Skip to content

Causal Revolution: 描述因果的数学语言

To appreciate the depth of this gap, imagine the difficulties that a scientist would face in trying to express some obvious causal relationships—say, that the barometer reading B tracks the atmospheric pressure P. We can easily write down this relationship in an equation such as B = kP, where k is some constant of proportionality. The rules of algebra now permit us to rewrite this same equation in a wild variety of forms, for example, P = B/k, k = B/P, or B–kP = 0. They all mean the same thing—that if we know any two of the three quantities, the third is determined. None of the letters k, B, or P is in any mathematical way privileged over any of the others. How then can we express our strong conviction that it is the pressure that causes the barometer to change and not the other way around? And if we cannot express even this, how can we hope to express the many other causal convictions that do not have mathematical formulas, such as that the rooster’s crow does not cause the sun to rise?

My college professors could not do it and never complained. I would be willing to bet that none of yours ever did either. We now understand why: never were they shown a mathematical language of causes; nor were they shown its benefits. It is in fact an indictment of science that it has neglected to develop such a language for so many generations. Everyone knows that flipping a switch will cause a light to turn on or off and that a hot, sultry summer afternoon will cause sales to go up at the local ice-cream parlor. Why then have scientists not captured such obvious facts in formulas, as they did with the basic laws of optics, mechanics, or geometry? Why have they allowed these facts to languish in bare intuition, deprived of mathematical tools that have enabled other branches of science to flourish and mature?

Part of the answer is that scientific tools are developed to meet scientific needs. Precisely because we are so good at handling questions about switches, ice cream, and barometers, our need for special mathematical machinery to handle them was not obvious. But as scientific curiosity increased and we began posing causal questions in complex legal, business, medical, and policy-making situations, we found ourselves lacking the tools and principles that mature science should provide.

Belated awakenings of this sort are not uncommon in science. For example, until about four hundred years ago, people were quite happy with their natural ability to manage the uncertainties in daily life, from crossing a street to risking a fistfight. Only after gamblers invented intricate games of chance, sometimes carefully designed to trick us into making bad choices, did mathematicians like Blaise Pascal (1654), Pierre de Fermat (1654), and Christiaan Huygens (1657) find it necessary to develop what we today call probability theory. Likewise, only when insurance organizations demanded accurate estimates of life annuity did mathematicians like Edmond Halley (1693) and Abraham de Moivre (1725) begin looking at mortality tables to calculate life expectancies. Similarly, astronomers’ demands for accurate predictions of celestial motion led Jacob Bernoulli, Pierre-Simon Laplace, and Carl Friedrich Gauss to develop a theory of errors to help us extract signals from noise. These methods were all predecessors of today’s statistics.

Ironically, the need for a theory of causation began to surface at the same time that statistics came into being. In fact, modern statistics hatched from the causal questions that Galton and Pearson asked about heredity and their ingenious attempts to answer them using cross-generational data. Unfortunately, they failed in this endeavor, and rather than pause to ask why, they declared those questions off limits and turned to developing a thriving, causality-free enterprise called statistics.

This was a critical moment in the history of science. The opportunity to equip causal questions with a language of their own came very close to being realized but was squandered. In the following years, these questions were declared unscientific and went underground. Despite heroic efforts by the geneticist Sewall Wright (1889–1988), causal vocabulary was virtually prohibited for more than half a century. And when you prohibit speech, you prohibit thought and stifle principles, methods, and tools.

Readers do not have to be scientists to witness this prohibition. In Statistics 101, every student learns to chant, “Correlation is not causation.” With good reason! The rooster’s crow is highly correlated with the sunrise; yet it does not cause the sunrise.

Unfortunately, statistics has fetishized this commonsense observation. It tells us that correlation is not causation, but it does not tell us what causation is. In vain will you search the index of a statistics textbook for an entry on “cause.” Students are not allowed to say that X is the cause of Y—only that X and Y are “related” or “associated.”

… I hope with this book to convince you that data are profoundly dumb. Data can tell you that the people who took a medicine recovered faster than those who did not take it, but they can’t tell you why. Maybe those who took the medicine did so because they could afford it and would have recovered just as fast without it.

Over and over again, in science and in business, we see situations where mere data aren’t enough. Most big-data enthusiasts, while somewhat aware of these limitations, continue the chase after data-centric intelligence, as if we were still in the Prohibition era.

As I mentioned earlier, things have changed dramatically in the past three decades. Nowadays, thanks to carefully crafted causal models, contemporary scientists can address problems that would have once been considered unsolvable or even beyond the pale of scientific inquiry. For example, only a hundred years ago, the question of whether cigarette smoking causes a health hazard would have been considered unscientific. The mere mention of the words “cause” or “effect” would create a storm of objections in any reputable statistical journal.

Even two decades ago, asking a statistician a question like “Was it the aspirin that stopped my headache?” would have been like asking if he believed in voodoo. To quote an esteemed colleague of mine, it would be “more of a cocktail conversation topic than a scientific inquiry.” But today, epidemiologists, social scientists, computer scientists, and at least some enlightened economists and statisticians pose such questions routinely and answer them with mathematical precision. To me, this change is nothing short of a revolution. I dare to call it the Causal Revolution, a scientific shakeup that embraces rather than denies our innate cognitive gift of understanding cause and effect.

Side by side with this diagrammatic “language of knowledge,” we also have a symbolic “language of queries” to express the questions we want answers to. For example, if we are interested in the effect of a drug (D) on lifespan (L), then our query might be written symbolically as: P(L|do(D)). In other words, what is the probability (P) that a typical patient would survive L years if made to take the drug? This question describes what epidemiologists would call an intervention or a treatment and corresponds to what we measure in a clinical trial. In many cases we may also wish to compare P(L|do(D)) with P(L |do(not-D)); the latter describes patients denied treatment, also called the “control” patients. The do-operator signifies that we are dealing with an intervention rather than a passive observation; classical statistics has nothing remotely similar to this operator.

We must invoke an intervention operator do(D) to ensure that the observed change in Lifespan L is due to the drug itself and is not confounded with other factors that tend to shorten or lengthen life. If, instead of intervening, we let the patient himself decide whether to take the drug, those other factors might influence his decision, and lifespan differences between taking and not taking the drug would no longer be solely due to the drug. For example, suppose only those who were terminally ill took the drug. Such persons would surely differ from those who did not take the drug, and a comparison of the two groups would reflect differences in the severity of their disease rather than the effect of the drug. By contrast, forcing patients to take or refrain from taking the drug, regardless of preconditions, would wash away preexisting differences and provide a valid comparison.

Mathematically, we write the observed frequency of Lifespan L among patients who voluntarily take the drug as P(L|D), which is the standard conditional probability used in statistical textbooks. This expression stands for the probability (P) of Lifespan L conditional on seeing the patient take Drug D. Note that P(L|D) may be totally different from P(L|do(D)). This difference between seeing and doing is fundamental and explains why we do not regard the falling barometer to be a cause of the coming storm. Seeing the barometer fall increases the probability of the storm, while forcing it to fall does not affect this probability.

Judea Pearl. 2018. The Book of Why