Theorem : Point Estimate from two sources

If we have data from multiple trials, irrespective of monotonicty, we can always arrive at a point estimate.

University of Michigan - Ann Arbor

The city of Los Angeles runs an RCT to determine effect of Russian Roulette on mortality. Death rate amongst Roulette players is 17.5%. Death rate among others is 1%.
The mayor of New York now wants to determine what the city's mortality rate would be if everyone plays Russian Roulette. New York starts with a mortality rate of 5% (vs LA's 1%).
How do we use the data from the LA RCT to determine this answer?

Motivating Example : Russian Roulette

Cinelli, Carlos, and Judea Pearl. “Generalizing Experimental Results by Leveraging Knowledge of Mechanisms.” Technical Report - UCLA, 10 Dec. 2019.

https://ftp.cs.ucla.edu/pub/stat_ser/r492.pdf

Generalizing Experimental Results by
Leveraging Knowledge of Mechanisms

X : X = 1 --> Plays Russian Roulette ;  X = 0 --> Doesn't play Russian Roulette
Y : Y = 1 --> Dies ;  Y = 0 --> Survives


Coarse Causal Diagram

X : X = 1 --> Plays Russian Roulette ;  X = 0 --> Doesn't play Russian Roulette
Y : Y = 1 --> Dies ;  Y = 0 --> Survives
S : S points to Y because of the disparity in the base mortality rate of the two cities.

Coarse Selection Diagram

Must take into account existing health condition disparities that cause difference in base mortality rate.
Thus we define : 
1. B : B=1-->Losing at Russian Roulette i.e. bullet in the barrel ; B=0--> No bullet in the barrel.
2. H : H=1-->All other health factors that cause death ; H=0--> other factors that cause survival


Building Correct Structural Diagram

We know, given the nature of the problem that 
$$Y = H \vee (X \land B) $$
This can be reframed as a more concrete counterfactual :
$$PS_{01} = P(Y_1 = 1 | Y_0 = 0 )$$
Where $Y_1$ is the event $H \land B$ and $Y_0$ is the event $H$. 
$PS_{01}$ is the probability that a death is caused by Russian Roulette alone.
Now let, 
$$PS_{10} = P(Y_1 = 0 | Y_0 = 1 )$$
By definition, since $Y_1$ contains $Y_0$, $PS_{10} = 0$
The law of total probability gives us : 
$$P(Y_1 = 1) = (1 - PS_{10})(P(Y_0 = 1)) + PS_{01}(1 - P(Y_0 = 1)) $$
We can then solve for $PS_{01}$ to get :
$$\frac{P(Y_1 = 1) + P(Y_0 = 1)}{1 - P(Y_0 = 1)} = \frac{1}{6}$$
Now we use the equation derived from the law of total probability and plug in values for NYC, to get the estimated mortality rate as $20.8%$

Thus using the structural diagram(s) and functional constraints, we've managed to solve this problem using only the two assumptions we began with.

Using Functional Constraints

While having a monotonic function/probability allows for precise computations, the absence of monotonicity does not hinder the calculations of estimates. Instead of a precise value, we then arrive at a tight bound for the estimate.

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