Bayesian Statistics: Marginal Posterior

Question

In a hierarchical model, the prior $\pi(\theta\mid\xi)$ for $\theta$ depends on hyperparameters $\xi$.

In my lecture notes, the following is now given:

$$ \pi(\xi \mid x) = { \pi(\theta,\xi\mid x)\over \pi(\theta\mid x,\xi)}.$$

I don't see how to derive this formula.

Answer 1

$$ \frac{\pi(\theta,\xi\mid x)}{\pi(\theta\mid x,\xi)}=\frac{\pi(\theta,\xi,x)}{\pi(x)}\cdot\frac{\pi(x,\xi)}{\pi(\theta,\xi,x)}=\frac{\pi(x,\xi)}{\pi(x)}=\pi(\xi\mid x)\;. $$