1 Introduction to Causal Inference
These notes were automatically generated by AI and lightly reviewed by the lecturer. They may still contain mistakes and factual errors.
1.1 Causality and inference
Let’s begin this lecture with four questions. Give yourself a moment to answer each in up to three words or phrases:
- What is mathematics?
- What is statistics?
- What is causality?
- What is design?
Causal inference draws on mathematics and statistics, but people have thought about its questions for much longer, and well beyond either discipline.
The opening questionnaire received 19 responses to each question. These clouds group related answers into themes. Each answer is assigned to one main theme; larger type means more answers in that group.
The answers suggest the following starting points:
- Mathematics: logic, axioms and deduction; abstract objects, structures and patterns; numbers and problem solving. Phrases included “reasoning about structures”, “Working from axioms” and “Art of problem solving”.
- Statistics: learning, inference and conclusions from data; probability, random processes and models; description and prediction. Examples included “Applied probability”, “Learning from data” and “Description of Reality”.
- Causality: links, influence and consequences; mechanisms behind observations; how one variable changes another. Answers ranged from “Not correlation” to “Here to learn”.
- Design: choice, construction and experimental setup; methods of observation and inference; invention for a purpose. “Choice” appeared twice; “No idea” and “not sure.” also appeared.
There was room for humour too: mathematics included “asking claude to do my sheets”, while statistics included “good to put on my cv” and “Potential future job”. These are starting points for discussion, rather than agreed definitions. The answers about causality in particular suggest quite varied understandings.
And what about proof? It seemed absent from the word cloud displayed in class. One response in the discussion was that proof was too obvious to mention; another was that it had indeed been submitted. The written responses include “Collection of ideas centered around deducing knowledge from only that which is already proved.” The display may not have shown everything.
We will ask these questions again later in the course. Linking answers over time will give us a longitudinal dataset to examine together. If you believe your answers improve, I shall try not to take full causal credit: it is not all about this course! It also matters how you think about the lectures, reflect on their ideas and make them your own.
1.1.1 Four words and four causes
The word histories provide an orientation:
| Word | Root | Meaning |
|---|---|---|
| Mathematics | Greek mathēma | That which is learned |
| Statistics | German Statistik | Describing the state |
| Causality | Latin causa | Cause, reason |
| Design | Latin designare | Mark out, designate |
The term statistics dates from the eighteenth century. It originally concerned official records: collecting government data and making sense of what was happening in a state.
Aristotle (384–322 BCE) distinguished four explanatory roles for a cause. Consider a bronze statue:
| Cause | What it explains | Example |
|---|---|---|
| Material | What it is made from | The bronze |
| Formal | What form it has | The figure represented |
| Efficient | What brings it about | The sculptor’s craft |
| Final | What it is for | Commemoration |
Our focus will be efficient causes: if we change one thing, does another change as a consequence? The other kinds of cause still matter in real problems. In particular, with AI, we may need to think even harder about the final cause: what is the purpose of studying something? Aristotle sets out the four causes in Physics, II.3.
1.1.2 Hume and the problem of induction
David Hume (1711–1776), the Scottish philosopher, places experience at the foundation of our knowledge of the world. This is empiricism. But what warrants extending past experience to something not yet observed? Consider the two illustrations:
“The sun always rises in the east.”
“When someone dies, it’s never me.”
Both appeal to past observations, yet the second clearly gives no guarantee of future survival. The fact that an inductive method has worked before cannot itself provide a non-circular justification for trusting it in the future: that argument already uses induction.
Hume puts the difficulty sharply:
“We are determined by custom alone to suppose the future conformable to the past.”3
Hume’s problem of induction is one of the most famous in philosophy. Immanuel Kant famously tried to answer Hume, but philosophers still debate exactly how Kant understood Hume’s challenge and what his proposed answer amounts to—not just whether it succeeds.4 Keep the question in mind as we reason from data: on what grounds can we justify moving from past observations to a prediction or a claim about cause and effect?
1.1.3 Peirce: three arrangements of the same statements
Now consider Charles Sanders Peirce (1839–1914), an American philosopher and a founder of pragmatism. His thinking is close to that of a modern statistician: he starts with a sampling problem. In “Deduction, Induction, and Hypothesis” (1878), he uses beans from a bag to distinguish three forms of inference. The same statements play different roles depending on what we know and what we are trying to infer.6
Deduction uses the bag’s contents and the sampled beans’ origin to deduce their colour. If the premises are true, the conclusion must be true.
Induction generalizes from the sample to the bag’s contents, much as statistical inference moves from a sample to a population. This conclusion is fallible: the beans we have not sampled might include other colours.
Abduction proposes an explanation for the beans’ colour. This is an educated guess, not a deductively valid conclusion: other bags may also contain white beans.
Much of what we do in causal inference resembles abduction. We make an educated guess about a causal effect, examine the assumptions and data supporting it, and try to put forward the strongest case we can. At the end of the day, the guess may still be wrong.
From an explanation to a test
In Peirce’s later account of scientific inquiry, these forms of inference work together. Abduction proposes a hypothesis; deduction works out what else we should observe if it is true; induction tests those predictions against observations. Successful checks lead us back to deduce and test further predictions. Failed checks can send us back to abduction for a new hypothesis. Here induction has a broader role than the simple sample-to-population argument above: it includes the testing of hypotheses.7
1.2 From the law of errors to causal questions
1.2.1 Astronomy, social regularities and Laplace’s demon
Many of the statistical methods we now learn developed in the past hundred years. Their nineteenth-century predecessors often addressed quite different problems. Astronomical measurements were one starting point. Legendre (1805) and Gauss (1809) developed least-squares methods for problems such as estimating an orbit from imperfectly measured positions.
The same ideas of least squares, probability and Gaussian distributions then found uses in social questions. Laplace observed regularities in government records of births, marriages and even misaddressed letters. Quetelet’s Sur l’homme (1835) developed “social physics”: perhaps social phenomena, too, followed laws. Darwin’s work on natural selection (1859) led on to Galton’s studies of heredity, variation and regression towards the mean (1886). We will encounter traces of these developments in the first few lectures.
One unsettling vision from this period is Laplace’s demon. In 1814, Laplace wrote:
We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past could be present before its eyes.
— A Philosophical Essay on Probabilities (1814).
Complete information and sufficient analytical power would leave nothing uncertain. This worldview haunted science for a long time, and part of the development of statistics can be seen as a response to it. Current discussions of AI and superintelligence bring us back to some of the same questions.
1.2.2 Yule: what does a regression coefficient mean?
George Udny Yule (1871–1951) was an early user of regression in social science. His 1899 investigation asked a politically charged question: did relief outside the workhouse increase pauperism? Poor people could receive accommodation and food in a workhouse in exchange for work, or receive support outside it. The out-relief ratio compared outside relief with workhouse relief.
Yule studied Poor Law unions, including London districts. “Pauperism” here concerns support at public expense, rather than a modern income-based poverty measure. His linear models for changes in pauperism gave these coefficients:
| Metropolitan unions | Constant | Out-relief ratio | Aged proportion | Population |
|---|---|---|---|---|
| 1871–1881 | 13.19 | 0.755 | −0.022 | −0.322 |
| 1881–1891 | 1.36 | 0.324 | 1.370 | −0.369 |
The out-relief coefficient is positive in both periods. Yule used this as support for a positive causal effect, and used causal language throughout his paper. Yet he also included a striking footnote:
“Strictly, for ‘due to’ read ‘associated with.’”
Yule clearly understood the distinction between association and causation. What, then, does a regression coefficient tell us about a causal effect? Giving a justified interpretation to such coefficients is one of the problems causal inference seeks to address.9
1.3 Two motivating studies
1.3.1 Coover: a randomized test of thought-transference
Before developing the mathematics, consider two real studies of controversial questions. Randomized experiments gave us a powerful device for causal inference, and we will spend about three lectures on randomization. An early example comes from John Edgar Coover’s research on telepathy, reported in 1917. Such a disputed claim calls for a carefully designed experiment.10
Could a sender mentally transmit the identity of a card to a receiver, without ordinary communication? The receiver sat facing away from the sender and could not see the target card. One trial proceeded as follows:
The sender shuffled and cut a 40-card pack, keeping the target concealed. A die roll selected the sender’s condition.
During the specified interval, the sender followed that condition:
Die face Sender’s condition 2, 4 or 6 The card remained unknown to the sender: control. 1 Look at the card. 3 Glance at it, conceal it, then imagine saying its name. 5 Look at it and imagine saying and hearing its name. After the interval, the receiver recorded a guess. The sender then recorded the card. The records were compared after the set of trials, rather than giving trial-by-trial correctness feedback.
Coover reported no evidence of thought-transference. The details of the procedure matter: if the sender could whisper a clue, we might find apparent evidence for quite a different reason. The instructions to imagine speech concerned mental imagery, not spoken clues.
Notice the two randomizations: selection of the card and selection of the sender’s condition. Keep the die roll in mind when we turn to the mathematics: it will help us distinguish assignment from exposure.
1.3.2 Smoking: the scale changes the comparison
The smoking–lung cancer debate motivated a different route to causal inference. We cannot randomly require people to smoke. We can randomize an incentive to quit, but people may continue smoking despite the offer. Such studies also tend to be smaller and more expensive. Randomizing encouragement is different from randomizing smoking itself.
Many studies contributed to the smoking–lung cancer debate, and much of the evidence was observational. Here we consider just one dataset from the famous British doctors study by Richard Doll and Austin Bradford Hill, which recorded smoking before subsequent mortality. Its 1956 report gives the following age-standardized annual death rates for male doctors aged 35 and over.11
| Cause of death | Non-smokers | 1–14 g/day | 15–24 g/day | 25+ g/day |
|---|---|---|---|---|
| Lung cancer | 0.07 | 0.47 | 0.86 | 1.66 |
| Other cancer | 2.04 | 2.01 | 1.56 | 2.63 |
| Other respiratory disease | 0.81 | 1.00 | 1.11 | 1.41 |
| Coronary thrombosis | 4.22 | 4.64 | 4.60 | 5.99 |
| Other causes | 6.11 | 6.82 | 6.38 | 7.19 |
Rates are deaths per 1,000 men per year, not percentages or cumulative risks. Amounts are grams of tobacco per day. Former smokers were classified by their amount at quitting. The displayed rates are rounded.
Now compare the 25+ g/day group with non-smokers. The following are calculations from those rates, not additional reported observations:
| Cause | Risk ratio | Risk difference |
|---|---|---|
| Lung cancer | 23.71 | 1.59 |
| Other cancer | 1.29 | 0.59 |
| Other respiratory disease | 1.74 | 0.60 |
| Coronary thrombosis | 1.42 | 1.77 |
| Other causes | 1.18 | 1.08 |
For lung cancer, \(1.66/0.07\approx23.71\) and \(1.66-0.07=1.59\). For coronary thrombosis, the ratio is only \(5.99/4.22\approx1.42\), but the difference is \(5.99-4.22=1.77\). Thus lung cancer has by far the largest proportional association in this table, while coronary thrombosis has the larger absolute difference.
This mattered to the historical argument that the smoking association was particularly specific to lung cancer. The apparent specificity depends partly on the scale of comparison. Smoking is harmful for other diseases too. We will return to this choice of scale when we study sensitivity analysis later in the course.
1.4 Potential outcomes and randomization
The examples show why informal causal arguments can be difficult to settle. Potential outcomes give us a language in which to state the question and the assumptions. We start broadly, before imposing the familiar binary-treatment model.
1.4.1 The experimenter’s perspective
Let \(Z\in\mathcal Z\) be the randomized treatment assignment. Often \(Z=(Z_1,\ldots,Z_n)\) is a vector of assignments to units, but the general formulation does not require that structure. In the simplified Coover example, the assignment records die rolls under a specified experimental protocol.
Let \[ Y=(Y_1,\ldots,Y_n)\in\mathbb R^n \] be the measured outcome vector. For simplicity, we assume that each measurement \(Y_i\) is a single real value; outcomes need not have this form in general. A unit can be a person or a trial, depending on the question. For example, a trial outcome could indicate whether the receiver’s guess matched the target card. Merely recording \(Z\) and \(Y\) does not yet specify a causal comparison.
For every possible assignment \(z\in\mathcal Z\), define the potential outcome vector \(Y(z)\): the outcomes that would occur under that assignment. The potential-outcome schedule is the whole map \[ Y(\cdot):\mathcal Z\longrightarrow\mathbb R^n. \] It includes outcomes under assignments that were not realized. A particular \(Y(z)\) is one value of the map; \(Y(\cdot)\) is the map itself.
Consistency links the schedule to what is observed: \[ Y=Y(Z),\qquad Y_i=Y_i(Z),\quad i=1,\ldots,n. \] Evaluating the schedule at the realized assignment returns the actual outcomes. The observed assignment is factual; the alternatives provide counterfactual comparisons.
The size of the missing information can be enormous. If we use a simplified experiment with 100 trials and six possible die faces per trial, there are \(6^{100}\) whole assignment vectors: six choices at each of 100 trials. Numerically, \(6^{100}\approx6.53\times10^{77}\). For comparison, the observable universe is estimated to contain of order \(10^{80}\) atoms—only about 150 times as many.12 Only one outcome vector is observed; the other \(6^{100}-1\) remain unobserved. How can we possibly learn about them from just one? We will see in what sense inference is possible.
1.4.2 The unit’s perspective: exposure maps
An experimental unit need not respond to every detail of the whole assignment. An exposure map \[ A:\mathcal Z\longrightarrow\mathcal A^n, \qquad A(z)=(A_1(z),\ldots,A_n(z)), \] summarizes the assignment from each unit’s perspective. It is valid for the schedule \(Y(\cdot)\) if \[ A_i(z)=A_i(z')\quad\Longrightarrow\quad Y_i(z)=Y_i(z') \] for every \(i\in\{1,\ldots,n\}\) and every \(z,z'\in\mathcal Z\). Equal exposure must imply equal potential outcome for that unit. Writing down a map does not establish its validity: validity is a modelling assumption.
For repeated trials, one might first suppose that later assignments cannot affect earlier outcomes. The outcome at trial \(i\) could still depend on the whole assignment history through trial \(i\). A stronger simplification is that only the current trial’s condition matters, or even just whether the sender saw the card. Each reduction makes a modelling assumption; it need not be true.
For Coover’s die assignment, the four sender conditions give the exposure map \[ A_i(z)= \begin{cases} \text{control},&z_i\in\{2,4,6\},\\ \text{visual},&z_i=1,\\ \text{inner speech},&z_i=3,\\ \text{combined},&z_i=5. \end{cases} \] The assignment records the die face; the exposure records the sender’s condition. Treating this map as valid says that the condition captures what matters for this trial’s outcome.
1.4.3 The Neyman–Rubin model and no interference
Now suppose \(Z=(Z_1,\ldots,Z_n)\) gives one treatment assignment per unit. In this course, the Neyman–Rubin causal model means consistency together with a valid identity exposure map: \[ A(z)=z,\qquad A_i(z)=z_i. \] Its unit-level implication is \[ z_i=z_i'\quad\Longrightarrow\quad Y_i(z)=Y_i(z'). \] Only unit \(i\)’s own assignment matters to its potential outcome. Changing the other assignments while holding \(z_i\) fixed cannot affect it. This is no interference. The vector map is an identity; its \(i\)th component selects the \(i\)th entry, not the entire vector.
Consider vaccination. A person’s infection outcome may depend both on their own vaccination and on vaccination among people around them. An exposure description may need to include that surrounding vaccination pattern. The general potential-outcome schedule allows such dependence; the no-interference model rules it out.
We will use “Neyman–Rubin model” in this specific sense throughout the course; the name has no single consistent meaning across the literature. It honours Neyman and Rubin’s seminal contributions. It is also a little unfair to attach their names to the restriction: Rubin (1980), in his comment on Basu’s paper, explicitly discussed no interference, citing David Cox’s Planning of Experiments (1958, p. 19).13 We should keep the assumption in view whenever we use the model.
1.4.4 The missing comparison
Suppose treatment is binary and the no-interference model holds. Consider an experiment with six units and the following observed assignments and outcomes:
| Unit \(i\) | \(Z_i\) | \(Y_i\) | \(Y_i(0)\) | \(Y_i(1)\) |
|---|---|---|---|---|
| 1 | 1 | 5 | ? | 5 |
| 2 | 0 | 4 | 4 | ? |
| 3 | 1 | 7 | ? | 7 |
| 4 | 0 | 3 | 3 | ? |
| 5 | 1 | 7 | ? | 7 |
| 6 | 0 | 4 | 4 | ? |
Consistency lets us fill in one potential-outcome entry in each row. For unit 1, we assigned treatment 1 and observed 5, so \(Y_1(1)=5\). But what would have happened under treatment 0? We do not know. The same difficulty leaves one question mark in every row.
The individual treatment effect is \[ Y_i(1)-Y_i(0). \] For the same unit in this experiment, only one of the two outcomes is observed. The individual effect is therefore not directly observable. This is the fundamental problem of causal inference, the terminology associated with Holland (1986).
For binary treatment, we have the useful identity \[ \begin{aligned} Y_i&=Y_i(Z) &&\text{by consistency},\\ &=Y_i(Z_i) &&\text{under no interference},\\ &=Z_iY_i(1)+(1-Z_i)Y_i(0) &&\text{for binary treatment}. \end{aligned} \] The last line simply selects the factual outcome.
Note that \(Y_i(Z_i)\) is shorthand: initially \(Y_i(z)\) takes a whole assignment vector as its argument. Under no interference it is constant over vectors sharing their \(i\)th entry, so we reuse \(Y_i\) for the resulting function of that single entry. This is an abuse of notation, justified by the model, and a convention we will use often in this course.
1.4.5 What does randomization assume?
We have been talking about randomized assignments. What does randomization mean mathematically? We require two properties:
- The assignment is independent of the potential-outcome schedule: \(Z\mathrel{\perp\!\!\!\perp}Y(\cdot)\).
- The distribution of \(Z\) is known.
Think again about rolling a die. Its result depends on how hard we throw it, how it spins and the angle at which it lands. That mechanism should be independent of how the sender and receiver would respond under the different assignments. We cannot predict the die’s result, but we specify its distribution: for a fair die, each face has probability \(1/6\).
If \(X\) collects covariates measured before randomization, the assignment may depend on \(X\). We then require \[ Z\mathrel{\perp\!\!\!\perp}Y(\cdot)\mid X, \] and the conditional distribution of \(Z\mid X\) must be known. Both requirements now hold given the covariates used in the design.
Next time, we will see how randomization can make population-level inference possible, even though individual treatment effects remain unobservable.
Course discussion and reading
The course webpage is the destination for course materials. On Zulip, use the announcements channel for updates and the course-discussion channel for questions, corrections and discussion with one another. Give each question its own topic, for example Lecture 1 - Potential outcomes. Invitations are available through Moodle; if you need to join the Moodle course, contact the undergraduate office.
The optional AI sandbox is a place to try questions such as “What is a potential outcome?” Mention @Teaching Assistant to get a response. The bot has course materials but no internet access, and its answers may be mistaken.
For further reading, try Hacking (1988), “Telepathy: Origins of Randomization in Experimental Design”; Menand (2001), The Metaphysical Club, chapter 8, “The Law of Errors”; and Salsburg (2001), The Lady Tasting Tea, opening chapter. For the mathematical framework, see Neyman (1923), “On the Application of Probability Theory to Agricultural Experiments”; Rubin (1974), “Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies”; and Holland (1986), “Statistics and Causal Inference”.
© 2026 Qingyuan Zhao. Except where otherwise noted, these lecture notes are licensed under Creative Commons Attribution 4.0 International (CC BY 4.0). Third-party material is excluded from this licence; see the individual credits and rights notices. Public-domain material remains in the public domain.
Roman marble bust after a Greek original, Palazzo Altemps. Photograph by Jastrow (2006), public domain; image source.↩︎
Allan Ramsay, David Hume (1766), Scottish National Portrait Gallery. Public-domain painting and reproduction; image source.↩︎
Hume (1748), An Enquiry concerning Human Understanding, §IV, especially ¶¶2, 6 and 18–21. The quotation about custom is from his Abstract (1740), ¶15. The sun and mortality examples are illustrations, not quotations from Hume.↩︎
“Kant and Hume on Causality”, Stanford Encyclopedia of Philosophy, introduction and §§1–2.↩︎
Napoleon Sarony, Charles Sanders Peirce (1891), New York Public Library. Public domain; image source.↩︎
Peirce (1878), “Deduction, Induction, and Hypothesis”, Popular Science Monthly 13, 470–482; the three bean arguments are on p. 472.↩︎
“Charles Sanders Peirce”, §3, “Deduction, Induction, and Abduction”, Stanford Encyclopedia of Philosophy.↩︎
Church of Sweden population register, nineteenth century. Public-domain reproduction; image source. The image illustrates administrative records, not Laplace’s or Quetelet’s own data.↩︎
Yule (1899), “An Investigation into the Causes of Changes in Pauperism in England, Chiefly During the Last Two Intercensal Decades (Part I.)”, Journal of the Royal Statistical Society 62, 249–286. The coefficients are from Table C, p. 259; see also Freedman’s discussion, pp. 3–4. The quotation is on p. 270, footnote 25.↩︎
Coover (1917), Experiments in Psychical Research at Leland Stanford Junior University, pp. 51–54 for the procedure; the conclusion is reported on p. 124.↩︎
Doll and Hill (1956), “Lung Cancer and Other Causes of Death in Relation to Smoking”, British Medical Journal 2, 1071–1081, Table V, p. 1073. The table uses the most recent amount smoked and standardizes rates to the 1951 UK male population aged 35+. See also Berkson (1958), “Smoking and Lung Cancer: Some Observations on Two Recent Reports”.↩︎
For this rough scale comparison, see Peter Norvig (2016), “On the (Small) Number of Atoms in the Universe”. The estimate concerns the observable universe, not the entire universe.↩︎
Rubin (1980), Comment on “Randomization Analysis of Experimental Data: The Fisher Randomization Test” by D. Basu, Journal of the American Statistical Association 75, 591–593. Cox (1958), Planning of Experiments, Wiley, §2.4, p. 19.↩︎
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