topics in probability

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course website for ubc stat 547c fall 2019-20

Syllabus

Description: A graduate-level course in probability with an emphasis on how the theory is applied in statistical science and related areas. For example, construction of statistical models (the role of independence, multivariate random variables, and stochastic processes), ways to invert them (illustrative analysis of statistical estimators; Monte Carlo applications), and analysis of their asymptotic properties (laws of large numbers, CLT). We will cover a range of topics mainly from a user’s point of view, starting from language and foundational topics, and then focusing on statistics-relevant topics selected from: convergence theorems, Poisson processes, Gaussian processes, martingales, Markov chains, stationary processes, probabilistic symmetry. Examples and exercises will focus on applications in statistics and machine learning.

Co-requisite: STAT 460/560

Pre-requisites: Ideally, one upper-division undergraduate course in probability and one in analysis. (If you’re not sure, come talk to me after one or two lectures.)

Lectures: Monday/Wednesday 3:00 - 4:30 pm, ESB 4192

Instructor: Ben Bloem-Reddy

TA: Saif Syed

Policy on concessions

If circumstances arise that prevent you from attending class, completing an assignment, or taking an exam, please let me know as soon as possible. UBC’s policy on academic concessions is here. If you have grounds for academic concession, we will work together to find something that works.

The default policy for assignments: you have two “late days” to be used at your discretion during the term. When you have run out of late days, any further late days will result in the grade of the late work to be multiplied by 0.75 each day that it is late.

The default policy for exams: except in very special circumstances, make-up exams will not be offered. In the case that you miss an exam, its contribution to your overall grade will be redistributed to the other exam(s). If you miss all exams, you will not be given a passing grade; depending on the circumstances other concessions may be available (e.g., late withdrawal).

Statement on UBC’s values and policies

UBC provides resources to support student learning and to maintain healthy lifestyles but recognizes that sometimes crises arise and so there are additional resources to access including those for survivors of sexual violence. UBC values respect for the person and ideas of all members of the academic community. Harassment and discrimination are not tolerated nor is suppression of academic freedom. UBC provides appropriate accommodation for students with disabilities and for religious and cultural observances. UBC values academic honesty and students are expected to acknowledge the ideas generated by others and to uphold the highest academic standards in all of their actions. Details of the policies and how to access support are available here: https://senate.ubc.ca/policies-resources-support-student-success.

Course objectives

This course will prepare you to: understand probabilistic language and ideas in research papers; apply ideas from probability in your coursework; conduct research using probabilistic methods; study advanced topics in probability.

Knowledge: language and mathematical foundations of probability as it is used in modern statistics, machine learning, and related fields; special topics in modern probability.

Skills: techniques for analyzing probabilistic problems; technical tools for proving relevant properties.

Learning activities

Class meetings will be our primary mode of interaction. However, most of your learning will occur outside of class through (see below for details on each):

Reading and lecture notes

I will post (and update) my lecture notes throughout the course. They may be found on the Files page.

Disclaimer: These really are just notes to keep me organized during lecture. They will be a good reference for what we’ve covered, but they are not intended as a substitute for the textbooks.

Textbooks

Please note: This is different from the course description document previously posted on my website. If you have already purchased Grimmett & Stirzaker, it’s a good book to have, and I will list supplemental readings from it on the schedule. Alternatively, you can return the book and, if necessary, borrow/photocopy/scan from my copy.

Primary textbook: E. Çinlar, Probabiilty and Stochastics, available as a PDF through the UBC library.

Complements and references:

A. Gut, Probability: A Graduate Course, available as a PDF through the UBC library.

G. Grimmett and D. Stirzaker, Probability and Random Processes, 3rd ed.

R. Durrett, Probability: Theory and Examples, 5th ed. available as a PDF on the author’s website

Assessment

If I had my way, all assessments would be formative. However, the university (and others) require summative assessments (i.e., your final grade). Your final grade will be calculated as follows:

Category Contribution Notes
Exercises 5% See description below
Participation 10% See description below
Assignments 25% Four assignments; see description below
Midterm 20% In-class on October 16
Final 40% Format to be determined (project or take-home exam)

My primary concern is that you learn probability theory to the level of the course objectives. If you work hard and demonstrate what you are learning (via exercises, assignments, learning logs, in-class participation, office hours attendance, etc.), you will do fine.

Exercises

Every lecture will have exercises scattered throughout. These are typically not as challenging as the assignments, and are meant to get you comfortable thinking about the basic ideas, manipulating the relevant mathematical objects, and presenting some intermediate results and their proofs. Material from the exercises may appear on assignments and exams, though for any of these I will state the result in lecture and the exercise will consist of proving said result.

Solutions will be posted, and we may go over some of them in class. Depending on demand/availability, Saif may go over some of them during a problem session/office hours.

Exercises should not consume huge amounts of your time, but working through problems is the best way to master the material. The exercises are meant to keep you engaged with the material in a regular way that assignments aren’t typically able to do. Your attempts at the exercises will be marked on a binary scale that aims to capture effort/engagement: 0 = no or hardly any attempt, or wildly off the mark conceptually; 1 = good attempt, at the very least on the right track conceptually. If you try and get stuck, explain where you got stuck and what you (think you) would need to know in order to get unstuck (this may get you a 1).

Your solutions don’t need to be LaTeXed (though it would be better if they were). Your solutions can be submitted at any time before the final exam, uploaded as a PDF via Canvas. Please upload your work throughout the term. (It may be the case that the solution to an exercise is made available on the course website before you attempt the problem. Additionally, the solutions to many of the exercises can be found online. You will benefit far more from attempting to do the problem without consulting the solution.)

Cumulative points as a percentage of total possible points will determine the contribution to the overall grade. If you get at least 50% of the possible points for exercises, it will be treated as 100%, and decrease from there:

Percentage of possible points Score on Exercises portion of overall grade
>= 50 100
45 90
40 80
35 70
30 60
25 50
20 40
15 30
10 20
5 10
< 5 0

Participation

Participation will be assessed through in-class participation, attendance at office hours, and regular submission of learning logs.

Learning logs

Often, we don’t take the time for self-reflection during a course. This amounts to wandering through a forest without keeping track of where you’ve been and where you’re going. (Which can be nice, but can be harmful when trying to learn.) Especially when trying to learn conceptually/technically challenging material, I have found it helpful to step back to assess my understanding (or lack thereof).

To this end, I am experimenting with something new: learning logs. At the end of each week, you will upload to Canvas a plain-text file in which you reflect on your efforts, progress, and challenges over the week. These are a way to keep track of your learning and to keep in touch with me throughout the course. Grading will be binary (0 = no submission; 1 = submission) and count towards the Participation portion of your final grade. Feel free to discuss with classmates in order to get started.

If you’re putting in the work, it will be clear here. If you’re struggling with something, writing that out can help clarify where you’re stuck and what steps you need to take. If you feel like you understand something, trying to distill it into simple prose often reveals a gap in understanding.

Some prompts (these are just to get you thinking; feel free to use your own):

Assignments

There will be four assignments, roughly scheduled as: out on a Monday, due in two weeks. Solutions must be LaTeXed (I will post a template for you to use), submitted as a PDF via Canvas before class on the due date.

These will be challenging, with the objective of building and deepening your understanding/mastery. You will (most likely) learn something new—not covered in lecture—in the course of doing the assignment.

Typically, there will be some required questions, along with a bonus question or two—these are optional, but making a good attempt at the bonus questions will go a long way toward mastery of the concepts and methods.

I encourage you to discuss assignment problems with your classmates. Solutions must be written up independently. Additionally, please state who and/or what materials you consulted while working on the assignment.

Final exam

There will be an in-class written exam: “What I have learned in this course.” (I will provide more details later in the term.) This will count for 10% of your final grade.

The final 30% of your grade will come from a final project. It may include updating a classic paper, writing a technical report on a topic we don’t cover in class, or proposing/starting a research project. We will talk more about this in the coming weeks.