topics in probability for statistics

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course website for ubc stat 547c fall 2022 (Winter Term 1)

Syllabus

Acknowledgement

UBC’s Point Grey Campus is located on the traditional, ancestral, and unceded territory of the xʷməθkʷəy̓əm (Musqueam). The land it is situated on what has always been a place of learning for the Musqueam, who for millennia have passed on their culture, history, and traditions from one generation to the next on this site. We are fortunate to study, work, live, and play in this place.

More information: https://indigenous.ubc.ca/

Teaching team

Instructor: Ben Bloem-Reddy

TA: Johnny Xi

Teaching approach

I love measure theory and probability, and I hope you will, too!

I approach this class knowing that every student enters with different preparation, and with different goals. I will do my best to support your learning, regardless of from where you’ve come or where you’re going. This course can be challenging, and I am striving to make it a collaborative endeavour in which we work together to learn with and from each other. There will be plenty of opportunities for help and advice from peers and from the teaching team. Please reach out if there is something we could do better.

I am in a multi-year process of trying to incorporate more in-class learning activities and examples to bring these topics to life. Suggestions, ideas, etc., are always welcome.

Commitment to equity and inclusion

I am committed to supporting an inclusive learning environment, and I am continually learning how best to do so. If you have concerns that I or someone else may not be upholding this commitment, I invite you to either talk with me if you feel comfortable, or share your thoughts on an anonymous feedback survey. If in class discussions there are derogatory, harassing or hateful statements made I will intervene to help prevent further harm and uphold a respectful class environment. My pronouns are he/him/his, and I invite you to use the option on Canvas to provide your pronouns (find out how in the Canvas Student Guide).

Course overview

Description: Probability is the most widely used mathematical model of randomness and uncertainty. Measure theory is a branch of analysis upon which much of probability is built. In the first part of STAT 547C, we will build probability from the bottom up: beginning by asking how to construct collections of sets from which we will eventually build probabilistic queries; studying how properties of those collections might be transferred between spaces used in statistical models; and then figuring out how to assign (probability) measure in a coherent way. By the end of the first part, we will have done enough to build the probability spaces and random variables that are the primary objects in statistics and related fields. In the second half of the course, we will build on the foundations to explore in depth familiar concepts from undergraduate probability: expectation, independence, conditioning, and convergence. Along the way, we will see examples of how and where these abstract ideas appear in statistics, machine learning, and related fields.

Learning in this course is a collaborative effort led by you, with support from the teaching team. Learning will be evaluated based on a combination of individual and group work emphasizing regular practice, resourcefulness, and engaging with the concepts in ways that are meaningful to you. (Details are in Assessments.)

Formal pre-requisites: Ideally, one upper-division undergraduate course in probability and one in analysis. More details and some references can be found here (If you’re not sure, come talk to me after one or two class meetings.)

Informal pre-requisites: At least one of the following.

Class meetings: Regular attendance and participation are an important part of your learning in the course. Class meets in person in Room 208 of the Neville Scarfe building every Monday and Wednesday, 12:30 - 2:00 pm. On rare occasions, class may be held virtually on Zoom.

The scheduled class time can be a bit snoozy…

Communications

Please pay attention to announcements posted to the course home page.

Please use email for course-related communications (or talk to me before/after class).

Course objectives

If you are willing and able to meet the requirements, by the end of this course you will be able to:

More specific learning objectives can be found at the beginning of each section of the lecture notes.

Learning activities

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

Class meetings

It is helpful to read (textbook and/or notes) prior to class. I will cover certain important results in detail and devote time to examples. We will also work on and share our approaches to in-class exercises.

Reading

Textbooks

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

Complements and references:

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
Learning logs 10% See description below
Participation 10% See description below
Assignments 50% (roughly) weekly assignments; see description below
Final project 30% Project and final reflection; more details to follow

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 assignments, learning logs, in-class participation, office hours attendance, etc.), you will do fine.

Policy on concessions

If circumstances arise that prevent you from attending class or completing an assignment, 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 three “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 0.8 each day that it is late. If you are using a late day, you must let me know before the assignment is due.

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 ask that you regularly reflect on your efforts and progress in a weekly learning log. At the end of each week, you will upload to Canvas a PDF 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 your final grade. Feel free to discuss with classmates in order to get started.

Learning logs should be uploaded to Canvas weekly on Mondays. The LaTeX template to use is on the Assignments page.

If you’re putting in the work and thinking carefully, it will be clear here. If you’re struggling with something, writing it 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):

Participation

In each class meeting we will have at least one “think-pair-share” activity: I describe a problem, everyone thinks/works independently for a few minutes, we discuss our thoughts/work in pairs or small groups, and then someone volunteers to share with the entire class. The point is not necessarily to get the exercise correct; it is to practice thinking through a problem and communicating your thinking. Participating at each point of the activity is important for your learning. As a small incentive, your participation grade is based on you sharing with the entire class at least twice during the term; hopefully you will find the activities useful enough to not need additional incentive.

Assignments

There will be weekly(-ish) assignments, roughly scheduled as: out on a Wednesday, due in one week. Solutions must be LaTeXed (template on the Assignments page), submitted as a PDF via Canvas before class on the due date.

These will be a mix of exercises from the textbook and more challenging problems. Most of the grading will be binary (you make a good effort at the problem or not), and one or two problems will be graded in detail. You will (hopefully) learn something new—not covered in lecture—in the course of doing the assignment.

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; feedback is optional and welcome.

In-person class

Covid Safety in the Classroom

Masks: Masks are no longer required for indoor public spaces on campus, including classrooms. For our in-person meetings in this class, it is important that all of us feel as comfortable as possible engaging in class activities while sharing an indoor space. Non-medical masks that cover our noses and mouths are a primary tool to make it harder for Covid-19 to find a new host. If you choose to, please wear a mask during our class meetings. If you have not yet had a chance to get vaccinated against Covid-19, vaccines are available to you, free and on campus (http://www.vch.ca/covid-19/covid-19-vaccine). Please pay attention to trends in community transmission; mask-wearing may be recommended if infection rates rise.

Your personal health

If you’re sick, it’s important that you stay home, no matter what you think you may be sick with (e.g., cold, flu, other). You can do a self-assessment for Covid symptoms here: (https://bc.thrive.health/covid19/en) Do not come to class if you have Covid symptoms, have recently tested positive for Covid, or are required to quarantine. This precaution will help reduce risk and keep everyone safer. You can check this website to find out if you should self-isolate or self-monitor: (http://www.bccdc.ca/health-info/diseases-conditions/covid-19/self-isolation#Who).

If you do miss class because of illness:

Instructor health

I will do my best to stay well, but if I am ill, develop Covid symptoms, or test positive for Covid, then I will not come to class. If that happens, then either class will be held on Zoom or there will be a temporary replacement instructor. Our classroom will still be available for you to sit and attend an online session, in this (hopefully rare) instance.

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.