Using the above DAG I need to simplify the following conditional probabilities:
$$i) \quad p(x_4|x_1,x_2)$$
For this one I guess I can just remove the conditioning on $x_1$ (using the DAG) and the most simplified conditional probability would be $p(x_4|x_2)$
$$ii) \quad p(x_4,x_5|x_1,x_2)$$
Again removing the condition on $x_1$ we get $p(x_4,x_5|x_2)$ which can be further simplified as:
$$ p(x_4,x_5|x_2) = \frac{p(x_2,x_4,x_5)}{p(x_2} = \frac{p(x_2)p(x_4|x_2)p(x_5|x_4)}{p(x_2)} $$
Here $p(x_2)$ cancel and we are left with:
$$p(x_4,x_5|x_2) = p(x_4|x_2)p(x_5|x_4)$$
Essentially my question is can we ignore the conditioning on an ancestor that is above another ancestor that is being conditioned on?