those taken in previous price-raising situations—unless we replicate precisely the market conditions that existed when the price reached double its current value. Finally, the top level invokes Counterfactuals, a mode of reasoning that goes back to the philosophers David Hume and John Stuart Mill and that has been given comput-er-friendly semantics in the past two decades. 1, 18 A typical question in the counterfactual category is: “What if I had acted differently?” thus necessitating retrospective reasoning.

I place Counterfactuals at the top of the hierarchy because they subsume interventional and associational questions. If we have a model that can answer counterfactual queries, we can also answer questions about interventions and observations. For example, the interventional question: What will happen if we double the price? can be answered by asking the counterfactual question: What would happen had the price been twice its current value? Likewise, associational questions can be answered once we answer interventional questions; we simply ignore the action part and let observations take over. The translation does not work in the opposite direction. Interventional questions cannot be answered from purely observational information, from statistical data alone. No counterfactual question involving retrospection can be answered from purely interventional information, as with that acquired from controlled experiments; we cannot re-run an experiment on human subjects who were treated with a drug and see how they might behave had they not been given the drug. The hierarchy is therefore directional, with the top level being the most powerful one.

Counterfactuals are the building blocks of scientific thinking, as well as of legal and moral reasoning. For example, in civil court, a defendant is considered responsible for an injury if, but for the defendant’s action, it is more likely than not the injury would not have occurred. The computational meaning of “but for” calls for comparing the real world to an alternative world in which the defendant’s action did not take place.

Each layer in the hierarchy has a
syntactic signature that characterizes
the sentences admitted into that layer.
For example, the Association layer is
characterized by conditional prob-
ability sentences, as in P(y|x) = p, stating
that: The probability of event Y = y, given
that we observed event X = x is equal to
p. In large systems, such evidentiary
sentences can be computed efficiently
through Bayesian networks or any num-
ber of machine learning techniques.

At the Intervention layer, we deal with sentences of the type P(y|do(x), z) that denote “The probability of event Y = y, given that we intervene and set the value of X to x and subsequently observe event Z = z. Such expressions can be estimated experimentally from randomized trials or analytically using causal Bayesian networks. 18 A child learns the effects of interventions through playful manipulation of the environment ( usually in a deterministic playground), and AI planners obtain interventional knowledge by exercising admissible sets of actions. Interventional expressions cannot be inferred from passive observations alone, regardless of how big the data.

Finally, at the Counterfactual level, we deal with expressions of the type P(yx |x′,y′) that stand for “The probability that event Y = y would be observed had X been x, given that we actually observed X to be x′ and Y to be y′.” For example, the probability that Joe’s salary would be y had he finished college, given that his actual salary is y′ and that he had only two years of college.” Such sentences can be computed only when the model is based on functional relations or structural. 18

This three-level hierarchy, and the formal restrictions it entails, explains why machine learning systems, based only on associations, are prevented from reasoning about (novel) actions, experiments, and causal explanations.b b One could be tempted to argue that deep learning is not merely “curve fitting” because it attempts to minimize “overfit,” through, say, sample-splitting cross-validation, as opposed to maximizing “fit.” Unfortunately, the theoretical barriers that separate the three layers in the hierarchy tell us the nature of our objective function does not matter. As long as our system optimizes some property of the observed data, however noble or sophisticated, while making no reference to the world outside the data, we are back to level- 1 of the hierarchy, with all the limitations this level entails.

ponent in support of strong AI.

In the next section, I describe a three-level hierarchy that restricts and governs inferences in causal reasoning. The final section summarizes how traditional impediments are circumvented through modern tools of causal inference. In particular, I present seven tasks that are beyond the reach of “ associational” learning systems and have been (and can be) accomplished only through the tools of causal modeling.

 

The Three-Level Causal Hierarchy A useful insight brought to light through the theory of causal models is the classification of causal information in terms of the kind of questions each class is capable of answering. The classification forms a three-level hierarchy in the sense that questions at level i (i = 1, 2, 3) can be answered only if information from level j (j > i) is available.

Figure 1 outlines the three-level hierarchy, together with the characteristic questions that can be answered at each level. I call the levels 1. Association, 2. Intervention, and 3. Counterfactual, to match their usage. I call the first level Association because it invokes purely statistical relationships, defined by the naked data.a For instance, observing a customer who buys toothpaste makes it more likely that this customer will also buy floss; such associations can be inferred directly from the observed data using standard conditional probabilities and conditional expectation. 15 Questions at this layer, because they require no causal information, are placed at the bottom level in the hierarchy. Answering them is the hallmark of current machine learning methods. 4 The second level, Intervention, ranks higher than Association because it involves not just seeing what is but changing what we see. A typical question at this level would be: What will happen if we double the price? Such a question cannot be answered from sales data alone, as it involves a change in customers’ choices in reaction to the new pricing. These choices may differ substantially from

 

a Other terms used in connection with this layer include “model-free,” “model-blind,” “black-box,” and “data-centric”; Darwiche5 used “function-fitting,” as it amounts to fitting data by a complex function defined by a neural network architecture.