I struggle to understand a basic equation about marginalization and conditioning and would appreciate your help. $x=(x_1,x_2)^T$ here
Give $x\mid\xi \sim N_d(\mu,\Sigma \xi \upsilon) $ and $ \xi \sim IG( \frac{\upsilon}{2},\frac{1}{2})$ (IG = Inverted Gamma), why is it that this equation holds (generally):
$$ f_x(x)=\int f_{x_1,x_2\mid\xi}(x_1,x_2\mid\xi)f_\xi(\xi) d \xi=\int f_{x_2\mid\xi}(x_2\mid\xi,x_1)f_{x_1\mid\xi}(x_1\mid\xi)f_\xi(\xi)\,d \xi $$
I am not understanding how we get
$$ f_{x_1,x_2\mid\xi}(x_1,x_2\mid\xi)=f_{x_2|\xi}(x_2\mid\xi,x_1)f_{x_1\mid\xi}(x_1\mid\xi) $$