What's the relationship between statement "Z causes both X and Y" and "X and Y are independent given Z"? Suppose I have two statements:
Statement 1: Random variable Z is the common cause for random variable X and Y (Z causes both X and Y)
Statement 2: Random variable X and Y are (conditionally) independent given Z.
What's the relationship between Statement 1 and Statement 2? Does Statement 1 imply statement 2, or does statement 2 imply statement 1, or they have other relationships?
Example: Z could be the grade level(or age) for a primary school student, X could be his height, Y could be his math ability.
Thanks!
 A: Statement 1 is a scientific statement rather than a mathematical relationship. The idea of one random variable "causing" another doesn't have a strict mathematical definition, rather the role of the mathematician or statistician is to posit a mathematical model that captures the relationship in a way that is relevant to a particular scientific problem.
For example, a mathematician might suppose that X and Y both have functional relationships with Z but not with each other, and that would capture the scientific idea of "causation".
Statement 2 is a mathematical statement, and it is one sensible and useful way to translate the first statement into mathematical terms.
It is a very strong statement because it implies that Z accounts for all possible dependencies between X and Y.
There cannot be any other common causes of X and Y that are not mediated by Z or correlated with Z.
On the other hand, Statement 2 is more general than Statement 1 in the sense that X and Y could be conditionally independent given Z without being direct functions of Z.
