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

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

Conditional probability with do() versus without

Causal Definition of Confounder

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. 

University of Michigan - Ann Arbor

It is a language of queries, similar to algebra in mathematics, which we use to communicate our causal research questions. For example, if we are interested in studying the probability $P$ that you come up with $n$ number of creative ideas when you $do$ use 1Cademy ($I$), then we can represent this research question as $P(n|do(I))$.

Causal symbolic language

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

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

The Book of Why

`do()` operator

In a causal diagram, an instrumental variable is a variable that satisfies the following properties:

- The instrumental variable has a causal effect on the treatment variable.  
- The instrumental variable is not associated with unmeasured confounder(s).
- The instrumental variable affects the outcome only via the treatment variable; i.e., there is no direct effect of the instrumental variable on the outcome variable.
- Assumptions: linearity or monotonicity

Instrumental Variable

In the context of clinical trials, the monotonicity means there are no subjects who do not take the experimental treatment when in the experimental group but do take the experimental treatment when in the control group.  Similarly, there are no subjects who take the experimental treatment when in the control group but do not take the experimental treatment when in the experimental group.  

Monotonicity Assumption in Clinical Trials

If sprinklers are seasonally on (that is they are only turned on during spring time to water plants), then the sprinklers being on would give you the abililty to make a backward inference about what season it is (seen in the causal model on the left by the arrow connecting season with sprinkler). Contrastly, if a person were to turn on the sprinkler on random days throughout the year the same backward inference can not be made as to what season it is (seen in the causal model on the left by the absence of an arrow between season and sprinkler). This is the model representation of do(sprinkler = on) as opposed to sprinkler = on.

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