Course Syllabus

This page exists to make the following information accessible to the full MIT community. Enrolled students should access this information via the internal pages.

Course Structure

18.05 is taught in a Technology Enabled Active Learning (TEAL) classroom to facilitate discussion, group problem solving, and mentored coding. You may be familiar with these rooms and active learning from 8.01 / 8.02. We really like this format and have found that most students do too as a hands-on introduction to probability, Bayesian inference, and frequentist statistics.

We meet three times a week in 32-082 (basement of Stata):

• Class is TR 1-2:30 or 2:30-4pm. Classes are a blend of lecture, discussion, concept questions, and group problem solving.
• Studio is open F 2 - 5pm and should take one hour. Students should arrive at 2pm or 3:30pm, depending on their section. Studios are guided, long-form coding exercises that explore and apply the concepts from reading and class.

Before class

In our flipped classroom, you will thoughtfully read assigned materials before class (there is no external textbook). We do not expect that you to have mastered the material on first reading. Rather, the goal is to make learning more productive in class and save you time overall.

Each reading is accompanied by online reading questions hosted on MITx. These will help you reflect on the reading and prepare you for more challenging problems during class and on the problem sets. Reading questions are due before class and graded.

During class

Class includes concept questions and group problems.

Participation on concept questions is graded and only accessible during class.

Group problems are solved at the whiteboard with mentorship from staff. You should form three groups of 3 from the 9 students at your table. Let staff know if you need help.

Studio

Broad Learning Goals

• Learn the language and core concepts of probability theory
• Understand the principles of statistical inference, both Bayesian and frequentist
• Build a starter statistical toolbox with appreciation for both its utility and limitations
• Use software and simulation to do probability and statistics (R)
• Become an informed consumer of statistical information in research and media
• Appreciate growing real-world challenges related to privacy, fairness, and causality in policy, health, and climate.
• Prepare for further coursework or on-the-job learning in statistics and machine learning

Here's one student's feedback from the 2024 subject evaluation:

Super interesting subject! I loved the applications and how interactive it was through R studio and the TEAL format. I just opened my Options/Derivatives Textbook for my summer job and nearly every statistical concept was covered in this class in great detail with further applications in R. Feeling super prepared and happy about taking this class!!

Curriculum

In support of these learning goals, the curriculum consists of three units as reflected in the course calendar.

1. Probability (5 weeks)
   1. Counting and sets
   2. Random variables, distributions, quantiles, mean, variance
   3. Conditional probability, Bayes theorem, base rate fallacy
   4. Joint distributions, covariance, correlation, independence
   5. Law of large numbers and central limit theorem
2. Bayesian inference (3 weeks)
   1. Bayesian inference with known priors, odds, credible intervals
   2. Bayesian inference with unknown priors
3. Frequentist statistics (5 weeks)
   1. Significance tests
   2. Confidence intervals
   3. Bootstrapping
   4. Linear and logistic regression

Pedagogically, the order of these units is motivated by another natural division (this will make a lot more sense by the end of the course). Regarding statistics as applied probability, we formulate this division as:

1. Pure probability: uncertain world, perfect knowledge of the uncertainty
   • Unit 1. This is just math, like solving an algebra or calculus problem
2. Pure applied probability: data in an uncertain world, perfect knowledge of the uncertainty
   • Unit 2A. This is still just math, now using Bayes theorem to update our knowledge of the uncertainty based on data
3. Applied probability: data in an uncertain world, imperfect knowledge of the uncertainty
   • Unit 2B and Unit 3. Now one leaves the mechanics of math and reaches a fork in the road with two paths forward:
     1. Bayesian: make up knowledge of the uncertainty (via a prior) and apply Bayes theorem to the observed data
     2. Frequentist: do one's best using the probability of observed and potential data under hypothetical values of the uncertainty (via the likelihood function)

We take the Bayesian path first because it is a smoother transition from probability and far more intuitive than the frequentist path. Having seen Bayesian posteriors and credible intervals, students are more prepared to interpret frequentist p-values and confidence intervals correctly (or at least, to know what they are not).

We also develop principles of statistical thinking across the curriculum by:

• Using R and applets to compute, simulate and visualize statistical information
• Reflecting on statistical analyses in media and research papers
• Engaging with issues of privacy, fairness, and causality in policy, health, and climate.

Specific Learning Goals

Probability

Students completing the course will be able to:

1. use basic counting techniques (multiplication rule, combinations, permutations) to compute probability and odds
2. use R to run basic simulations of probabilistic scenarios
3. compute conditional probabilities directly and using Bayes theorem, and check for independence of events
4. set up and work with discrete random variables, including Bernoulli, binomial, geometric and Poisson
5. set up and work with continuous random variables, including uniform, normal, exponential, and beta.
6. explain what expectation and variance mean and be able to compute them
7. understand the law of large numbers and the central limit theorem
8. compute the covariance and correlation between jointly distributed variables
9. use available resources (books, wikipedia, AI) to learn about and use other distributions as they arise

Statistics

Students completing the course will be able to:

1. create and interpret scatter plots and histograms
2. understand the difference between probability and likelihood functions and find the maximum likelihood estimate for a model parameter
3. do Bayesian updating with discrete and continuous priors to compute posterior distributions and posterior odds
4. construct estimates, credible intervals, and predictions using the posterior distribution
5. understand the framework of null hypothesis significance tests (NHST) and p-values
6. select and apply NHST, including the z-test, t-tests, chi-squared test
7. find confidence intervals for parameter estimates, including order statistics
8. use bootstrapping to estimate confidence intervals
9. set up a least squares fit of data to a model
10. fit and interpret simple linear regression between two variables
11. fit and interpret multivariate linear and logistic regression (using R)
12. understand the difference between correlation and causation, including Simpson's paradox, and the importance of experimental design

Students will be familiar with the following distributions:

• Discrete: Bernoulli, binomial, geometric, hypergeometric, Poisson, and categorical.
• Continuous: uniform, exponential, normal, beta, t, chi-squared, F, gamma, Dirichlet, Cauchy, Laplace, Pareto, and Weibell.
• Bivariate normal and the notion of a high-dimensional normal with covariance matrix.

Grading and policies

Grades are a weighted average of the following seven components:

1. Reading questions, 5%
   • Due before class (12pm)
   • Autograded on MITs for correctness
   • Multiple attempts allowed
2. Concept questions, 5%
   • Only accepted during class 
   • Autograded on MITx for completion (not correctness)
   • We will discuss them in class
3. Problem sets, 25%
   • Due most Mondays by 9pm
   • Numerical checker on MITx allows you to check and discover mistakes ahead, multiple attempts allowed, ungraded
   • Submitted on Gradescope, graded for explanations
4. Studios, 10%
   • Designed to be completed during Studio but may be submitted by 6pm on Friday
   • Submitted on Gradescope, graded for clarity of code and correctness on additional test cases
5. Exam 1, 12.5%
   
   • In-class on Friday, March 7
   • 1 page cheat sheet, turned in, counts for 5%
6. Exam 2, 12.5%
   • In-class on Friday, April 18
   • 2 page cheat sheet, turned in, counts for 5%
7. Final Exam, 30%
   • In 50-340 on Tuesday, May 20, 9-12
   • 3 page cheat sheet, turned in, counts for 5%

Sometimes the workload at MIT piles on all at once or life throws an emotional curveball. To give you some breathing room, we will drop your lowest-scoring reading questions, concept questions, problem set, and studio.

Students generally score higher on the non-exam components than exams. Think of the problem sets, studios, reading questions, and attendance as putting money in the bank.

Grade cutoffs are as follows:

• A range: At least 90% overall.
• B range: At least 80% overall and 70% weighted exam average.
• C range: At least 70% overall and 60% weighted exam average.
• D range: At least 60% overall and 50% weighted exam average.

"Overall" includes the final exam. No rounding beyond the 64-bit precision afforded by modern computers.

Furthermore, in-person attendance is mandatory for classes (TR) and highly encouraged for Studios (F). Multiple unexcused absences will result in a warning. Additional unexcused absences will result in a full-grade deduction.

Collaboration and Academic Integrity

You are expected to adhere to the standards explained here: MIT Academic Integrity. If you have any questions, ask a staff member!

Collaboration on assignments

MIT has a culture of teamwork. We encourage you to work with study partners. However:

• You must write your solutions yourself, in your own words.
• You must list all collaborators and outside sources of information.
• You may not search for solutions in 18.05 materials from previous years.
• You may not use AI to generate solutions.

Piazza discussion board

Piazza can be a great resource for helping each other understand the material and problem sets. We encourage collaboration and learning communities, with the following ground rules:

• You may not provide solutions to assignments, even partially.
• You may help clarify what's being asked, shed light on a concept, or direct others to relevant materials.
• Please be kind and respectful.

In-class concept questions

For the sake of your learning and respect for those around you, devices must be put away except during concept questions. Repeated violations may result in loss of attendance credit.

Do not fake attendance by answering concept questions outside of class, or by having another student answer them for you. We will handle this as a major violation of academic integrity with serious consequences, including the loss of all concept question credit for all parties involved.