Is there an intuitive way to understand the Rubin Causal Model and the potential outcomes framework model? I am currently learning about the Rubin Causal Model and the potential outcomes framework and think I have the general gist of it. However, when the model starts to talk about superpopulations, I invariably get confused. I was wondering if anyone knew of a nice example or way of understanding the RCM as a whole. Thanks!
 A: The main idea in the Rubin causal model --- or the potential outcomes model, is the idea of potential outcomes itself.
For example, consider a individual with headache. Let Y represent the "headache" variable, which can take the values "headache" and "no headache".
Now let's consider a treatment X, such as taking "aspirin", X  can take the values "aspirin" and "no aspirin". 
The potential outcomes framework associates to each value of X a conterfactual statement for Y. Using Imbens' and Rubin's book picture:

That is, in this case we have two potential outcomes. 
Potential outcome under treatment: Y(X = "aspirin"), which means the state of your headache when you take the aspirin. For this particular individual the aspirin works. For other individuals it could be different.
Potential outcome under no treatment (aka control): Y(X = "no aspiring"), which means the state of your headache when you do not take the pill. For the individual in the example he would not heal naturally. For other individuals it could be different.
This is the individual treatment effect --- or unit level treatment effect. But usually you will be interested in (sub)population effects. 
So now let's consider a population of $N$ individuals. Let $Y_i(1)$ be the potential outcome of individual $i$ under treatment and $Y_i(0)$ be the potential outcome of individual $i$ under control. We can define the finite sample average treatment effect as:
$\tau_{fs} = \frac{1}{N}\sum_{i=1}^{N}\left(Y_i(1) - Y_i(0)\right)$
So this is the treatment effect on the concrete population you observe. But you could imagine that the population you observe is actually a sample of a "super-population" --- that is, ou don't want to measure the average treatment effect on the particular population you happen to have, but on a more general very large population like "adults in general". 
In that case, your causal estimand is now:
$\tau_{sp} =  \frac{1}{N_{sp}}\sum_{i=1}^{N_{sp}}\left(Y_i(1) - Y_i(0)\right)$ 
Which you can obtain by taking the expectation of $\tau_{fs}$, if you consider the sample you have as coming from the super population.
