conditional independence

Conditional independence

Two random variables $X$ and $Y$ are conditionally independent given a third random variable $Z$:

$$\begin{array}{rl}X\perp Y\mid Z⟺p(X,Y\mid Z)& =p(X|Y,Z)p(Y|Z)\\ & =p(X\mid Z)p(Y\mid Z)\end{array}$$

This means that once we know $Z$, knowing $Y$ doesn’t give us any more information about $X$ (and vice versa).

Independence != conditional independence

They don’t imply eachother:

Independent but not conditionally independent: …

Conditionally independent but not independent: …

Common cause vs Common effect

$A\leftarrow C\to B$
$A\to C\leftarrow B$
https://chatgpt.com/c/68c1cfdb-2234-8332-804e-3e1731d9f686