Block I · Foundations of Knowledge & Reasoning · Day 05 / 180
Causation
Ice cream and drowning rise and fall together. Nobody drowned because of a sundae. So what is the difference between a pattern and a cause?
Every summer, two numbers climb in lockstep. As ice-cream sales rise, so do drownings; as the cones stop selling in autumn, the drownings taper off too. Plot them and you get a clean, confident upward line — the kind of correlation that would make a careless analyst reach for a headline. Ban ice cream, save lives. And yet you already know, in your bones, that this is nonsense. No one has ever drowned because of a sundae.
What you know in your bones is one of the hardest things to write down in science. There is a third character lurking off-stage — summer — and it is pulling both strings at once. Heat drives people to buy ice cream, and heat drives people into lakes and oceans where some of them drown. The two visible numbers dance together only because an invisible one is conducting. Today is about the machinery for catching that invisible conductor — and the modern discovery that causation is not just a stronger correlation. It lives on a higher rung of a ladder you cannot climb by staring at data alone.
◆ Where we are
Four days in, the toolkit is starting to interlock. Day 1 warned us about beliefs that are true but only by luck — the stopped clock. A spurious correlationA spurious correlation is a real statistical association caused by something other than a direct causal link between the two measured variables. is exactly that at population scale: a number that's "right" for entirely the wrong reason. Day 2 introduced Hume and the problem of induction; today we meet the same Hume, because his attack on causation and his attack on induction are one and the same blade. Day 3 sorted reasoning into deduction, induction, and abduction — and causal discovery, we'll see, is abduction with teeth: inference to the best causal explanation. And Day 4 gave us the do-versus-see distinction's parent, probability: today's punchline, P(y|do(x)) ≠ P(y|x), is the most important inequality in the course so far.
The oldest problem
Hume kicks out the leg
Start where the trouble starts. In 1739, in A Treatise of Human Nature, David Hume asked a deceptively simple question: when one billiard ball strikes another and the second rolls away, what exactly do you see? You see the first ball move. You see them touch. You see the second ball move. What you never see — look as hard as you like — is the causing itself: the necessary connection, the hidden force, the little arrow of because linking the two events.
All we ever actually observe, Hume argued, is constant conjunctionConstant conjunction means one kind of event has repeatedly been followed by another kind of event.: events of this type are reliably followed by events of that type. Add that the cause comes first (priority) and that the two events touch in space and time (contiguity), and you have everything experience delivers. The sense of necessity — the feeling that the second ball had to move — is not out in the world at all. It is a habit of mind, a customary expectation built up by repetition and then projected back onto the world like a film onto a screen. Hume found two definitions of "cause" tangled together in our heads: one about the world (constant conjunction) and one about us (the mind's practiced leap from one to the other).
This should feel familiar. It is the problem of induction from Day 2, wearing a different coat. If causation is just "this has always been followed by that," then claiming the next collision will behave like the last is precisely the unprovable bet on the uniformity of nature that Hume showed can never be justified non-circularly. Causation and induction are the same wound. For two centuries philosophy basically picked at it.
Why "the cement of the universe"?
The phrase gets attached to Hume, but it is better treated as Mackie's title and image: J. L. Mackie's 1974 book is called The Cement of the Universe. The irony Hume relished remains: this "cement" is the one thing we can never see. We infer the glue only from the fact that the bricks keep ending up stuck together.
Four repairs
Saying what "more" there is
If causation is more than constant conjunction, the obvious move is to say what the "more" is. The twentieth century produced several serious answers — different ways to finish the sentence "C causes E means...". They are not simple rivals so much as lenses that modern causal modeling keeps borrowing from.
do(C) and graph surgery.Notice how Cartwright (1979) sharpened the probabilistic story, because her fix is the hinge of the whole day. Causes raise the probability of effects, yes — but only inside a "causally homogeneous" background, with all the other causes held fixed. Forget that proviso and you walk straight into the most beautiful trap in statistics.
The trap
Simpson's paradox: when the numbers literally reverse
Here is a fact that sounds impossible until you've seen it: a treatment can be better for small stones, better for large stones, and yet worse overall. Not "looks worse" — is, on the pooled numbers, worse. The real kidney-stone dataset below lets you watch the bars flip.
The mechanism is always a lurking variable that's distributed unevenly between the groups you're comparing. In the kidney-stone data (Charig et al., BMJ, 1986), surgeons gave the gentler treatment B mostly to the easy small stones and reserved old-fashioned open surgery A for the hard big ones. So B's overall success rate is flattered by an easier caseload. Split by stone size — hold the confounderA confounder is a third variable that influences both the apparent cause and the apparent effect, making them move together. fixed, exactly as Cartwright demanded — and A pulls ahead in both rooms.
A: 273/350 = 78.0%B: 289/350 = 82.6%
pooled winner: B
A: 81/87 = 93.1%B: 234/270 = 86.7%
conditioned winner: A
A: 192/263 = 73.0%B: 55/80 = 68.8%
conditioned winner: A
Read the three squares as the first version of conditioning. "All patients" mixes easy and hard cases into one bucket. "Small stones" and "large stones" compare like with like by holding stone size fixed. The whole foundation is here: the right causal comparison often appears only after you choose the right variable to condition on.
Interactive · watch it flip
The Reversal Machine
Real 1986 kidney-stone data, 350 patients per treatment. Toggle between the pooled view and the split-by-stone-size view. The split view conditions on stone size: it asks the same treatment question inside comparable cases instead of mixing easy and hard patients.
Reference table
Kidney-Stone Simpson Reversal
Treatment B looks better when every patient is pooled, but Treatment A does better inside each stone-size group.
| View | Treatment A | Treatment B | Apparent winner |
|---|---|---|---|
| All patients | 273 / 350 = 78.0% | 289 / 350 = 82.6% | B |
| Small stones | 81 / 87 = 93.1% | 234 / 270 = 86.7% | A |
| Large stones | 192 / 263 = 73.0% | 55 / 80 = 68.8% | A |
The pooled comparison reverses because B was used much more often on easier small-stone cases.
The lesson lands hard: you cannot read causation off a table of numbers. The very same digits support opposite conclusions depending on a variable that may not even be in the spreadsheet. Which raises the question that organizes everything after this — if the data alone won't tell you, what will? The answer, from a computer scientist who spent the 1980s building machines that reason under uncertainty, is that you need to add something the data doesn't contain: a model of which arrows point where.
The causal revolution
Pearl's ladder, and the verb that changed everything
Judea Pearl won the 2011 Turing Award — often called computing's Nobel — "for fundamental contributions to artificial intelligence through the development of a calculus for probabilistic and causal reasoning." His central image, popularized in The Book of Why (2018), is a Ladder of CausationPearl’s ladder separates association, intervention, and counterfactual reasoning into increasingly demanding levels. with three rungs, each demanding a stronger kind of question and a stronger kind of evidence than the one below. Most of statistics and nearly all of machine learning, Pearl likes to needle, never leaves the bottom rung.
Climb it yourself:
Interactive · climb the ladder
The Three Rungs of Causation
Tap a rung. Each adds a verb, a piece of notation, and a question the rung below cannot answer. The jump from Rung 1 to Rung 2 is the entire subject of today.
Reference table
Pearl's Ladder of Causation
| Rung | Verb | Notation | Question |
|---|---|---|---|
| 1. Association | Seeing | P(Y | X) | How do outcomes differ among cases where X is observed? |
| 2. Intervention | Doing | P(Y | do(X)) | What would happen if X were set by intervention? |
| 3. Counterfactuals | Imagining | P(Yx | X′, Y′) | What would have happened in a different world, given what actually happened? |
In the last line, Yx means "Y in the world where X is set to x"; X′ and Y′ are the actual facts you condition on.
The do-operator: seeing is not doing
Here is the conceptual hinge of the entire modern field, and it is worth saying slowly. There are two very different things you can do with a variable X.
You can conditionConditioning means narrowing attention to cases where some variable already has a given value. on it — written P(Y | X = x). This means: among all the cases where X happened to equal x, what's the distribution of Y? You're filtering an existing population. This is seeing.
Or you can intervene — written, in Pearl's notation, P(Y | do(X = x)). This means: reach in, force X to equal x for everyone, snipping X away from whatever usually causes it, and then watch Y. This is doing — it's what a randomized experiment performs.
When there's a confounder, these two numbers come apart, and the gap between them is the bias. Among people who happen to buy lots of ice cream, drowning really is more common (because they're the summer people) — so the seeing quantity is high. But if you force a random sample of people to buy ice cream in, say, evenly distributed weather, drownings don't budge — the doing quantity is flat. The machine below lets you turn a confounder up and down and watch the two quantities diverge.
Interactive · seeing vs doing
The do-operator vs the eyeball
A toy world: a hidden cause (summer) pushes up both ice-cream buying and swimming-deaths. There's also a tiny true direct effect of ice cream on drowning you can set yourself (a full stomach before swimming, say). Slide the confounder's strength and watch the naïve "seeing" estimate balloon while the honest "doing" estimate tracks only the direct effect you set.
Rung 1 · Seeing
P(drown | buy) − P(drown | no buy)
—looks like the effect
Rung 2 · Doing
P(drown | do(buy)) − P(drown | do(no buy))
—the real effect
The confounding gap
How the numbers are made: the left card compares observed buyers with observed non-buyers, so summer changes the mix of people in the two groups. The right card recomputes after cutting summer → ice cream; everyone keeps the same summer mix, and only the slider's direct effect remains.
Worked example
Seeing Is Not Doing
In the toy world, the observed association equals the true direct effect plus a bias term from summer. With the default settings, the true direct effect is +0.05, but the observed association is much larger because summer pushes both ice-cream buying and swimming deaths upward.
| Quantity | What it asks | Default reading |
|---|---|---|
| P(drown | buy) - P(drown | no buy) | Seeing: compare existing buyers with non-buyers. | True effect plus confounding bias. |
| P(drown | do(buy)) - P(drown | do(no buy)) | Doing: set buying directly and sever its usual causes. | Only the true direct effect. |
That severed arrow in the right-hand graph is the do-operator made visual. Intervening doesn't just look at X — it deletes the arrows pointing into X and replaces them with your hand. The summer→ice-cream link is cut, so summer can no longer use ice cream as a backdoor route to fake an effect on drowning. What survives is only the real thing.
Pearl gave us a grammar for doing this on paper. A structural causal modelA structural causal model represents how variables are generated from other variables and from background noise. draws the variables as a directed acyclic graphA directed acyclic graph, or DAG, is a network of arrows with no directed loops. — boxes and arrows, no loops — and three named patterns do most of the work. A fork (X ← Z → Y) is a confounder; block it by conditioning on Z. A chain (X → Z → Y) is a mediator. And a collider (X → Z ← Y) is the trap: X and Y are unrelated until you condition on their shared effect Z, at which point a phantom correlation springs into being. This is why "controlling for everything you can measure" is not cautious but reckless — condition on a collider and you manufacture the very bias you were trying to remove.
Reference table
Concrete DAG Examples
| Pattern | Concrete graph | What it means |
|---|---|---|
| Fork / confounder | Summer → ice cream; Summer → drowning | Condition on summer to compare like with like. |
| Chain / mediator | Smoking → tar → cancer | Tar carries part of the effect from smoking to cancer. |
| Collider | Skill → admission ← luck | Conditioning on admission can create a fake relationship between skill and luck. |
The front-door trick
Pearl's front-door criterionThe front-door criterion identifies a causal effect through a measured mediator when the direct cause-effect path is confounded. is one of the framework's neatest moves: sometimes you can estimate the effect of smoking on cancer even with an unmeasured genetic confounder, as long as you can measure a complete mediator in between (say, tar deposits in the lungs). The calculation has three moves: estimate how smoking changes tar; estimate how tar changes cancer while accounting for smoking; then average those pieces over the observed smoking mix. The hidden genetic confounder affects smoking and cancer, but it does not enter the measured smoking → tar and tar → cancer pieces in the final front-door expression. It is a genuine "get a causal answer from observational data" move — but only because you supplied the graph that says tar is a full mediator. No free lunch: the assumptions just moved from the spreadsheet into the diagram.
The frontier · 2026
Three live edges — and the hype filter
Now the question that has launched a thousand papers and at least one industry: can you infer causation from observation alone? The honest 2026 answer is a precise "partly — and there's a proven wall." Each claim below is tagged for how much weight it can bear.
The two theorems that fence the field
This is the most solid ground in the whole day — not empirical findings that could be overturned, but mathematical proofs. First, do-calculus is complete. Pearl's three rewrite rules turn do() expressions into ordinary probabilities whenever that's possible at all; and Shpitser & Pearl (2006) and, independently, Huang & Valtorta (2006) proved that if the rules cannot eliminate the do-operator, then no method can — the effect is genuinely unidentifiable from observational data plus that graph. Pearl called this closing "the chapter of nonparametric identification." A clean, permanent answer to "when can seeing substitute for doing?"
What a do-calculus rewrite looks like
- Delete an irrelevant observation: if Z already blocks every open path from X to Y, then learning X adds no more information:
P(Y | X, Z) = P(Y | Z). - Swap an action for an observation: after the right adjustment, setting X and observing X can answer the same question:
P(Y | do(X), Z) = P(Y | X, Z). - Delete an irrelevant action: if intervening on Z cannot reach Y once X is fixed, drop it:
P(Y | do(X), do(Z)) = P(Y | do(X)).
Those are toy examples, not the full formal side-conditions. The point is the flavor: the graph licenses exactly which symbols may be erased or exchanged.
Second, the wall on the other side: the Markov-equivalence ceiling. Using only the conditional-independence patterns in observational data, certain different causal graphs are provably indistinguishable. X→Y→Z, X←Y←Z, and X←Y→Z all imply the exact same single fact ("X and Z are independent once you know Y"), so no amount of that data can tell them apart. They form a Markov equivalence classA Markov equivalence class is a set of causal graphs that imply the same conditional independences, so observational data of that kind cannot choose among them.. Only the collider X→Y←Z stands out, because X and Z are independent in the raw data but become dependent after you condition on their shared effect Y. The takeaway is stark: observation alone, assumption-free, can never deliver a unique causal graph — only a class of candidates. Both results are ESTABLISHED in the strongest sense available: theorems.
Squeezing direction out of still data — and why experiments still reign
So is the ceiling the end? Not quite — you can climb over it by importing extra assumptions the bare independence-tests don't use. LiNGAMLiNGAM is a causal-discovery method whose name abbreviates linear non-Gaussian acyclic model. (Shimizu et al., JMLR, 2006) showed that if relationships are linear, there are no hidden confounders, and the noise is non-Gaussian, the full direction becomes identifiable — the asymmetry of non-Gaussian noise breaks the X→Y / Y→X tie. In the true direction, the leftover noise is independent of the cause; in the wrong direction, the residuals still carry a telltale dependence. Additive-noise modelsAdditive-noise models assume the effect equals a function of the cause plus independent noise, an asymmetry that can sometimes identify direction. (Hoyer, Janzing, Mooij, Peters & Schölkopf, NeurIPS 2009) extended this to nonlinear cause-effect pairs. On the standard Tübingen Cause-Effect PairsThe Tübingen benchmark is a collection of real-world variable pairs with expert-labeled causal directions. benchmark — 108 pairs with known ground truth, like altitude→temperature — one strong system reported about 83% accuracy (Mosaic, Wu & Fukumizu 2020), evidence that something "often thought impossible" is partly doable. But note the shape of the trick: you escape the ceiling only by assuming non-Gaussianity or additivity. Those assumptions are not testable from the same observational distribution alone — and the load-bearing one for the whole constraint-based program, faithfulnessFaithfulness says the independences in the data reflect graph structure, not exact cancellations among causal paths., can fail silently in finite samples. PROMISING / works under assumptions.
Which is why the gold standard remains brutally simple: run the experiment. A randomized controlled trialA randomized controlled trial assigns participants to conditions by chance so background causes are balanced on average. physically performs do(X) by assigning X at random, severing every backdoor at a stroke. When you can't randomize, economics' credibility revolutionThe credibility revolution in economics moved emphasis from clever regressions to research designs that approximate randomized experiments: instruments, discontinuities, and policy changes that assign exposure as-if at random. hunts for "natural experiments" that mimic randomization — instrumental variables, regression discontinuity, difference-in-differences. That program earned the 2021 Nobel in economics for David Card, Joshua Angrist, and Guido Imbens (we'll spend real time here on Day 152). ESTABLISHED.
Can the machines do it? Causal ML and the "causal parrot"
The hottest and haziest edge. Causal representation learningCausal representation learning tries to learn useful causal variables from raw data such as pixels, text, or sensor streams. (Schölkopf et al., Proceedings of the IEEE, May 2021) asks the deep question: classical causal discovery assumes the variables are handed to you, but the real world arrives as pixels and words. Can a network learn the high-level causal variables — and would that buy robustness to distribution shiftDistribution shift means the data a model sees after deployment differ from the data it learned from., which today's models often lack? It's a serious, active program whose biggest promises remain PROMISING HINT, not yet delivered at scale.
Then the lightning rod: can large language models reason about cause and effect? Kıcıman, Ness, Sharma & Tan (2023) reported GPT-4 hitting 97% on the Tübingen pairwise causal-direction task — a 13-point jump over the prior best — plus strong counterfactual scores, and argued memorization alone can't explain it. The rebuttal came fast: Zečević et al., "Causal Parrots" (TMLR, 2023), argued LLMs talk causality without being causal — they recite causal facts marinating in their training text rather than performing Pearl-style inference. The 2024–25 synthesis (e.g., Jin et al., ICLR 2024, "Can LLMs Infer Causation from Correlation?") splits the difference: models are strong causal-knowledge retrievers and often shine on Rung-1 questions, but the hard test is genuinely novel Rung-2/3 interventional and counterfactual structure. Recall the Day 1 Gettier trap in a new guise: an LLM that outputs a true causal claim for reasons that have nothing to do with the causal structure is right, but does it know? And recall Day 3: reciting a memorized fact isn't abduction. Verdict on "LLMs reason causally in Pearl's sense": CONTESTED / HYPE. Useful for setting up a causal analysis; not established as a causal reasoner. The open-source tooling underneath causal ML (DoWhy, EconML) is genuine and built on exactly the theorems above, but the sales pitch that causal AI will soon replace ordinary correlational ML is still ahead of the evidence.
Open questions
What's genuinely unsettled
- Is there a "right" theory of causation at all, or are counterfactual, probabilistic, and interventionist accounts each capturing a different facet — with none reducible to the others? No analysis has escaped all the counterexamples (preemption, overdetermination, contextual unanimity).
- How far can we trust faithfulness? The assumption that real systems never have exactly-cancelling causal paths is convenient and untestable — and biology, with its feedback and homeostasis, may violate it routinely.
- Can causal variables be learned from raw data (pixels, language) rather than handed to the algorithm — and is that even well-posed, since the "right" carve-up of the world into variables may itself be perspective-dependent?
- Do large models build internal causal world-models, or only ever statistics of causal talk? The answer reaches straight into Days 138–145 and the question of whether prediction can ever amount to understanding.
- Where do the arrows come from? Every method here needs some causal input — a graph, an assumption, an experiment. Hume's ghost still asks whether that input is ever read off the world, or always brought to it.
◆ The day in three sentences
- Big idea
- Causation is not a stronger correlation but a different kind of thing, living on higher rungs of Pearl's ladder — doing and imagining above mere seeing — so that
P(Y|do(X)) ≠ P(Y|X)whenever a confounder lurks, and no amount of staring at observational data alone can close that gap without importing causal assumptions. - Best analogy
- Ice cream and drowning marching upward together while summer, off-stage, pulls both strings — and the do-operator as a pair of scissors that cuts the strings into a variable so you can see what it really drives.
- Live controversy
- Whether causation can be inferred from observation alone — provably "only up to a Markov-equivalence class" without extra assumptions, partly recoverable with them (LiNGAM, additive noise) — and the noisy 2026 fight over whether LLMs genuinely reason causally or merely parrot causal talk.
Threads today › information (a graph is the extra information data lacks; the do-operator quantifies evidence vs intervention) · computation (do-calculus as a complete algorithm; causal discovery as search) · emergence (causal structure as a higher-level layer over raw correlation) — with callbacks to Day 1 (true-by-luck), Day 2 (Hume), Day 3 (abduction), and Day 4 (P(y|x)).
Tomorrow → Day 06
Statistics & the Art of Not Fooling Yourself
Today we saw how a confounder can flip a conclusion. Tomorrow we meet the subtler enemy: yourself. p-hacking a coin flip, the garden of forking paths, effect sizes versus the worship of "significance" — and the collider trap we just met returns as one of the easiest ways an honest researcher fools an honest audience. Bring today's instinct to ask, every time, "what's the variable that isn't on this chart?"
Sources
Sources & further reading
- Hume, D. (1739–40). A Treatise of Human Nature, Book I, Part III; and (1748) An Enquiry Concerning Human Understanding, §VII.
- Mackie, J. L. (1974). The Cement of the Universe: A Study of Causation. Oxford University Press.
- Lewis, D. (1973). "Causation." Journal of Philosophy 70(17): 556–567. doi.org/10.2307/2025310 See also Lewis, Counterfactuals (Blackwell, 1973); revised "influence" account (2000).
- Reichenbach, H. (1956). The Direction of Time. University of California Press.
- Suppes, P. (1970). A Probabilistic Theory of Causality. North-Holland.
- Cartwright, N. (1979). "Causal Laws and Effective Strategies." Noûs 13(4): 419–437. overview
- Woodward, J. (2003). Making Things Happen: A Theory of Causal Explanation. Oxford University Press.
- Charig, C. R., Webb, D. R., Payne, S. R. & Wickham, J. E. A. (1986). "Comparison of treatment of renal calculi by open surgery, percutaneous nephrolithotomy, and extracorporeal shockwave lithotripsy." British Medical Journal 292: 879–882.
- Simpson, E. H. (1951). "The Interpretation of Interaction in Contingency Tables." JRSS B 13: 238–241. Blyth, C. R. (1972), JASA 67: 364–366 (coins "Simpson's paradox"). Yule, G. U. (1903) on spurious correlation.
- Pearl, J. (2009). Causality: Models, Reasoning, and Inference, 2nd ed. Cambridge University Press. And Pearl, J. & Mackenzie, D. (2018). The Book of Why. Basic Books.
- Shpitser, I. & Pearl, J. (2006). "Identification of Joint Interventional Distributions in Recursive Semi-Markovian Causal Models." AAAI. & Huang, Y. & Valtorta, M. (2006). "Pearl's Calculus of Intervention Is Complete." UAI.
- ACM (2012). 2011 A.M. Turing Award — Judea Pearl. amturing.acm.org/award_winners/pearl
- Spirtes, P., Glymour, C. & Scheines, R. (2000). Causation, Prediction, and Search, 2nd ed. MIT Press.
- Shimizu, S., Hoyer, P. O., Hyvärinen, A. & Kerminen, A. (2006). "A Linear Non-Gaussian Acyclic Model for Causal Discovery." JMLR 7: 2003–2030. reference code
- Hoyer, P., Janzing, D., Mooij, J., Peters, J. & Schölkopf, B. (2009). "Nonlinear causal discovery with additive noise models." NeurIPS. Mooij, J. et al. (2016). "Distinguishing Cause from Effect Using Observational Data." JMLR 17(32): 1–102. Wu, P. & Fukumizu, K. (2020). "Causal Mosaic: Cause-Effect Inference via Nonlinear ICA and Ensemble Method." Proceedings of Machine Learning Research 108: 1157–1167.
- Schölkopf, B., Locatello, F., Bauer, S., Ke, N. R., Kalchbrenner, N., Goyal, A. & Bengio, Y. (2021). "Toward Causal Representation Learning." Proceedings of the IEEE 109(5): 612–634.
- Kıcıman, E., Ness, R., Sharma, A. & Tan, C. (2023). "Causal Reasoning and Large Language Models." & Zečević, M., Willig, M., Dhami, D. S. & Kersting, K. (2023). "Causal Parrots: Large Language Models May Talk Causality But Are Not Causal." TMLR. & Jin, Z. et al. (2024). "Can Large Language Models Infer Causation from Correlation?" ICLR.
- The Royal Swedish Academy of Sciences (2021). The Sveriges Riksbank Prize in Economic Sciences — Card, Angrist & Imbens. nobelprize.org/prizes/economic-sciences/2021
- Stanford Encyclopedia of Philosophy: "Causation" / "Counterfactual Theories of Causation" / "Causal Models" / "Probabilistic Causation." plato.stanford.edu/entries/causation-counterfactual
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