Let (EX, ED, S) represent years of experience, years of education, and salary. Suppose ED $\in$ $\{0,1,2\}$, where 0 = high school, 1 = college, 2 = graduate school. 

If we are given information about one of the employees, we only have the amount of their salary, and the level of schooling they received. For example, Alice has only a high school degree, we have $S_1(u)$, but we want to find out $S_2(u)$, that is the potential salary Alice would have if she got a college degree. 

Pearl explains typical data-driven imputation techniques for finding these missing values, such as linear regression, or pattern matching. However, these methods fall short of doing actual causal analysis.


Problem setup for Pearl’s Potential Outcome Example

When writing the relations in a causal diagram, an arrow indicates a causal function of the input variables mapping to the output variable. The absence of an arrow indicates the absence of a causal function mapping those variables to one or the other. 

In this new context, we can write an explicit function of variables to model an outcome variable, but we are doing so with a strict “structural” assumption, that our model is the model occuring in nature. When we run a linear regression in this context we already know that X, Z cause Y. The regression will now tell us how.

By varying the input variables, we are assessing the many alternate worlds that occur by changing what we have observed. 


Structural Modeling for Counterfactuals

“The value Y would have taken for individual u, had X been assigned the value x”.

$Y_X=x^{(u)}$, or $Y_X(u)$. Note that these “potential outcomes” are defined at the individual level, not the population. 


University of Michigan - Ann Arbor

Lewis was also intrigued by human’s ability to ideate counterfactuals. He described the ability as being able to come up with an alternative universe where our variable of concern is the only thing not held constant. 

For example, one world where Joe took the aspirin and another where he didn’t. At the point that Joe did/didn’t take the aspirin, what might be the effect on that universe?

Pearl notes that in order to judge the causal strength of a variable, we need to assess the “closeness” of the world resulting from a counterfactual to our world. 


Lewis’s Definition for Counterfactuals

Pearl, J., & Mackenzie, D. (2018). $\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$ Basic Books.

The Book of Why

Pearl evaluates the counterfactual “had X been x” by deleting arrows in a causal diagram, or equations in a structural formula. That is, the minimal alteration on a causal diagram to cause X to assume the value x. 

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