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

Scenario: Berkeley Admissions Paradox

Setup:
We include three variables in our definition: 
a treatment $$X$$
an outcome $$Y$$
a mediator $$M$$

Formal Definition:
$$CDE(0) = P(Y = 1 | do(X = 1), do(M = 0)) – P(Y = 1 | do(X = 0), do(M = 0)) $$   


University of Michigan - Ann Arbor

The direct effect means that a variable (X) does not go through (without the intervention of) a mediator (M) to produce a causal impact on the outcome (Y).  That is: 

$$X -> Y$$   ( as opposed to   $$X -> M -> Y$$)$$

Direct Effect

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

The Book of Why

Goal: To reduce the effect of the overcontrolled mediator value, we can use natural direct effect to model the effect. 

Scenario: Berkeley Admissions Paradox

Definition:
$$NDE = P(Y_{M = M_0} = 1 | do(X = 1)) – P(Y_{M = M_0} = 1 | do(X = 0)) $$

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