Implications and interpretation of conditional independence assumption I am currently studying Treatment Effect Analysis and I am reading about the Conditional Independence Assumption (CIA):
\begin{align}
(Y_1,Y_0)\perp D|X
\end{align}
So the outcomes are independent of the treatment, conditional on X.
First Question: Can someone explain this better? So give a real interpretation of this and not just the mathematical formulation verbally?
The CIA implies mean independence.
Second Question: What does this implication mean? So what is the connection about the CIA and the measurement of the treatment effect?
I am not used to the topic and tried to read typical literature, but they are not really giving an intutively interpretation. 
 A: Regarding the first part of your question. I believe it is more correct to say that $D$ is an indicator for assignment of treatment. Thus the assumption is that the choice, whether an individual gets treated or not, is not correlated to possible outcomes. 
The problem is the possible selection into treatment. It may be that treatment is assigned (or there is self-selection into) to those who are going to benefit most from it. For instance, suppose there is some training that improves academic achievement and you want to measure its impact. However, students are not assigned randomly for this course, but it is chosen mostly by those who have excellent computer literacy. In this case, if you estimate treatment effect of this particular training, you compare the outcomes of participants (mostly computer literate) to non-participants (mostly not computer literate). Thus the result would be biased as the outcome is partly dependent on selection of individuals to be treated. Selection bias should be evident if assignment to treatment correlates with outcome. 
If there is such non-random assignment to treatment and you know that the assignment depends only on characteristic $X$ (in this case computer literacy), you make an assumption that after controlling for $X$ both the treated and non-treated groups are equivalent in their remaining characteristics, except that some of them got treated and others not. So the difference between outcomes of the treated and non-treated can be attributed only to the fact of being treated, not that the individuals in groups were different from the beginning. Conditional on $X$ you assume that assignment to treatment is random, so it cannot correlate with possible outcomes. 
A: First Question: Can someone explain this better? So give a real interpretation of this and not just the mathematical formulation verbally?
It means that if we control for X, the treatment assignment is independent of the potential outcomes. For example, for the study of class size on learning outcome, STAR program randomly assign students in particular schools into small size classrooms or middle size classrooms. In this case, if we control for X, which is belonging to schools of experiment and in the grade of experiments, then the assignment of whether or not someone is in the small or middle size classroom has nothing to do with the future outcome of each class. The future outcome may be confusing. Let's suppose if it is not independent, then what can happen is that parents who care about their children may push them to study harder and also pulling strings to get them into smaller classes. Reverse for the kids with careless parents. So Y(1)-Y(0) is not just the difference due to class assignment, but also the effect of careful parents. So when CIA is satisfied, if you use regression to calculate, the coefficient of W would be the causal effect of interest. (I guess I also answered question II oops)
