Pearl notes that experience likely depends on education. If someone spends 4 years getting a college degree, then they will lose 4 years of potential experience. Thus for Carol and Bert, the example goes that Carol makes more money than Bert, and has a graduate degree, while Bert has a college degree. 

The potential outcome model claims that Carol would make more money than Bert had she only had a college degree, because she would have had the same education and **more** experience than Bert. 


Why Having a Model is important for Potential Outcomes

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.


University of Michigan - Ann Arbor

“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. 


Rubin's Definition of Counterfactuals

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

The Book of Why

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. 


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