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.  

University of Michigan - Ann Arbor

In 1973, Eugene Hammel, an associate dean at the University of California, noticed a worrisome trend in the university’s admission rates for men and women. According to his data, it is shown that 44 percent of the men who applied to graduate school at Berkeley had been accepted, compared to only 35 percent of the women.
After a careful investigation, Eugene found out that even though for each apartment, the admissions decisions were consistently more favorable to women than to men, the admission rate for women is lower than that of men.
The causal story Eugene discovered is that women were rejected in greater numbers because they applied to harder departments to get into, like history and literature, therefore having a lower acceptance rate.

Berkeley Admissions Paradox

Intervention refers to evaluating possible outcomes of a situation with some imposed condition.

Intervention

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

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

The Book of Why

Diagram for Berkeley Admissions Paradox

In the context of Berkeley's admission paradox, Bickel contends that bias is to do with a pattern of association between a particular decision and particular sex of the applicant.  The words "pattern" and "association" are indications that bias belongs to the rung one of the Ladder of Causation, association.  

Bickel's Definition of Bias

Bickel's Definition of Discrimination

We would like to examine the direct effect of $$X$$ on $$Y$$ without allowing $$M$$ to change. 
In the context of the Berkeley example, X represents the gender, Y represents the admission result, M represents the mediator(department candidates apply to).
We force everybody to apply to the history department by setting $$do(M = 0)$$. People are randomly asked to report their sex as male ($$do(X = 1)$$) or as female ($$do(X = 0)$$), regardless of their actual genders. We obtain CDE through the difference in admission rates between the two groups.
In symbols, $$CDE(0)$$ indicates that we forced the mediator to take on the value zero, i.e. the history department. We could also force everybody to apply to the engineering department: $$do(M = 1)$$. We would then denote the resulting controlled direct effect as $$CDE(1)$$. 


Apply CDE to Berkeley Admissions Paradox

The formal definition indicates that NDE requires counterfactuals. We ask the following questions in the definition: what will happen had X been set to 1? What will happen had X been set to 0?

In the Berkeley scenario, X represents the "reported" sex, which means a woman can also report her sex as a male in the what-if scenario. In this way, the admission is decided by the reported(fake) sex rather than the actual sex, invoking the counterfactual subscript.

Apply NDE to Berkeley Admissions Paradox

"What will happen if we *change* the environment?"
"What if we do..."
Ex: What is the probability we get an A on our next Python exam given we increase our daily study hours?
$ P(AGrade| do(hours))  $


"How?"
Ex: "How many more daily hours should I dedicate to exam study?"

Common Questions of Intervention

The intervention causes the variable to become unrelated to factors that usually affect it so the arrows between these factors and the variables will not exist in the causal diagram with the intervention.

Intervention in Causal Diagrams

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

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. 

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”

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

Examining a counterfactual using a structural causal model involves three steps. The approach is valid regardless of the form of the model relationships (linear or otherwise).

 - Abduct
Apply abductive reasoning (logical inference that uses observation to find the simplest/most likely explanation) to estimate $u$, the proxy for the unobserved variables on the specific observation that supports the counterfactual.

 - Act
For a specific observation, use the do operator to establish the counterfactual (e.g., m=0), modifying the equations accordingly.

 - Predict
Calculate the values of the output (y) using the modified equations.

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