Registration Deadline: January 24, 2027

Instructor: Professor Sebastian Ferrando, Toronto Metropolitan University

Course Dates: TBA
Mid-Semester Break: February 15-19, 2027
Lecture Time: TBA
Office Hours: TBA

Registration Fee:

• Students from our Principal Sponsoring & Affiliate Universities: Free
• Other Students: CAD$500

Capacity Limit: 15 students (Auditing students require prior approval from the instructor. If you would like to audit this course, please get permission from Professor Ferrando first and then email Shu-Chen Kuo for manual enrolment.)

Format: Online via Zoom

Course Description

This course provides a mathematically grounded introduction to causal inference centered on structural causal models and directed acyclic graphs, with particular emphasis on the framework developed by Judea Pearl. The presentation begins with concrete causal situations and asks how a physical or scientific description is represented by a DAG, what the graph asserts, what is observable, and what causal conclusions can actually be justified.

The mathematical core develops conditional independence, graphical separation, interventions, observational and interventional distributions, and the identification problem. Special attention is given to the exact hypotheses used in each result and to the distinction between defining a causal quantity and determining it from observable information.

A substantial part of the course is devoted to probabilistic identification: truncated factorization, fully observed systems, back-door and front-door identification, do-calculus, non-identifiability, and partial identification by bounds. A later module develops dynamic observable identifiability, moving from distributional identification to state-by-state questions about what can be learned from factual observable information as it accumulates.

A third module consolidates these ideas through selected examples and exercises from Pearl, Glymour, and Jewell, and introduces counterfactual reasoning as a conceptual bridge to Module 4. Additional advanced directions may be introduced as time permits.

Learning Objectives:

• Formulate causal questions precisely within a structural causal model and distinguish observational, interventional, and counterfactual quantities.
• Read and analyze DAGs using paths, colliders, conditional independence, and d-separation.
• Explain what identifiability means relative to an ambiguity class of causal models sharing observable information.
• Derive and justify standard identification formulas, including truncated factorization and back-door and front-door adjustment.
• Recognize when a causal effect is not point identified and understand the role of partial identification and bounds.
• Work through realistic examples in which causal interpretation, rather than formal manipulation alone, determines the appropriate mathematical question.
• Understand the distinction between probabilistic, pathwise, and dynamic observable identifiability, including the role of future factual information.
• Read advanced causal-inference literature critically and present a self-contained technical topic with a developed example.

Proposed Prerequisites: Mathematical maturity and a solid course in probability are expected, e.g. a clear understanding of conditional independence and conditional expectation is important. No prior course in causal inference is required. The course is intended to be mathematically self-contained on the causal concepts it uses.

Course Structure and Contents

The following structure records the present course plan. Modules 1–4 form the core. Selected further material will depend on available time and may also be covered through student presentations.

Module 1 — Structural causal models and probabilistic foundations

• Finite structural causal models: DAGs, exogenous and endogenous variables, structural equations, and induced probability laws.
• Observed and hidden variables, observational information, and ambiguity classes of observationally indistinguishable causal models.
• Joint and conditional laws, independence, conditional independence, sigma-algebras, and conditional expectation as needed for the causal theory.
• Recursive structural generation on a DAG; structural versus probabilistic dependence; Markov factorization under independent local exogenous inputs.
• Chains, forks, colliders, active paths, d-separation, and the resulting conditional-independence statements.
• Construction and interpretation of causal DAGs from physical or scientific descriptions; limitations of graphical information and a brief discussion of Markov equivalence.

Module 2 — Interventions and probabilistic identifiability

• Natural trajectories versus interventions; interventions as modifications of structural mechanisms rather than conditioning on observed events.
• Observational equivalence, ambiguity classes, and identifiability as uniqueness of a causal quantity across compatible models; identification versus estimation and partial identification.
• Truncated factorization and complete identification in fully observed systems, including the non-confounded case.
• Hidden variables and the limitations of direct recovery from observational kernels.
• Back-door identification: criterion, adjustment theorem, proof, successful and failed adjustment, and worked examples.
• Front-door identification and do-calculus as a general language for transforming interventional expressions.
• Non-identifiability and causal bounds: identified sets, finite optimization and linear-programming formulations, imperfect compliance, and valid versus sharp bounds.

Module 3 — Applications, exercises, and a bridge to counterfactual reasoning

• Selected examples and exercises from Chapters 1–3 of Pearl, Glymour, and Jewell, used to consolidate the material of Modules 1–2 in a different notation, vocabulary, and style of presentation.
• Worked problems emphasizing causal interpretation, graphical analysis, interventions, confounding, adjustment, and identification.
• From Chapter 4, introduction to counterfactual reasoning, especially the structural interpretation of counterfactuals and the abduction–action–prediction procedure, as a conceptual bridge toward Module 4.

Module 4 — Dynamic observable identifiability

This module develops new material that is not part of the standard causal-inference literature. It introduces a pathwise and dynamic approach to causal ambiguity and observable information, with close mathematical connections to incompleteness, hedging, and superhedging in financial mathematics.

• From probabilistic identification to the state-by-state question of determining an intervened outcome Y^x(ω) from factual observable information.
• Causal completions, factual observable histories and fibres, causal payoffs, and pathwise identifiability.
• Current lower and upper causal envelopes and finite methods for computing them over compatible causal scenarios.
• Future factual observability: what ambiguity can disappear through ordinary observation as the system evolves, and the corresponding future-observable residual.
• Eventual factual identification and observable reductions in ambiguity, with computations based on fibres, finite optimization, and dynamic programming.
• Probabilistic counterparts based on conditional expectation and the evolution of causal assessments as observable information accumulates.
• Distinguishing uncertainty that future observation can resolve from uncertainty that requires an additional intervention or experimental inquiry.

Selected further directions — time permitting

• Mediation, direct and indirect effects, and more advanced counterfactual questions.
• Dynamic or policy interventions and sequential decision problems.
• Connections with causal discovery, adversarial learning, game theory, and modern AI.
• Further current research topics suggested by student interests and available time.

Assessment

Assessment is entirely project-based. Each student gives two technical presentations during the term. Each presentation is accompanied by a written note. The two presentation projects have equal weight.

Presentation project requirements:

• Oral component: approximately 15 minutes of prepared presentation, followed by questions and discussion.
• Written component: approximately 5–10 pages.
• The presentation and written note must define all important terminology used and make the causal question mathematically precise.
• Each project must contain at least one well-developed physical, scientific, or empirical example. Purely formal examples should not be the sole content.
• A substantial result should be stated and explained. When feasible, the proof or the main proof idea should be presented rather than cited without explanation.
• The student should explain explicitly how the topic connects with material already developed in the course.
• Sources must be identified clearly. The written note should be sufficiently self-contained to be useful to another student in the class.

Topic allocation:

Students will choose one topic from the first set and one topic from the second set. More than one student may present the same topic; duplication is expected and is not a problem. Students who share a topic should develop their presentations independently and are encouraged to use different examples or complementary viewpoints. Final scheduling and topic assignment will be confirmed after enrollment is known.

The list of approved presentation topics will be distributed separately in class.

Course Logistics and Working Policies

• Delivery: The course will be delivered online to allow real-time participation by students from different universities. Fields will provide the Zoom account used for the lectures.
• Weekly pattern: The planned format is 3 contact hours per week over two days: one 2-hour meeting and one 1-hour meeting. Exact days and times are TBA.
• Recordings: Fields plans to record, edit, and post lectures as unlisted videos accessible to registered students. Any later public release of the recordings is subject to the instructor’s consent.
• Registration: Students, including TMU students, register through the Fields Institute registration process. The registration deadline is January 24, 2027; late registration requires instructor approval.
• Auditing: Auditing requires prior instructor permission and direct arrangements before registration.
• LMS: The Fields LMS (Moodle) will be used for course materials and communication. A university-restricted LMS will not be used because students from multiple institutions will participate.
• Attendance and Participation: Because the course is discussion-based and includes student presentations, regular attendance and active participation are expected. Students should ask questions when definitions, assumptions, or arguments are unclear.
• Credit: Students seeking credit must arrange accreditation with their home department. Fields does not itself arrange university credit.

Academic expectations for presentation:

Presentation work must represent the student’s own understanding and synthesis. Collaboration and discussion with classmates are encouraged, especially when students select the same topic, but each student must prepare an independent oral presentation and written note and must be able to answer questions about the mathematical and causal content.

Principal Readings and References

• J. Pearl, M. Glymour, and N. P. Jewell, Causal Inference in Statistics: A Primer, Wiley, 2016. Primary source for many examples and exercises.
• J. Pearl, Causality: Models, Reasoning, and Inference, 2nd ed., Cambridge University Press, 2009. Main advanced reference for structural causal models, identification, interventions, counterfactuals, and related topics.
• Instructor course notes: Structural causal models and probabilistic foundations; Interventions and probabilistic identifiability; Dynamic Observable Identifiability; additional notes distributed during the term.
• Additional research papers will be assigned selectively for presentation topics and advanced modules.