Suppose $X$ and $Y$ are independent random variables with prior distributions $p_X(x)$ and $p_Y(y)$. Suppose now that we have data $D$ that are characterized by $X,Y$ and that after observing the data, we have a posterior distribution:
$$ p(X,Y|D) $$
Why is it that two independent priors do not result in a posterior distribution that is also independent, or that $p(X,Y|D) \neq p(X|D)p(Y|D)$?
Is there a further implication here in terms of real-life applications? Thanks.