In note 1, we introduce the identification-estimation flowchart. And in note 2, we introduce the corresponding concepts in graphs.
Now we will introduce causal models and data into prior identification-estimation flowchart:
The $do-$operator
Firstly we differ the conditioning and intervening:
The main effect of introducing causal models is finding all the confounders $X$, by which we can have unconfounderness assumption and do identification:
Modularity assumption
Nextly, we introduce the modularity assumption:
More formaly:
which means that
Take a simple example of identifying $P(y \mid do(t))$:
And then we can find the key difference between causation and association in formula:
which is the sample probability over the confounderness $X$.