Question

consider the dag in figure 17.6 with a collider, prove that x z and that x and z are depedent given y

Answer 1

In order to prove that X and Z are dependent given Y, we need to show that P(X|Y) ≠ P(X) and P(Z|Y) ≠ P(Z). If we have a collider in the directed acyclic graph (DAG), it means that there is a direct causal relationship between X and Z through the variable Y. Therefore, if we can prove the presence of a collider in the DAG, we can conclude that X and Z are dependent given Y.

Step-by-step explanation:

In order to prove that X and Z are dependent given Y, we need to show that P(X|Y) ≠ P(X) and P(Z|Y) ≠ P(Z). If X and Z are independent given Y, then P(X, Z|Y) = P(X|Y)P(Z|Y). However, if X and Z are dependent given Y, then P(X, Z|Y) ≠ P(X|Y)P(Z|Y).

If we have a collider in the directed acyclic graph (DAG), it means that there is a direct causal relationship between X and Z through the variable Y. This implies that X and Z are not independent given Y.

Therefore, if we can prove the presence of a collider in the DAG, we can conclude that X and Z are dependent given Y.