The Book of Why: The New Science of Cause and Effect (Penguin Science)

by Judea Pearl

Finally—and here comes Wright’s ingenuity—he showed that if we knew the causal quantities in Figure 2.7, we could predict correlations in the data (not shown in the diagram) by a simple graphical rule. This rule sets up a bridge from the deep, hidden world of causation to the surface world of correlations. It was the first bridge ever built between causality and probability, the first crossing of the barrier between rung two and rung one on the Ladder of Causation.

This idea must have seemed simple to Wright but turned out to be revolutionary because it was the first proof that the mantra “Correlation does not imply causation” should give way to “Some correlations do imply causation.”

When you take apart the diagram arrow by arrow in this way, I think you will find that every one of them makes perfect sense. Note also that each arrow is accompanied by a small letter (a, b, c, etc.). These letters, called path coefficients, represent the strength of the causal effects that Wright wanted to solve for. Roughly speaking, a path coefficient represents the amount of variability in the target variable that is accounted for by the source variable.

Wright’s paper was a tour de force and deserves to be considered one of the landmark results of twentieth-century biology. Certainly it is a landmark for the history of causality. Figure 2.7 is the first causal diagram ever published, the first step of twentieth-century science onto the second rung of the Ladder of Causation. And not a tentative step but a bold and decisive one!

Of course, at times scientists do not know the entire web of relationships between their variables. In that case, Wright argued, we can use the diagram in exploratory mode; we can postulate certain causal relationships and work out the predicted correlations between variables. If these contradict the data, then we have evidence that the relationships we assumed were false. This way of using path diagrams, rediscovered in 1953 by Herbert Simon (a 1978 Nobel laureate in economics), inspired much work in the social sciences.

He wishes to submit that the combination of knowledge of correlations with knowledge of causal relations to obtain certain results, is a different thing from the deduction of causal relations from correlations implied by Niles’ statement.”

Lesson two, whether you followed the mathematics or not: in path analysis you draw conclusions about individual causal relationships by examining the diagram as a whole. The entire structure of the diagram may be needed to estimate each individual parameter.

I previously quoted Yule on how relations with Pearson became strained if you disagreed with him and impossible if you criticized him. Exactly the same thing could be said about Fisher. The latter carried out nasty feuds with anyone he disagreed with, including Pearson, Pearson’s son Egon, Jerzy Neyman (more will be said on these two in Chapter 8), and of course Wright.

The fates of path analysis in economics and sociology followed different trajectories, each leading to a betrayal of Wright’s ideas. Sociologists renamed path analysis as structural equation modeling (SEM), embraced diagrams, and used them extensively until 1970, when a computer package called LISREL automated the calculation of path coefficients (in some cases). Wright would have predicted what followed: path analysis turned into a rote method, and researchers became software users with little interest in what was going on under the hood. In the late 1980s, a public challenge (by statistician David Freedman) to explain the assumptions behind SEM went unanswered, and some leading SEM experts even disavowed that SEMs had anything to do with causality.

As late as 1995, most economists refrained from explicitly attributing causal or counterfactual meaning to their equations. Even those who used structural equations for policy decisions remained incurably suspicious of diagrams, which could have saved them pages and pages of computation. Not surprisingly, some economists continue to claim that “it’s all in the data” to this very day.

Wright, to his great credit, understood the enormous stakes and stated in no uncertain terms, “In treating the model-free approach (3) as preferred alternative … Karlin et al. are urging not merely a change in method, but an abandonment of the purpose of path analysis and evaluation of the relative importance of varying causes. There can be no such analysis without a model. Their advice to anyone with an urge to make such an evaluation is to repress it and do something else.” Wright understood that he was defending the very essence of the scientific method and the interpretation of data. I would give the same advice today to big-data, model-free enthusiasts. Of course, it is okay to tease out all the information that the data can provide, but let’s ask how far this will get us. It will never get us beyond the first rung of the Ladder of Causation, and it will never answer even as simple a question as “What is the relative importance of various causes?” E pur si muove!

For Wright, drawing a path diagram is not a statistical exercise; it is an exercise in genetics, economics, psychology, or whatever the scientist’s own field of expertise is. Second, Wright traces the allure of “model-free” methods to their objectivity. This has indeed been a holy grail for statisticians since day one—or since March 15, 1834, when the Statistical Society of London was founded. Its founding charter said that data were to receive priority in all cases over opinions and interpretations. Data are objective; opinions are subjective. This paradigm long predates Pearson. The struggle for objectivity—the idea of reasoning exclusively from data and experiment—has been part of the way that science has defined itself ever since Galileo.

Bayesian statistics give us an objective way of combining the observed evidence with our prior knowledge (or subjective belief) to obtain a revised belief and hence a revised prediction of the outcome of the coin’s next toss. Still, what frequentists could not abide was that Bayesians were allowing opinion, in the form of subjective probabilities, to intrude into the pristine kingdom of statistics. Mainstream statisticians were won over only grudgingly, when Bayesian analysis proved a superior tool for a variety of applications, such as weather prediction and tracking enemy submarines. In addition, in many cases it can be proven that the influence of prior beliefs vanishes as the size of the data increases, leaving a single objective conclusion in the end. Unfortunately, the acceptance of Bayesian subjectivity in mainstream statistics did nothing to help the acceptance of causal subjectivity, the kind needed to specify a path diagram. Why? The answer rests on a grand linguistic barrier. To articulate subjective assumptions, Bayesian statisticians still use the language of probability, the native language of Galton and Pearson. The assumptions entering causal inference, on the other hand, require a richer language (e.g., diagrams) that is foreign to Bayesians and frequentists alike. The reconciliation between Bayesians and frequentists shows that philosophical barriers can be bridged with goodwill and a common language. Linguistic barriers are not surmounted so easily.

So the savvy reader will probably not be surprised to find out that I arrived at the theory of causality through a circuitous route that started with Bayesian probability and then took a huge detour through Bayesian networks. I will tell that story in the next chapter.

However, in recent years experts in artificial intelligence (AI) have made considerable progress toward automating the process of reasoning from evidence to hypothesis and likewise from effect to cause. I was fortunate enough to participate in the very earliest stages of this progress by developing one of its basic tools, called Bayesian networks. This chapter explains what these are, looks at some of their current-day applications, and discusses the circuitous route by which they led me to study causation.

In this chapter I will tell the story of Bayesian networks from their roots in the eighteenth century to their development in the 1980s, and I will give some more examples of how they are used today. They are related to causal diagrams in a simple way: a causal diagram is a Bayesian network in which every arrow signifies a direct causal relation, or at least the possibility of one, in the direction of that arrow. Not all Bayesian networks are causal, and in many applications it does not matter. However, if you ever want to ask a rung-two or rung-three query about your Bayesian network, you must draw it with scrupulous attention to causality.

University of Scotland,

For Bayes, this assertion provoked a natural, one might say Holmesian question: How much evidence would it take to convince us that something we consider improbable has actually happened? When does a hypothesis cross the line from impossibility to improbability and even to probability or virtual certainty?

This is perhaps the most important role of Bayes’s rule in statistics: we can estimate the conditional probability directly in one direction, for which our judgment is more reliable, and use mathematics to derive the conditional probability in the other direction, for which our judgment is rather hazy.

The recognition that the relation “given that” deserves its own symbol evolved only in the 1880s, and it was not until 1931 that Harold Jeffreys (known more as a geophysicist than a probability theorist) introduced the now standard vertical bar in P(S | T).

Now let me discuss the practical objection to Bayes’s rule—which may be even more consequential when we exit the realm of theology and enter the realm of science. If we try to apply the rule to the billiard-ball puzzle, in order to find P(L | x) we need a quantity that is not available to us from the physics of billiard balls: we need the prior probability of the length L, which is every bit as tough to estimate as our desired P(L | x). Moreover, this probability will vary significantly from person to person, depending on a given individual’s previous experience with tables of different lengths.

In many ways, Bayes’s rule is a distillation of the scientific method. The textbook description of the scientific method goes something like this: (1) formulate a hypothesis, (2) deduce a testable consequence of the hypothesis, (3) perform an experiment and collect evidence, and (4) update your belief in the hypothesis. Usually the textbooks deal with simple yes-or-no tests and updates; the evidence either confirms or refutes the hypothesis. But life and science are never so simple! All evidence comes with a certain amount of uncertainty. Bayes’s rule tells us how to perform step (4) in the real world.

The idea did not come to me in a dream; it came from an article by David Rumelhart, a cognitive scientist at University of California, San Diego, and a pioneer of neural networks. His article about children’s reading, published in 1976, made clear that reading is a complex process in which neurons on many different levels are active at the same time (see Figure 3.4).

Reading Rumelhart’s paper, I felt convinced that any artificial intelligence would have to model itself on what we know about human neural information processing and that machine reasoning under uncertainty would have to be constructed with a similar message-passing architecture.

More precisely, I assumed that the network would be hierarchical, with arrows pointing from higher neurons to lower ones, or from “parent nodes” to “child nodes.” Each node would send a message to all its neighbors (both above and below in the hierarchy) about its current degree of belief about the variable it tracked (e.g., “I’m two-thirds certain that this letter is an R”). The recipient would process the message in two different ways, depending on its direction. If the message went from parent to child, the child would update its beliefs using conditional probabilities, like the ones we saw in the teahouse example. If the message went from child to parent, the parent would update its beliefs by multiplying them by a likelihood ratio, as in the mammogram example.

Bayes’s rule tells us how to reverse the procedure, specifically by multiplying the prior probability by a likelihood ratio.

These three junctions—chains, forks, and colliders—are like keyholes through the door that separates the first and second levels of the Ladder of Causation. If we peek through them, we can see the secrets of the causal process that generated the data we observe; each stands for a distinct pattern of causal flow and leaves its mark in the form of conditional dependences and independences in the data.

In 1993, an engineer for France Telecom named Claude Berrou stunned the coding world with an error-correcting code that achieved near-optimal performance. (In other words, the amount of redundant information required is close to the theoretical minimum.) His idea, called a “turbo code,” can be best illustrated by representing it with a Bayesian network.

The term “confounding” originally meant “mixing” in English, and we can understand from the diagram why this name was chosen. The true causal effect X → Y is “mixed” with the spurious correlation between X and Y induced by the fork X ← Z → Y. For example, if we are testing a drug and give it to patients who are younger on average than the people in the control group, then age becomes a confounder—a lurking third variable. If we don’t have any data on the ages, we will not be able to disentangle the true effect from the spurious effect.

Klein raises a valid concern. Statisticians have been immensely confused about what variables should and should not be controlled for, so the default practice has been to control for everything one can measure.

The textbook approach of statisticians to confounding is quite different and rests on an idea most effectively advocated by R. A. Fisher: the randomized controlled trial (RCT). Fisher was exactly right, but not for exactly the right reasons. The randomized controlled trial is indeed a wonderful invention—but until recently the generations of statisticians who followed Fisher could not prove that what they got from the RCT was indeed what they sought to obtain. They did not have a language to write down what they were looking for—namely, the causal effect of X on Y.

That is because it has nothing to do with data or statistics. Confounding is a causal concept—it belongs on rung two of the Ladder of Causation.

There is now an almost universal consensus, at least among epidemiologists, philosophers, and social scientists, that (1) confounding needs and has a causal solution, and (2) causal diagrams provide a complete and systematic way of finding that solution. The age of confusion over confounding has come to an end!

I like the image that Fisher Box provides in the above passage: Nature is like a genie that answers exactly the question we pose, not necessarily the one we intend to ask. But we have to believe, as Fisher Box clearly does, that the answer to the question we wish to ask does exist in nature. Our experiments are a sloppy means of uncovering the answer, but they do not by any means define the answer. If we follow her analogy exactly, then do(X = x) must come first, because it is a property of nature that represents the answer we seek: What is the effect of using the first fertilizer on the whole field? Randomization comes second, because it is only a man-made means to elicit the answer to that question. One might compare it to the gauge on a thermometer, which is a means to elicit the temperature but is not the temperature itself.

Around 1923 or 1924, Fisher began to realize that the only experimental design that the genie could not defeat was a random one.

However, according to historian Stephen Stigler, the second benefit was really Fisher’s main reason for advocating randomization. He was the world’s master of quantifying uncertainty, having developed many new mathematical procedures for doing so. By comparison, his understanding of deconfounding was purely intuitive, for he lacked a mathematical notation for articulating what he sought. Now, ninety years later, we can use the do-operator to fill in what Fisher wanted to but couldn’t ask. Let’s see, from a causal point of view, how randomization enables us to ask the genie the right question.

Fortunately, the do-operator gives us scientifically sound ways of determining causal effects from nonexperimental studies, which challenge the traditional supremacy of RCTs. As discussed in the walking example, such causal estimates produced by observational studies may be labeled “provisional causality,” that is, causality contingent upon the set of assumptions that our causal diagram advertises.

How was confounding defined then, and how should it be defined? Armed with what we now know about the logic of causality, the answer to the second question is easier. The quantity we observe is the conditional probability of the outcome given the treatment, P(Y | X). The question we want to ask of Nature has to do with the causal relationship between X and Y, which is captured by the interventional probability P(Y | do(X)). Confounding, then, should simply be defined as anything that leads to a discrepancy between the two: P(Y | X) ≠ P(Y | do(X)). Why all the fuss? Unfortunately, things were not as easy as that before the 1990s because the do-operator had yet to be formalized.

Few moments in a scientific career are as satisfying as taking a problem that has puzzled and confused generations of predecessors and reducing it to a straightforward game or algorithm. I consider the complete solution of the confounding problem one of the main highlights of the Causal Revolution because it ended an era of confusion that has probably resulted in many wrong decisions in the past. It has been a quiet revolution, raging primarily in research laboratories and scientific meetings. Yet, armed with these new tools and insights, the scientific community is now tackling harder problems, both theoretical and practical, as subsequent chapters will show.

On the other hand, the triumph is incomplete. The period it took to reach the above conclusion, roughly from 1950 to 1964, might have been shorter if scientists had been able to call upon a more principled theory of causation. And most significantly from the point of view of this book, the scientists of the 1960s did not really put together such a theory.

Before cigarettes, lung cancer had been so rare that a doctor might encounter it only once in a lifetime of practice.

Of course Hill knew that an RCT was impossible in this case, but he had learned the advantages of comparing a treatment group to a control group. So he proposed to compare patients who had already been diagnosed with cancer to a control group of healthy volunteers. Each group’s members were interviewed on their past behaviors and medical histories. To avoid bias, the interviewers were not told who had cancer and who was a control. The results of the study were shocking: out of 649 lung cancer patients interviewed, all but two had been smokers. This was a statistical improbability so extreme that Doll and Hill couldn’t resist working out the exact odds against it: 1.5 million to 1.

By the end of the decade, the accumulation of so many different kinds of evidence had convinced almost all experts in the field that smoking indeed caused cancer. Remarkably, even researchers at the tobacco companies were convinced—a fact that stayed deeply hidden until the 1990s, when litigation and whistle-blowers forced tobacco companies to release many thousands of previously secret documents. In 1953, for example, a chemist at R.J. Reynolds, Claude Teague, had written to the company’s upper management that tobacco was “an important etiologic factor in the induction of primary cancer of the lung,” nearly a word-for-word repetition of Hill and Doll’s conclusion.

For all these reasons, the link between smoking and cancer remained controversial in the public mind long after it had ended among epidemiologists. Even doctors, who should have been more attuned to the science, remained unconvinced: a poll conducted by the American Cancer Society in 1960 showed that only a third of American doctors agreed with the statement that smoking was “a major cause of lung cancer,” and 43 percent of doctors were themselves smokers.

The paper by Cornfield and Lilienfeld had paved the way for a definitive statement by health authorities about the effects of smoking. The Royal College of Physicians in the United Kingdom took the lead, issuing a report in 1962 concluding that cigarette smoking was a causative agent in lung cancer. Shortly thereafter, US Surgeon General Luther Terry (quite possibly on the urging of President John F. Kennedy) announced his intention to appoint a special advisory committee to study the matter (see Figure 5.3).

In The Direction of Time, published posthumously in 1956, philosopher Hans Reichenbach made a daring conjecture called the “common cause principle.” Rebutting the adage “Correlation does not imply causation,” Reichenbach posited a much stronger idea: “No correlation without causation.” He meant that a correlation between two variables, X and Y, cannot come about by accident. Either one of the variables causes the other, or a third variable, say Z, precedes and causes them both.

Reichenbach’s error was his failure to consider collider structures—the structure behind the data selection. The mistake was particularly illuminating because it pinpoints the exact flaw in the wiring of our brains. We live our lives as if the common cause principle were true. Whenever we see patterns, we look for a causal explanation.

For almost twenty years, I have been trying to convince the scientific community that the confusion over Simpson’s paradox is a result of incorrect application of causal principles to statistical proportions. If we use causal notation and diagrams, we can clearly and unambiguously decide whether Drug D prevents or causes heart attacks. Fundamentally, Simpson’s paradox is a puzzle about confounding and can thus be resolved by the same methods we used to resolve that mystery. Curiously, three of the four 2016 papers that I mentioned continue to resist this solution.

Because Simpson’s paradox has been so poorly understood, some statisticians take precautions to avoid it. All too often, these methods avoid the symptom, Simpson’s reversal, without doing anything about the disease, confounding. Instead of suppressing the symptoms, we should pay attention to them. Simpson’s paradox alerts us to cases where at least one of the statistical trends (either in the aggregated data, the partitioned data, or both) cannot represent the causal effects.

In this chapter we finally make our bold ascent onto the second level of the Ladder of Causation, the level of intervention—the holy grail of causal thinking from antiquity to the present day. This level is involved in the struggle to predict the effects of actions and policies that haven’t been tried yet, ranging from medical treatments to social programs, from economic policies to personal choices.