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.  

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

Association refers to linking variables together solely based on their perceived outcomes and is not influenced by reasoning. 

Association

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 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

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

-Conditional Probability 
- What is the probability of Z happening given the event of X? This is also written as P(Z|X)

-Correlation/Regression 
        -Shows how the data is related or Associated with eachother 

 -Machine Learing and Deep learning occours at this level because they only predict outcomes they dont reason why things are occouring 
 
 -Statistics occurs on the association rung on the ladder (what data tells us about the result)

Association Examples

A real life example of a question that can be answered through association is: "How likely is a customer who bought toothpaste to also buy dental floss?"

What does a symptom tell you about a disease?

What does wearing glasses tell you about a person's eyesight?

Real life example of association questions/application

A genome-wide association study (GWAS) is a statistical genomic analysis technique to associate specific genetic variations with particular diseases by looking for patterns and frequencies so that certain genetic variations can be identified to associate with and predict disease.  It is important to note that GWAS cannot tell causality since it can only be used to make associations between variables, -hence the first ladder of causation, Association.  

Genome-Wide Association Study (GWAS)

A Bayesian network's structure is made up of nodes (which can be multi-variate), edges between paired nodes, and a conditional probability distribution in each nodes. The nodes in a Bayesian network represent random variables and edges represent the conditional dependence of these nodes. We try to use this structure to model the conditional probability distribution of outcome variables (which are often causal) after observing new evidence.

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