning objectives are to develop a rigorous understanding of mathematical probability and to understand the deep connections between probability and statistics.
Teaching Method
Interactive Lectures
Evaluation Method
There will be homework problem sets throughout the semester (approximately weekly), as well as a midterm and a final exam. Your final score is a combination of your performance in these, with the following breakdown: HW 30%, midterm 30%, final 40%.
Class Materials (Required)
Probability Theory: lecture notes by Amir Dembo (for a similar course at Stanford), 2021. Freely available online: https://adembo.su.domains/stat-310c/lnotes.pdf
Class Materials (Suggested)
There are many texts that cover first year graduate probability. While the focus and scope of this course is slightly different, these texts can be valuable resources. David Aldous has an extensive annotated list here and here; in particular, consider consulting:
- Probability: Theory and Examples (5th Edition) by R. Durrett, 2019. Freely available online: https://services.math.duke.edu/~rtd/PTE/pte.html
- Probability and Measure (3rd Edition) by P. Billingsley, 1995.
Class Notes
Please email the instructor with any questions.