Grasping Reality on TypePad, by Brad DeLong

“Our brains are just not wired to do probability problems very well, so I’m not surprised there were mistakes,” said Persi Diaconis, a statistician at Stanford University, in a 1991 interview with the New York Times. True, but there’s more to it. Our brains are not wired to do probability problems, but they are wired to do causal problems. And this causal wiring produces systematic probabilistic mistakes, like optical illusions.

Because there is no causal connection between My Door and Location of Car [in the Monty Hall Problem], either directly or through a common cause, we find it utterly incomprehensible that there is a probabilistic association. Our brains are not prepared to accept causeless correlations, and we need special training—through examples like the Monty Hall paradox or the ones discussed in Chapter 3—to identify situations where they can arise. Once we have “rewired our brains” to recognize colliders, the paradox ceases to be confusing...

This confusion between seeing and doing has resulted in a fountain of paradoxes, some of which we will entertain in this book. A world devoid of P(L | do(D)) and governed solely by P(L | D) would be a strange one indeed. For example, patients would avoid going to the doctor to reduce the probability of being seriously ill; cities would dismiss their firefighters to reduce the incidence of fires; doctors would recommend a drug to male and female patients but not to patients with undisclosed gender; and so on. It is hard to believe that less than three decades ago science did operate in such a world: the do-operator did not exist...

One of the crowning achievements of the Causal Revolution has been to explain how to predict the effects of an intervention without actually enacting it. It would never have been possible if we had not, first of all, defined the do-operator so that we can ask the right question and, second, devised a way to emulate it by noninvasive means.... If we are in possession of a causal model, we can often predict the result of an intervention from hands-off, intervention-free data....

Another advantage causal models have that data mining and deep learning lack is adaptability. Note that in Figure I.1, the estimand is computed on the basis of the causal model alone, prior to an examination of the specifics of the data. This makes the causal inference engine supremely adaptable, because the estimand computed is good for any data that are compatible with the qualitative model...

Counterfactuals are not products of whimsy but reflect the very structure of our world model. Two people who share the same causal model will also share all counterfactual judgments....

These queries take us to the top rung of the Ladder of Causation, the level of counterfactuals, because to answer them we must go back in time, change history, and ask, “What would have happened if I had not taken the aspirin?” No experiment in the world can deny treatment to an already treated person and compare the two outcomes, so we must import a whole new kind of knowledge...

In fact, almost all of [these "paradoxes"] represent clashes with causal intuition and therefore reveal the anatomy of that intuition. They were canaries in the coal mine that should have alerted scientists to the fact that human intuition is grounded in causal, not statistical, logic. I believe that the reader will enjoy this new twist on his or her favorite old paradoxes...

Deep learning has succeeded primarily by showing that certain questions or tasks we thought were difficult are in fact not. It has not addressed the truly difficult questions that continue to prevent us from achieving humanlike AI....

This leap forward in cognitive ability was as profound and important to our species as any of the anatomical changes that made us human. Within 10,000 years after the Lion Man’s creation, all other hominids (except for the very geographically isolated Flores hominids) had become extinct. And humans have continued to change the natural world with incredible speed, using our imagination to survive, adapt, and ultimately take over. The advantage we gained from imagining counterfactuals was the same then as it is today: flexibility, the ability to reflect and improve on past actions, and, perhaps even more significant, our willingness to take responsibility for past and current actions....

Probabilistic causality has always foundered on the rock of confounding. Every time the adherents of probabilistic causation try to patch up the ship with a new hull, the boat runs into the same rock and springs another leak. Once you misrepresent “probability raising” in the language of conditional probabilities, no amount of probabilistic patching will get you to the next rung of the ladder. As strange as it may sound, the notion of probability raising cannot be expressed in terms of probabilities...

Philosophers were too quick to commit to the only uncertainty-handling language they knew, the language of probability. They have for the most part gotten over this blunder in the past decade or so, but unfortunately similar ideas are being pursued in econometrics even now, under names like “Granger causality” and “vector autocorrelation”...

Lesson one from this example: causal analysis allows us to quantify processes in the real world, not just patterns in the data. The pups are growing at 3.34 grams per day, not 5.66 grams per day. 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....

In a world where science progresses logically, Wright’s response to Niles should have produced a scientific excitement followed by an enthusiastic adoption of his methods by other scientists and statisticians. But that is not what happened. “One of the mysteries of the history of science from 1920 to 1960 is the virtual absence of any appreciable use of path analysis, except by Wright himself and by students of animal breeding,” wrote one of Wright’s geneticist colleagues, James Crow. “Although Wright had illustrated many diverse problems to which the method was applicable, none of these leads was followed.”....

In the 1960s, things began to change. A group of social scientists, including Otis Duncan, Hubert Blalock, and the economist Arthur Goldberger (mentioned earlier), rediscovered path analysis as a method of predicting the effect of social and educational policies. In yet another irony of history, Wright had actually been asked to speak to an influential group of econometricians called the Cowles Commission in 1947, but he utterly failed to communicate to them what path diagrams were about. Only when economists arrived at similar ideas themselves was a short-lived connection forged...

As we saw, Bayes’s rule is formally an elementary consequence of his definition of conditional probability. But epistemologically, it is far from elementary. It acts, in fact, as a normative rule for updating beliefs in response to evidence. In other words, we should view Bayes’s rule not just as a convenient definition of the new concept of “conditional probability” but as an empirical claim to faithfully represent the English expression “given that I know.” It asserts, among other things, that the belief a person attributes to S after discovering T is never lower than the degree of belief that person attributes to S AND T before discovering T. Also, it implies that the more surprising the evidence T—that is, the smaller P(T) is—the more convinced one should become of its cause S.

No wonder Bayes and his friend Price, as Episcopal ministers, saw this as an effective rejoinder to Hume. If T is a miracle (“Christ rose from the dead”), and S is a closely related hypothesis (“Christ is the son of God”), our degree of belief in S is very dramatically increased if we know for a fact that T is true. The more miraculous the miracle, the more credible the hypothesis that explains its occurrence. This explains why the writers of the New Testament were so impressed by their eyewitness evidence. 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. A person who has never in his life seen a snooker table would be very doubtful that L could be longer than ten feet. A person who has only seen snooker tables and never seen a billiard table would, on the other hand, give a very low prior probability to L being less than ten feet. This variability, also known as “subjectivity,” is sometimes seen as a deficiency...

Causal diagrams make possible a shift of emphasis from confounders to deconfounders. The former cause the problem; the latter cure it. The two sets may overlap, but they don’t have to. If we have data on a sufficient set of deconfounders, it does not matter if we ignore...

Mistaking a mediator for a confounder is one of the deadliest sins in causal inference and may lead to the most outrageous errors. The latter invites adjustment; the former forbids it...

In fact, Cornfield’s method planted the seeds of a very powerful technique called “sensitivity analysis,” which today supplements the conclusions drawn from the inference engine described in the Introduction. Instead of drawing inferences by assuming the absence of certain causal relationships in the model, the analyst challenges such assumptions and evaluates how strong alternative relationships must be in order to explain the observed data. The quantitative result is then submitted to a judgment of plausibility, not unlike the crude judgments invoked in positing the absence of those causal relationships. Needless to say, if we want to extend Cornfield’s approach to a model with more than three or four variables, we need algorithms and estimation techniques that are unthinkable without the advent of graphical tools...

Glynn and Kashin’s results show why the front-door adjustment is such a powerful tool: it allows us to control for confounders that we cannot observe (like Motivation), including those that we can’t even name. RCTs are considered the “gold standard” of causal effect estimation for exactly the same reason. Because front-door estimates do the same thing, with the additional virtue of observing people’s behavior in their own natural habitat instead of a laboratory, I would not be surprised if this method eventually becomes a serious competitor to randomized controlled trials...

A climate scientist can say, “There is a 90 percent probability that man-made climate change was a necessary cause of this heat wave,” or “There is an 80 percent probability that climate change will be sufficient to produce a heat wave this strong at least once every 50 years.” The first sentence has to do with attribution: Who was responsible for the unusual heat? The second has to do with policy. It says that we had better prepare for such heat waves because they are likely to occur sooner or later. Either of these statements is more informative than shrugging our shoulders and saying nothing about the causes of individual weather events...

Aristotle set up a whole taxonomy of causation, including “material causes,” “formal causes,” “efficient causes,” and “final causes.” For example, the material cause of the shape of a statue is the bronze from which it is cast and its properties; we could not make the same statue out of Silly Putty. However, Aristotle nowhere makes a statement about causation as a counterfactual, so his ingenious classification lacks the simple clarity of Thucydides’s account of the cause of the tsunami...

Even today, many people seeing the [Monty Hall] puzzle for the first time find the result hard to believe. Why? What intuitive nerve is jangled? There are probably 10,000 different reasons, one for each reader, but I think the most compelling argument is this: vos Savant’s solution seems to force us to believe in mental telepathy. If I should switch no matter what door I originally chose, then it means that the producers somehow read my mind. How else could they position the car so that it is more likely to be behind the door I did not choose?....

When we condition on a collider, we create a spurious dependence between its parents. The dependence is borne out in the probabilities: if you chose Door 1, the car location is twice as likely to be behind Door 2 as Door 1; if you chose Door 2, the car location is twice as likely to be behind Door 1. It is a bizarre dependence for sure, one of a type that most of us are unaccustomed to. It is a dependence that has no cause. It does not involve physical communication between the producers and us. It does not involve mental telepathy. It is purely an artifact of Bayesian conditioning: a magical transfer of information without causality.

Our minds rebel at this possibility because from earliest infancy, we have learned to associate correlation with causation. If a car behind us takes all the same turns that we do, we first think it is following us (causation!). We next think that we are going to the same place (i.e., there is a common cause behind each of our turns). But causeless correlation violates our common sense. Thus, the Monty Hall paradox is just like an optical illusion or a magic trick: it uses our own cognitive machinery to deceive us....

Notice that I have really given two explanations of the Monty Hall paradox. The first one uses causal reasoning to explain why we observe a spurious dependence between Your Door and Location of Car; the second uses Bayesian reasoning to explain why the probability of Door 2 goes up in Let’s Make a Deal.

Both explanations are valuable.

The Bayesian one accounts for the phenomenon but does not really explain why we perceive it as so paradoxical. In my opinion, a true resolution of a paradox should explain why we see it as a paradox in the first place. Why did the people who read her column believe so strongly that vos Savant was wrong? It wasn’t just the know-it-alls. Paul Erdos, one of the most brilliant mathematicians of modern times, likewise could not believe the solution until a computer simulation showed him that switching is advantageous. What deep flaw in our intuitive view of the world does this reveal?

“Our brains are just not wired to do probability problems very well, so I’m not surprised there were mistakes,” said Persi Diaconis, a statistician at Stanford University, in a 1991 interview with the New York Times. True, but there’s more to it. Our brains are not wired to do probability problems, but they are wired to do causal problems. And this causal wiring produces systematic probabilistic mistakes, like optical illusions.

Because there is no causal connection between My Door and Location of Car, either directly or through a common cause, we find it utterly incomprehensible that there is a probabilistic association. Our brains are not prepared to accept causeless correlations, and we need special training—through examples like the Monty Hall paradox or the ones discussed in Chapter 3—to identify situations where they can arise. Once we have “rewired our brains” to recognize colliders, the paradox ceases to be confusing...