The first release
This edition contains twelve compact chapters, three linked browser experiments, worked numerical examples, exercises with expandable solutions, references, and a teaching plan. The capstone connects a synthetic training decision to an explicit sensitivity calculation.
The mathematical route moves from expectation and projection to identification, inference, and decisions. Later chapters introduce instruments, panels, local designs, orthogonal scores, dependence, and generalization. Those chapters are entry points to substantial literatures. They cannot replace the proofs and practice of a full graduate sequence.
The browser experiments cover:
- Causal worlds. The same observed relation is compatible with different intervention responses. Its allowed parameter range is chosen for teaching; it is not a data-derived identified set.
- Repeated sampling. The experiment studies confidence-interval behavior in a stated Gaussian model with known variance. Extensions to estimated variance, dependence, and misspecification require additional arguments.
- Design sensitivity. A bound on an untreated-trend difference produces a range of effects. That range represents the chosen identifying restriction and excludes sampling uncertainty.
The measurement investigation
The measurement extension adds an original narrative essay, a worked undergraduate field guide, and a doctoral technical companion. Two additional browser experiments investigate ordinal recoding and robust comparisons under empirical and value uncertainty. The same hypothetical service-design case connects all three reading experiences.
The companion states and proves elementary teaching propositions, identifies their established lineage, and formulates research and educational hypotheses. Those proposals are untested. The synthetic rubrics are assumed for instruction and are not validated instruments.
Where the material comes from
The prose, worked teaching cases, and browser experiments in this course were prepared for this edition. Established results are attributed through chapter-level references and the reading room. Author-hosted texts and original papers provide the route to fuller statements and proofs.
The course links earlier public work as companion material. Pictures of Inference is a separate, coauthored project by Ian Helfrich and Elizaveta Gonchar. Its authorship and licensing remain attached to that work. Links to older sites do not imply that their full contents were incorporated or independently revalidated here.
No downloaded textbook PDF is distributed with this edition. No private student exercise, submission, or dataset is used as public course evidence. Synthetic examples are labeled where they appear, and invented business quantities should never be quoted as empirical findings.
Verification
The public repository contains the source and verification code. Numerical checks can establish that an implementation satisfies specified identities and reproduces its stated examples. They cannot establish the validity of an empirical identifying assumption.
This is a development edition. Its pedagogy has not been evaluated in a controlled teaching study, and it has not undergone external scholarly peer review as a complete book.
Coverage of likelihood, Bayesian modeling, structural estimation, time series, spatial methods, and current causal research is selective. Additional readings are needed for specialist work. A recent paper is marked as a reading path with its status; inclusion does not imply that its method is implemented in the browser.
Notation
The table fixes the role of each main variable across the twelve chapters. Coefficients, errors, functions, and other symbols are defined where they appear and can recur with a local meaning; in chapter 9, for example, K(·) is a kernel weight function, and W denotes the controls in chapter 3, a weighting matrix in chapter 7, and the observation (Y, D, X) in chapter 10.
| Symbol | Role |
|---|---|
| Y | Outcome: output, or whether an employee meets the output target. |
| D | Training, the treatment of interest. Participation (0 or 1) in most chapters; intensity or hours in chapters 1, 3, 6, and 10. |
| Y(d) | Potential outcome under training status d; Y(1) with training and Y(0) without. |
| X | Pre-treatment covariates. In general regression and likelihood formulas (chapters 3, 4, 7, and 11), the regressors. Exercise 2 in chapters 2 and 3 uses X and Y as generic random variables. |
| K | Prior experience, a binary pre-treatment covariate (chapters 2 and 3). |
| U | An unobserved characteristic or confounder. |
| Z | An instrument (chapter 7). |
| R | The running variable of a regression discontinuity design (chapter 9). |
| S | Inclusion in the analyzed data: an observed outcome (chapter 6) or trial participation (chapter 11). |
| E[·] | Expectation, the population average. No variable is named E. |
| τ | A treatment effect. Each chapter states whether it is an individual effect or an average, and over which population: all employees, compliers, adopting offices, or a target population. |
Conventions
- Population quantities, sample estimators, and realized estimates have distinct roles. Each worked argument defines the relevant objects.
- A fitted association receives a causal interpretation only through an explicit model or design.
- A confidence interval, a prediction interval, and an identified set answer different questions. The accompanying assumptions determine which guarantee applies.
- An identified set is always taken relative to the maintained model class. Chapter 1 defines it, and chapters 8 and 12 state the restrictions behind each set they report.
- Consistency has two meanings. An estimator is consistent when it converges in probability to its target; the causal consistency assumption equates an observed outcome with the potential outcome under the treatment received. Chapter 1 defines estimator consistency; chapters 2 and 5 define the causal assumption and mark the difference.
- A numerical decision requires an objective, units, and a target population. Statistical significance supplies none of these by itself.
- Optional reading progress and notes are stored in the browser. No account is required. Keep a separate copy of notes you need to retain.
Using and improving the course
You can link to a chapter, work through the public examples, and inspect the chapter source when preparing a session. Source availability does not change the separate rights attached to linked books, papers, data, or companion projects. Consult the repository’s license terms before redistributing or adapting material.
For a correction, send the page address, the disputed equation or sentence, and a small reproducible example if computation is involved. Explain the assumptions under which you think the result fails. A proposed counterexample is especially useful.
Development dates and substantive corrections should remain visible as the edition changes. Links and publication status can age; the reading room records when its sources were checked. The source history provides the record of changes to this site.
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