In the Rubin causal model, a potential outcome of a variable $$Y$$ is simply “the value that $$Y$$ would have taken for individual $$u$$, had $$X$$ been assigned the value $$x$$.” 
It’s often convenient to write this quantity more compactly as $$Y_X = _x(u)$$. Often we abbreviate this further as $$Y_x(u)$$ if it is apparent from the context what variable is being set to the value x. 

University of Michigan - Ann Arbor

Counterfactuals refer to situations that contrast actual events in reality. To answer counterfactual questions, we must go back in time and answer what would have happened if an event happened differently. Assessing counterfactuals is difficult because it is impossible to replicate them in reality. 

Counterfactuals

We have a causal belief that studying lecture notes the previous day caused the outcome of passing a major exam. If we our correct in our causal belief, then studying lecture notes will change not pass to pass on the exam. 
"What would have happened if I did not study lecture notes last night  for that exam?"
"Why did I pass that exam?" "Did I pass because I slept well?" "Because I did practice problems?"

Examples of Counterfactual Questions

In Hume’s philosophizing, we are given a new definition of causation depending on counterfactuals.

Hume defines causation between object 1 and 2 as: “if the first object had not been, the second had never existed”. 

This new definition depends on humans' ability to ideate an alternate reality where a single event occured, while the rest of the world was held constant. 

Hume's Redefinition of Causation

The Model Penal Code expresses the “but-for” test as follows: “Conduct is the cause of a result when: (a) it is an antecedent but for which the result in question would not have occurred.” If the defendant fired a gun and the bullet struck and killed the victim, the firing of the gun is a but-for, or necessary, cause of the death, since the victim would be alive if not for the firing. But-for causes can also be indirect. If Joe blocks a building’s fire exit with furniture, and Judy dies in a fire after she could not reach the exit, then Joe is legally responsible for her death even though he did not light the fire.
(Symbolic representation see the child node)

Counterfactuals and the law

Rubin Casual Model

ETT is a counterfactual quantity that has been used to evaluate whether people who got the treatment are those who would benefit most from it. For example, ETT can be used to measure the effectiveness of a program on its participants.
ETT can be represented by the expression:
$$ETT = P(Y_x = y|X = {x}')$$
Here we ask for the value of an outcome variable Y after setting the value of another variable X to x, but also knowing that X attains value x′ naturally. Where $$Y_x$$ is a random variable representing the behavior of Y after X is set to value x.  

Effect of Treatment on the Treated (ETT)

Link to a video that gives a great description and useful examples of counterfactuals
https://www.youtube.com/watch?v=9j_HWkrSxzI


Counterfactual youtube “bootcamp”

In the context of Berkeley's admission paradox, Bickel contends that discrimination is to do with making decisions that are influenced by the sex of the applicant, even if it is immaterial for the admission decision.  The words like "exercise of decision," "influence," and "immaterial" are suggestive of causation, either belonging to the second (intervention)or third (counterfactuals) Ladder of Causation.  

Bickel's Definition of Discrimination

If there is a model of the universe that perfectly maps how a variable effects and is effected by its neighbors you can generate all possible counterfactuals. Under this model a counterfactual can always be represented by this equation. $$Y_x(u)$$ is the event if it were to have occured and $$Y_{M_x}(u)$$ is the event as a counterfactual.

$$Y_x(u) = Y_{M_x}(u)$$

The Law of Counterfactuals

Evaluating a counterfactual using a Structural Causal Model (SCM) involves three sequential steps, which are valid regardless of whether the functional relationships are linear or non-linear:

1. **Abduction**: Use evidence (observations) to estimate the values of the unobserved background (exogenous) variables $$U$$.
2. **Action**: Use the do-operator to intervene on the model, setting the value of the antecedent variable (e.g., setting $$M = m$$), which modifies the structural equations by removing the equations that originally determined that variable.
3. **Prediction**: Use the modified structural equations and the estimated values of $$U$$ to calculate the counterfactual values of the outcome variables (e.g., $$Y$$).

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