Heading ##I am wondering whether these two properties are equivalent:
$X$ is conditionally independent of $Y$ given $Z$
$X$ is conditionally independent of $Y$ given $a^T Z$, $\forall a \in R^p$
Actually they are not eqivalent.
Suppose $X = Z_1+Z_2+\epsilon_1,Y = Z_1+Z_2+\epsilon_2, Z=(Z_1,Z_2)^T,\epsilon_1 \bot Z,\epsilon_2 \bot Z$. It's obvious that $X \bot Y |Z$. But if we take $a=(1,0)^T$ , $X \not \bot Y|a^TZ$. So maybe the first property is a lot weaker than the second property.