When does independence imply conditional independence, and vice versa? Given a subalgebra $\mathcal G$ of the probability space, if two random variables $X$ and $Y$ are independent, when will $E(X\mid \mathcal G)$ and $E(Y\mid \mathcal G)$ are also independent?
When will the converse be true?
Given a random variable $X$, if two subalgebra $\mathcal G$ and $\mathcal F$ of the probability space are independent, when will $E(X\mid \mathcal G)$ and $E(X\mid \mathcal F)$ are also independent?
When will the converse be true?
Thanks and regards!

This is only a partial answer to the "vice-versa" part.  Say a number $p\in(0,1)$ is chosen randomly from some nondegenerate distribution, and $X,Y$ are i.i.d. and
$\Pr(X=1\mid p)=p$ and $\Pr(X=0\mid p)=1-p$.  The $\Pr(X=1\mid Y=1)>\Pr(X=1)$, since the observation that $Y=1$ makes it more probable that $p$ is big.  So that's one sort of circumstance in which conditional independence holds and marginal independence does not hold.
