Suppose we have a given patient avoids the flue (A) given the flu shot (FS). 
- $P(A|do(FS))$
    - The probability that a typical patient would avoid the flu if the patient got the shot without choice.
- $ P(A|FS) $
    - The probability a patient avoided the flu given they were observed to take the flue shot. This probability asks the observed frequency of patients who avoided the flu when they *volunteered* to get a flu shot

University of Michigan - Ann Arbor

In conditional probability, when we want to find the likelihood of an occurence given we **know** an *intervention* occured and the value of that intervention, we use the do() operator. 

`do()` operator

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

The Book of Why

The following are different ways the do() operator can be used
- P(A | do(B), C)
    Intervening on B, seeing C
    
- P(A | do(B), do(C))
    Multiple interventions
    
- P(A | do(P(B) = P’))
    A change on the distribution of B (not only a point
    mass distribution)

Different uses of do() operator

Pearl, Judea. “On the Interpretation of Do(x).” Technical Report - UCLA, 13 Feb. 2019.

https://ftp.cs.ucla.edu/pub/stat_ser/r486.pdf


On the Interpretation of do(x)


Pearl (1995) uses expression of the form Q = E[Y |do(X = x)] to denote the causal effect of X on Y .  Where do(x) is a minimally invasive intervention that sets the value of X to x and leaves other relationships unchanged.

However, when X is non-manipulable for experimenters (for example, gender/race are non-manipulable), critics raise concern about the interpretation of the casual effect Q. 


Question of do(x) in Structural Causal Modeling (SCM) 


Assuming an observational study guided by model M, we identify and evaluate the Q with the expression Q = E[Y |do(x)] = q(x), where q(x) is a function of x computed from the joint distribution of observed variables in the model. 

One can use this information in three ways:
1. Q represents a limit on pending interventions
2. Q tells us about the effects of feasible interventions
3. Q serves as an auxiliary mathematical operation in the derivation of causal effects of manipulable variables.


The Semantics of Q 

When X is non-manipulable, although direct tests of Q(x) are infeasible, indirect tests are available.We can use back-door adjustment or front-door adjustment to test Q.

For example, if a set of variable z satisfies the back door criterion relative to (X,Y),  Q(x) is identifiable and is given by the formula: $$ P(y|do(x))=\sum_{z} P(y|x,z)P(z)$$. 
Equating the adjustment formulae generated by each of those sets provides a test for M, and hence a test for Q.

Fig.1 shows a front-door model where I is manipulable, X is non-manipulable, and U is an unobserved confounder. X satisfies the front door criterion relative to (I,Y). Randomizing I yields an estimate of P(y|do(I)), which should be equal to the estimand of P(y|do(I)) obtained through the front-door formula. The causal effect of I on Y is identifiable and is given by the formula:  $$Q(x)=\sum_{i} P(y|x, I = i)P(I = i)$$


Testing do(x) Claims

Conditional probability with do() versus without

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