Inferring Causation from Correlation: Why Hume Says You Can’t, and How Science Does It Anyway

Last updated: November 4, 2026

You cannot infer causation from correlation because a correlation between A and B fits at least three other stories besides “A causes B”: B causes A, a third factor causes both, or the match is a coincidence. The Internet Encyclopedia of Philosophy files the mistake under the false cause fallacies, among them post hoc ergo propter hoc, “after this, therefore because of this”, and cum hoc ergo propter hoc, “with this, therefore because of this”. Hume dug one level lower. In An Enquiry Concerning Human Understanding (1748), David Hume argued that we do not observe a cause producing an effect at all: watched closely, two events “seem conjoined, but never connected” (Section VII, paragraph 26).

The meme is a Halloween pumpkin. Someone carved “Inferring causation from correlation” into it, lit it from inside and put it on a dining table, the scariest thing they could think of. Statisticians laugh, the rest of us ask why, and the answer runs through an eighteenth-century Scottish philosopher, a London doctor, an economist turned epidemiologist and a long quarrel about cigarettes.

Correlation vs Causation: The Difference in Plain Words

Correlation means two things move together; causation means changing one would change the other. The first is a pattern in data, the second is a claim about how the world works.

Statisticians usually measure correlation with a coefficient that runs from -1 to +1. A value near +1 says that when one variable is high, the other tends to be high as well; near -1, one rises as the other falls; near zero, there is no straight-line relationship at all. The idea has a precise birthday. Francis Galton presented a paper called “Co-relations and their measurement, chiefly from anthropometric data” to the Royal Society in 1888, using body measurements such as height and the length of the forearm, and his younger colleague Karl Pearson turned it into the coefficient still taught today. Pearson also gave the problem its name: in 1897 he published a memoir on “spurious correlation”, the kind that appears in the numbers without any real connection behind it.

Causation asks a different question. Not “do these move together?” but “if I reached in and changed A, would B change?” A thermometer and the heating bill are strongly correlated in winter, and you will not save any money by breaking the thermometer. That is the whole difference in one sentence: correlation describes what you see, causation predicts what happens when you act. The table below collects the usual ways a correlation can look causal without being so. Each of them is covered in more detail further down.

Three Ways a Correlation Lies, and the Fallacy With a Latin Name

A correlation only tells you that two things vary together; reverse causation, a hidden common cause or pure chance can produce exactly the same pattern.

The IEP gives a tidy example of the hidden common cause: car accidents rise and fall with windshield wiper use, but the wipers cause nothing, because the rain is the common cause of both. Statisticians call such a third factor a confounder. Its sibling is post hoc ergo propter hoc, the inference that because B followed A, A caused B; the IEP’s example is a fan who decides his good seat must be causing his team’s wins. The IEP also lists reversed causation and regression to the mean.

Then there is chance. In 2014, Tyler Vigen, then a Harvard law student, started a website that pairs unrelated data series, and he discussed with NPR how the number of films Nicolas Cage appeared in tracked swimming pool drownings from 1999 to 2009. His site still lists pairings such as per capita margarine consumption against the divorce rate in Maine from 2000 to 2009, and became a 2015 book, Spurious Correlations. Compare enough short data series and some will match closely by luck, so a tight curve on a chart proves little until you know how many curves were thrown away.

Hume’s Billiard Balls: Where the Feeling of Necessity Comes From

Hume’s claim is that we do not perceive the “necessary connection” between a cause and its effect; we only see one event regularly followed by another, and the feeling of necessity comes from our own habit of expecting it.

Take the billiard balls. In a single collision, Hume says, all we observe is that “the impulse of one billiard-ball is attended with motion in the second” (E 7.6). After we have seen the same sequence many times, though, the mind starts to expect the second event when it sees the first, and Hume locates the idea of necessary connection in that expectation: “this customary transition of the imagination” (E 7.28). As the Stanford Encyclopedia of Philosophy puts it, repetition adds nothing to the balls; it produces a feeling in us, which we project onto the world. Hume’s broader verdict, from Section V of the Enquiry, is that “custom, then, is the great guide of human life.”

Hume then gave two definitions of cause in the same paragraph (E 7.29). The first is regularity: one object followed by another, where all similar objects are followed by similar ones. The second is psychological: the appearance of one always conveys the thought to the other. Tucked between them is a rewording of the first that sounds like a different idea altogether: “if the first object had not been, the second never had existed.” That is a counterfactual, a claim about what would have happened otherwise. The SEP notes that scholars still argue whether the two definitions fit together at all.

The Problem of Induction: Why the Sun Might Not Rise Tomorrow

The problem of induction is Hume’s argument that no reasoning can justify our expectation that the future will resemble the past, the expectation that causal claims rely on.

In Section IV of the Enquiry Hume points out that “the sun will not rise to-morrow” is a perfectly intelligible proposition, no more contradictory than its opposite, so logic cannot rule it out. Experience cannot prove it either, because any argument from past experience already assumes that the future will be conformable to the past, which is the point in question. The SEP sums up the conclusion: reason does not ground our inferences from experience, custom does. Our post on epistemology and the theory of knowledge covers the older question of what counts as knowing anything at all.

Hume did not conclude that we should stop reasoning about causes. In his Treatise of Human Nature he set out eight rules by which to judge of causes and effects (Book 1, Part 3, Section 15). A cause must come before its effect, and when two similar objects produce different effects, the difference must come from the point in which they differ, which is the logic of a controlled comparison. He added that such rules are “very easy in their invention, but extremely difficult in their application.”

Simpson’s Paradox: The Berkeley Numbers That Flipped When You Split Them

Simpson’s paradox is the case where a correlation in a whole population vanishes or reverses once you split the population into groups.

The textbook case comes from the University of California, Berkeley. The SEP entry on Simpson’s paradox describes how a 1975 study by Bickel, Hammel and O’Connell found that men were more likely than women to be admitted to graduate programmes overall, yet the authors could not detect a bias towards men in any individual department. Women had applied more often to the departments with lower acceptance rates. The overall number was true and the obvious causal story was far too simple, although, as the SEP adds, why women chose those departments is a further causal question. Bickel and colleagues’ paper is a reminder that the level at which you slice the data is itself a causal choice.

Missing Bombers and Lucky Cadets: Two Quieter Traps

Two further traps produce causal illusions without any hidden third factor: selection bias, where the data you see has been filtered, and regression to the mean, where extreme results drift back to normal on their own.

The classic case of selection bias comes from the Second World War. Abraham Wald, a mathematician who had fled Austria, worked for the Statistical Research Group at Columbia University, and one of his tasks was to estimate how vulnerable aircraft were to enemy fire. The obvious data were the damaged planes that came back. Wald’s insight, reconstructed by Marc Mangel and Francisco Samaniego in a 1984 paper in the Journal of the American Statistical Association, was that those planes were exactly the wrong sample. Holes in a plane that returned showed where a plane could be hit and still fly home. The planes hit in the places that mattered were at the bottom of the sea, and they were missing from the data. The popular retellings simplify his memoranda a great deal, but the core lesson is sound: before you read a pattern, ask what filtered the cases you can see. Startup success stories, miracle diets and “I smoked all my life and I’m fine” all come from samples that the failures never reached.

Regression to the mean is subtler, because it looks like a cause even in a perfectly clean sample. Daniel Kahneman tells the story in Thinking, Fast and Slow (2011). Teaching flight instructors in the Israeli Air Force, he explained that rewarding good performance works better than punishing mistakes. One instructor objected: when he praised a cadet for a superb manoeuvre, the next one was usually worse, and when he screamed at a cadet for a bad one, the next one was usually better. As Kahneman explains, the instructor had observed a real pattern and drawn the wrong causal conclusion. Any single performance mixes skill and luck. An unusually good flight probably had good luck in it, so the next one tends to be closer to average whatever the instructor says; the same holds, in reverse, for an unusually bad one. The yelling got the credit that belonged to statistics.

Galton, again, was there first. In his studies of height he noticed that very tall parents tend to have children who are tall, but less extremely tall than they are, and he called the effect “regression towards mediocrity”. The same mechanism explains why patients often feel better after seeing a doctor at their worst moment, why a sports team’s record season is often followed by a weaker one, and why a policy introduced after a bad year so often seems to work. If you start counting at an extreme, the next number will usually look like improvement or decline, with or without a cause.

Put the two traps together and you get a quick test for any causal headline. Ask who could not appear in the data, and ask whether the story started counting at an unusually good or bad moment. If the answer to either question is yes, the correlation may be real and the cause still imaginary. A surprising number of business books and health claims fail one of those two questions before they even reach confounders.

The Statistician Who Blamed Genes for Lung Cancer

Where experiments are impossible, scientists can still build a causal case from converging evidence, as the smoking and lung cancer story shows.

On 30 September 1950, Richard Doll and Austin Bradford Hill published a study in the British Medical Journal of 709 lung cancer patients from twenty London hospitals and concluded that smoking is “a factor, and an important factor” in the disease. It was a correlation, and Ronald A. Fisher said so. Fisher, whose 1935 book The Design of Experiments championed randomization as a principle of experimental design, argued in a July 1958 letter to Nature that smoking and lung cancer might both be influenced by “a common cause, in this case the individual genotype”: a confounder in its purest form. The same letter even tried reverse causation, suggesting smoking might be prompted by the “subconscious irritation of the postulated pre-cancerous condition”. Since nobody can ethically assign people to smoke for decades, the objection could not be tested the way Fisher preferred. According to Morabia and Abel, people who knew Fisher reported that he later admitted he had been wrong.

Hill’s broader answer came in 1965, in his presidential address to the Section of Occupational Medicine of the Royal Society of Medicine, “The Environment and Disease: Association or Causation?”. It listed nine viewpoints for judging whether an association is causal: strength, consistency, specificity, temporality, biological gradient, plausibility, coherence, experiment and analogy. They are often called the Bradford Hill criteria, but Hill insisted, as Jeffrey Aronson quotes him in The BMJ, that “none can be required as a sine qua non.” When an experiment is possible, it usually settles more than a checklist can: Hill had helped design the Medical Research Council’s streptomycin trial for tuberculosis, published in 1948, “a milestone in the history of clinical trials” according to the James Lind Library, in which envelopes ordered by a series of random numbers decided who got the drug. Randomization tends to balance confounders, known and unknown, between the groups, and it removes the experimenter’s hand from who gets what.

Natural Experiments: When History Runs the Trial for You

A natural experiment is a situation in which chance, geography or a policy change splits people into groups as if someone had randomized them, so researchers can compare outcomes without running a trial themselves.

The founding example is older than modern statistics. During the London cholera epidemic of 1854 the physician John Snow noticed that two water companies served the same streets south of the Thames: the Southwark and Vauxhall Company took its water from a stretch of the river polluted with sewage, while the Lambeth Company drew it from further upstream. Rich and poor, large houses and small were supplied by both, so the households differed in little except their water. Snow counted. As the University of Chicago’s data science course reproduces his figures, houses supplied by Southwark and Vauxhall suffered 315 cholera deaths per 10,000 houses, Lambeth’s only 37. Snow called it a “grand experiment”, and nobody had to drink sewage on purpose for it to happen.

Modern economics turned the idea into a toolkit. In 2021 David Card shared the economics prize in memory of Alfred Nobel with Joshua Angrist and Guido Imbens for work on natural experiments, as Scienceline reported. Card’s best-known study compared restaurants in New Jersey, which had raised its minimum wage from $4.25 to $5.05, with restaurants just across the border in eastern Pennsylvania, which had not. Employment in New Jersey did not fall. The finding remains debated, but the method changed the field: find a border, a lottery, a cut-off date or a quirk of the rules that sorts people for reasons unrelated to the outcome, and you have something close to Hill’s randomized envelopes.

Hume would recognise the move. His rule that a difference in effects must come from the point in which two similar cases differ is precisely what Snow and Card exploited. What they added was the search for cases where the similarity is real rather than assumed.

Pearl’s Ladder of Causation: Seeing, Doing, Imagining

Judea Pearl’s ladder of causation sorts causal questions into three levels, association, intervention and counterfactuals, and holds that data from a lower level cannot answer a question from a higher one.

Pearl, who won the 2011 Turing Award for work on probabilistic and causal reasoning, popularised the ladder in The Book of Why (2018, with Dana Mackenzie). In a 2018 paper he labels the rungs “Seeing”, “Doing” and “Imagining”. The first rung asks what a symptom tells you about a disease, the second what happens if I take an aspirin, the third whether it was the aspirin that stopped my headache. Pearl writes that interventional questions “cannot be answered from purely observational information”, however big the dataset. The ladder also puts Hume’s definitions at opposite ends. Regularity lives on the first rung; “if the first object had not been, the second never had existed” sits on the third.

Not every philosopher thought the effort was worth it. Bertrand Russell told the Aristotelian Society in 1912 that the law of causality is “a relic of a bygone age”, surviving like the monarchy only because people wrongly suppose it does no harm. One fair reply is that physics may manage without the word, but medicine, economics and anyone who wants to change something cannot, and Ockham’s principle of parsimony will not tell you which way the arrow points either. For more ways arguments go wrong in everyday life, C.M. Black’s guide to five frustrating fallacies picks up where the false cause family stops.

If the formal side of reasoning interests you, read When logic became formalized.