Base: Correlation does not mean casualty.
- If X and Y are statistically dependent, X does not necessarily cause Y (or Y cause X). 相关性不代表有因果性
- If X causes Y, then X & Y are very likely to be statistically dependent (but not always, there is extreme condition). 但是因果性代表相关性
Study 1. V Structure:
- Chain
$$ X\rightarrow Y\rightarrow Z $$
Z and X are likely dependent. However, Z and X are independent, conditional on Y.
$$ P(Z=z|X=x,Y=c)=P(Z=z|Y=c) $$i.e.
i.e.
\(f_x: X=u_x\)
\(f_y: Y=84-X+u_Y := c\)
\(f_z: Z=100\underbrace{Y}_{c}+u_z\)
Now, Z and X are independent.
Therefore, we know, in the Chain:
$$ X\equiv Z$$
$$ X\bot Z|Y $$
- Folk
$$ Y\leftarrow X\rightarrow Z $$
Y and Z are likely dependent. However, Y and Z are independent conditional on X.
$$P(Z=z|Y=y, X=c)=P(Z=z|X=c)$$
While conditioning on intermediate node X, then Z and Y are independent.
$$Y\bot Z|X$$
- Collider
$$ X\rightarrow Z\leftarrow Y $$
X and Y are independent. However, X and Y are dependent conditional on Z.
$$ P(X=x|Y=y, Z=c)\neq P(X=x|Z=c) $$
i.e.
If we know \(Z=X+Y+u_Z:=c\), then \( X=c-Y-u_Z\), and thus X and Y become dependent conditional on \(Z=c\). Otherwise, \(X=u_X\) and \(Y=u_Y\).
Once, conditioning on \(Z\), the way gets connected. Otherwise (unconditional), we get independent.
P.S. Descendent of Z:
$$ X (or\ Y)\rightarrow Z\rightarrow W $$
Similarly, we get in the Collider:
$$ X\bot Y $$
$$ X\equiv Y|Z $$
$$ X\equiv Y |W $$
- See notes for further studies.
Reference
Pearl, J., Glymour, M. and Jewell, N.P., 2016. Causal inference in statistics: A primer. John Wiley & Sons.