If sprinklers are seasonally active (only turned on during spring to water plants), the sprinklers being on allows a backward inference about the season (represented by the arrow from season to sprinkler). In contrast, if someone turns on the sprinkler on random days, this backward inference cannot be made (represented by the absence of that arrow). This illustrates the difference between the intervention $$do(\text{sprinkler} = \text{on})$$ and the observation $$\text{sprinkler} = \text{on}$$.

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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

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

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

Example of Seeing vs. Doing: Sprinklers

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

Causal Definition of Confounder

Suppose we want to find the probability that a patient avoids the flu ($$A$$) given they received the flu shot ($$FS$$). The difference between conditional probability with the $$do()$$ operator and without is:

*   **With the $$do()$$ operator ($$P(A|do(FS))$$)**: The probability that a typical patient would avoid the flu if they were assigned to get the shot (by intervention, removing personal choice).
*   **Without the $$do()$$ operator ($$P(A|FS)$$)**: The probability that a patient avoided the flu given they were observed to take the flu shot. This represents the observed frequency of flu avoidance among patients who volunteered to get the shot.

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