topics in probability for statistics

Logo

course website for ubc stat 547c fall 2020-21 (Winter Term 1)

Syllabus

Description: A graduate-level course in measure theoretic 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, uses of image measures, and stochastic processes), ways to invert them (illustrative analysis of statistical estimators; conditioning/disintegration), and analysis of asymptotic properties (laws of large numbers, CLT). We will cover a range of topics, starting from language and foundations, and then focusing on statistics-relevant topics.

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 lectures.)

Class meetings: Tuesday/Thursday 3:30 - 5 pm, Zoom. (See Canvas for Zoom details.)

Instructor: Ben Bloem-Reddy

TA: Saif Syed

Communications

This term we will be using Piazza for class discussion. The system is highly catered to getting you help fast and efficiently from classmates, the TA, and myself. Rather than emailing questions to the teaching staff, I encourage you to post your questions on Piazza. If you have any problems or feedback for the developers, email team@piazza.com.

The signup link can be found on Canvas (left-hand navigation bar).

Please post course-related questions to Piazza.

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 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.

Statement regarding online learning for international students

During this pandemic, the shift to online learning has greatly altered teaching and studying at UBC, including changes to health and safety considerations. Keep in mind that some UBC courses might cover topics that are censored or considered illegal by non-Canadian governments. This may include, but is not limited to, human rights, representative government, defamation, obscenity, gender or sexuality, and historical or current geopolitical controversies. If you are a student living abroad, you will be subject to the laws of your local jurisdiction, and your local authorities might limit your access to course material or take punitive action against you. UBC is strongly committed to academic freedom, but has no control over foreign authorities (please visit http://www.calendar.ubc.ca/vancouver/index.cfm?tree=3,33,86,0 for an articulation of the values of the University conveyed in the Senate Statement on Academic Freedom). Thus, we recognize that students will have legitimate reason to exercise caution in studying certain subjects. If you have concerns regarding your personal situation, consider postponing taking a course with manifest risks, until you are back on campus or reach out to your academic advisor to find substitute courses. For further information and support, please visit: http://academic.ubc.ca/support-resources/freedom-expression.

Course objectives

This course is a foundation that will prepare you to:

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

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

Learning activities

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

Class meetings

Class will meet on Zoom on Tuesdays and Thursdays, 3:30 - 5 pm. (Please see the Canvas course site for Zoom info.) Class will be recorded and posted to Canvas. If, because of time zone constraints, you are unable to attend class, please schedule time to meet with me one-on-one.

Class will be quasi-flipped: you are expected to do all reading (textbook and/or notes) prior to class, and we will spend significant portions of class working on exercises/problems in small groups. The reading schedule will be updated regularly on Canvas.

In-class collaboration

We will use Zoom breakout rooms to work on exercises in small groups. Starting the second week of class, please be ready to collaborate using a Zoom whiteboard on a tablet or using a webcam and pen/paper. See here for details on your options.

Reading and notes

I will post (and update) my notes throughout the course. They may be found here.

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

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
Exercises 5% See description below
Learning logs 5% See description below
Assignments 40% 4-5 assignments; see description below
New problems 15% See description below
Final project and “exam” 35% 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 exercises, assignments, learning logs, in-class participation, office hours attendance, etc.), you will do fine.

Exercises

During every class meeting, we will spend time working in small groups on exercises, and presenting our work to the rest of class. These typically are 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. There are additional exercises throughout the course notes (see the List of Exercises after the Table of Contents).

The 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 start of the exam period, uploaded as a PDF to Gradescope. (Please accurately mark the pages corresponding to the exercise.)

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

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.

If you’re putting in the work, 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):

Assignments

There will be four or five assignments, roughly scheduled as: out on a Thursday, due in two weeks. Solutions must be LaTeXed (I will post a template for you to use), submitted as a PDF via Gradescope before class on the due date. (More details about submitting to Gradescope will be sent with the first assignment.)

These will be challenging, with the objective of building and deepening your understanding/mastery. 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.

New problems

Rather than solve problems set for you, you will come up with new problems appropriate for assignments. Designing a new problem requires you to think about things differently, and to understand the concepts at a deep level. Each student will create two sets of problems; due dates and more details will be announced in class.

Final project

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 5% of your final grade.

The remaining 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.