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The calculus of causation consists of two languages: causal diagrams, to express what we know, and a symbolic language, resembling algebra, to express what we want to know. The causal diagrams are simply dot-and-arrow pictures that summarize our existing scientific knowledge. The dots represent quantities of interest, called “variables,” and the arrows represent known or suspected causal relationships between those variables—namely, which variable “listens” to which others. These diagrams are extremely easy to draw, comprehend, and use,
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 |
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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)).
Seeing the barometer fall increases the probability of the storm, while forcing it to fall does not affect this probability.
When the scientific question of interest involves retrospective thinking, we call on another type of expression unique to causal reasoning called a counterfactual. For example, suppose that Joe took Drug D and died a month later; our question of interest is whether the drug might have caused his death. To answer this question, we need to imagine a scenario in which Joe was about to take the drug but changed his mind. Would he have lived?
with humans about their own choices and
Two people who share the same causal model will also share all counterfactual judgments.
in the world of AI, you do not really understand a topic until you can teach it to a mechanical robot. That is why you will find me emphasizing and reemphasizing notation, language, vocabulary, and grammar. For example, I obsess over whether we can express a certain claim in a given language and whether one claim follows from others. It is amazing how much one can learn from just following the grammar of scientific utterances.
The inference engine is a machine that accepts three different kinds of inputs—Assumptions, Queries, and Data—and produces three kinds of outputs. The first of the outputs is a Yes/No decision as to whether the given query can in theory be answered under the existing causal model, assuming perfect and unlimited data. If the answer is Yes, the inference engine next produces an Estimand. This is a mathematical formula that can be thought of as a recipe for generating the answer from any hypothetical data, whenever they are available. Finally, after the inference engine has received the Data
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The listening pattern prescribed by the paths of the causal model usually results in observable patterns or dependencies in the data. These patterns are called “testable implications” because they can be used for testing the model. These are statements like “There is no path connecting D and L,” which translates to a statistical statement, “D and L are independent,” that is, finding D does not change the likelihood of L. If the data contradict this implication, then we need to revise our model. Such revisions require another engine, which obtains its inputs from boxes 4 and 7 and computes the
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“Estimand” comes from Latin, meaning “that which is to be estimated.” This is a statistical quantity to be estimated from the data that, once estimated, can legitimately represent the answer to our query.
contrary to traditional estimation in statistics, some queries may not be answerable under the current causal model, even after the collection of any amount of data. For example, if our model shows that both D and L depend on a third variable Z (say, the stage of a disease), and if we do not have any way to measure Z, then the query P(L | do(D)) cannot be answered. In that case it is a waste of time to collect data. Instead we need to go back and refine the model, either by adding new scientific knowledge that might allow us to estimate Z or by making simplifying assumptions (at the risk of
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data are profoundly dumb about causal relationships. They tell us about quantities like P(L | D) or P(L | D, Z). It is the job of the estimand to tell us how to bake these statistical quantities into one expression that, based on the model assumptions, is logically equivalent to the causal query—say, P(L | do(D)).
estimands and in fact the whole top part of Figure I does not exist in traditional methods of statistical analysis. There, the estimand and the query coincide. For example, if we are interested in the proportion of people among those with Lifespan L who took the Drug D, we simply write this query as P(D | L). The same quantity would be our estimand.
if our model is correct and our data are sufficient, we get an answer to our causal query, such as “Drug D increases the Lifespan L of diabetic Patients Z by 30 percent, plus or minus 20 percent.” Hooray! The answer will also add to our scientific knowledge (box 1) and, if things did not go the way we expected, might suggest some improvements to our causal model (box 3).
we collect data only after we posit the causal model, after we state the scientific query we wish to answer, and after we derive the estimand. This contrasts with the traditional statistical approach, mentioned above, which does not even have a causal model.
information about the effects of actions or interventions is simply not available in raw data, unless it is collected by controlled experimental manipulation.
this adaptability is important, compare this engine with a learning agent—in this instance a human, but in other cases perhaps a deep-learning algorithm or maybe a human using a deep-learning algorithm—trying to learn solely from the data. By observing the outcome L of many patients given Drug D, she is able to predict the probability that a patient with characteristics Z will survive L years. Now she is transferred to a different hospital, in a different part of town, where the population characteristics (diet, hygiene, work habits) are different. Even if these new characteristics merely
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Studying this engine will empower the reader to spot certain patterns in the causal diagram that deliver immediate answers to the causal query. These patterns are called back-door adjustment, front-door adjustment, and instrumental variables, the workhorses of causal inference in practice.
“We may define a cause to be an object followed by another, and where all the objects, similar to the first, are followed by objects similar to the second. Or, in other words, where, if the first object had not been, the second never had existed.”
Hume really gave two definitions, not one, the first of regularity (i.e., the cause is regularly followed by the effect) and the second of the counterfactual (“if the first object had not been …”). While philosophers and scientists had mostly paid attention to the regularity definition, Lewis argued that the counterfactual definition aligns more closely with human intuition: “We think of a cause as something that makes a difference, and the difference it makes must be a difference from what would have happened without it.”
compute an actual value (or probability) for any counterfactual query, no matter how convoluted. Of special interest are questions concerning necessary and sufficient causes of observed events. For example, how likely is it that the defendant’s action was a necessary cause of the claimant’s injury? How likely is it that man-made climate change is a sufficient cause of a heat wave?
Chapter 9 discusses the topic of mediation. You may have wondered, when we talked about drawing arrows in a causal diagram, whether we should draw an arrow from Drug D to Lifespan L if the drug affects lifespan only by way of its effect on blood pressure Z (a mediator). In other words, is the effect of D on L direct or indirect? And if both, how do we assess their relative importance?
causal reasoning is essential for machines to communicate with us in our own language about policies, experiments, explanations, theories, regret, responsibility, free will, and obligations—and, eventually, to make their own moral decisions.
you are smarter than your data. Data do not understand causes and effects; humans do.
The mental model is the arena where imagination takes place. It enables us to experiment with different scenarios by making local alterations to the model. Somewhere in our hunters’ mental model was a subroutine that evaluated the effect of the number of hunters. When they considered adding more, they didn’t have to evaluate every other factor from scratch. They could make a local change to the model, replacing “Hunters = 8” with “Hunters = 9,” and reevaluate the probability of success. This modularity is a key feature of causal models.
a causal learner must master at least three distinct levels of cognitive ability: seeing, doing, and imagining.
classify a cognitive system in terms of the queries it can answer.
one event is associated with another if observing one changes the likelihood of observing the other.
If, for example, the programmers of a driverless car want it to react differently to new situations, they have to add those new reactions explicitly. The machine will not figure out for itself that a pedestrian with a bottle of whiskey in hand is likely to respond differently to a honking horn. This lack of flexibility and adaptability is inevitable in any system that works at the first level of the Ladder of Causation.
We cannot answer questions about interventions with passively collected data, no matter how big the data set or how deep the neural network.
Why not just go into our vast database of previous purchases and see what happened previously when toothpaste cost twice as much? The reason is that on the previous occasions, the price may have been higher for different reasons. For example, the product may have been in short supply, and every other store also had to raise its price. But now you are considering a deliberate intervention that will set a new price regardless of market conditions.
If you had data on the market conditions that existed on the previous occasions, perhaps you could make a better prediction … but what data do you need? And then, how would you figure it out?
A very direct way to predict the result of an intervention is to experiment with it under carefully controlled conditions. Big-data companies like Facebook know this and constantly perform experiments to see what happens if items on the screen are arranged differently or the customer gets a different prompt (or even a different price).
successful predictions of the effects of interventions can sometimes be made even without an experiment.
the sales manager could develop a model of consumer behavior that includes market conditions. Even if she doesn’t have data on every factor, she might have data on enough key surrogates to make the prediction. A sufficiently strong and accurate causal model can allow us to use rung-one (observational) data to answer rung-two (interventional) queries.
P(floss | do(toothpaste)), which asks about the probability that we will sell floss at a certain price, given that we set the price of toothpaste at another price.
we have too much toothpaste in our warehouse. “How can we sell it?” he asks. That is, what price should we set for it? Again, the question refers to an intervention, which we want to perform mentally before we decide whether and how to do it in real life. That requires a causal model.
My headache is gone now, but why? Was it the aspirin I took? The food I ate? The good news I heard? 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.
The laws of physics, for example, can be interpreted as counterfactual assertions, such as “Had the weight on this spring doubled, its length would have doubled as well” (Hooke’s law).
This statement is, of course, backed by a wealth of experimental (rung-two) evidence, derived from hundreds of springs, in dozens of laboratories, on thousands of different occasions. However, once anointed as a “law,” physicists interpret it as a functional relationship that governs this very spring, at this very moment, under hypothetical values of the weight. All of these different worlds, where the weight is x pounds and the length of the spring is Lx inches, are treated as objectively knowable and simultaneously active, even though only one of them actually exists.
“What is the probability that a customer who bought toothpaste would still have bought it if we had doubled the price?” We are comparing the real world (where we know that the customer bought the toothpaste at the current price) to a fictitious world (where the price is twice as high).
Finding out why a blunder occurred allows us to take the right corrective measures in the future. Finding out why a treatment worked on some people and not on others can lead to a new cure for a disease. Answering the question “What if things had been different?” allows us to learn from history and the experience of others, something that no other species appears to do.
How can machines (and people) represent causal knowledge in a way that would enable them to access the necessary information swiftly, answer questions correctly, and do it with ease, as a three-year-old child can? In fact, this is the main question we address in this book. I call this the mini-Turing test. The idea is to take a simple story, encode it on a machine in some way, and then test to see if the machine can correctly answer causal questions that a human can answer. It is “mini” for two reasons. First, it is confined to causal reasoning, excluding other aspects of human intelligence
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When we began, the vaccination rate was 99 percent. We now ask the counterfactual question “What if we had set the vaccination rate to zero?” Using the probabilities I gave you above, we can conclude that out of 1 million children, 20,000 would have gotten smallpox, and 4,000 would have died.
a causal model entails more than merely drawing arrows. Behind the arrows, there are probabilities. When we draw an arrow from X to Y, we are implicitly saying that some probability rule or function specifies how Y would change if X were to change. We might know what the rule is; more likely, we will have to estimate it from data.
in many cases we can leave those mathematical details completely unspecified. Very often the structure of the diagram itself enables us to estimate all sorts of causal and counterfactual relationships: simple or complicated, deterministic or probabilistic, linear or nonlinear.
if we reversed the arrow Vaccination → Smallpox, we would get the same associations in the data but would erroneously conclude that smallpox affects vaccination.
the idea of causes and effects is much more fundamental than the idea of probability. We begin learning causes and effects before we understand language and before we know any mathematics. (Research has shown that three-year-olds already understand the entire Ladder of Causation.)
