Derivation of posterior for Bayesian hierarchical models

Question

In Bayesian hierarchical models, the following posterior is used:

$$p(\theta,\phi|y)\propto p(y|\theta)p(\theta|\phi)p(\phi)$$

I'm trying to derive this myself but when I use Bayes' rule, I get the following.

$$p(\theta,\phi|y)=\frac{p(y|\theta,\phi)p(\theta|\phi)p(\phi)}{p(y)}$$

Are we asserting that $p(y|\theta,\phi)=p(y|\theta)$ during the derivation or is there something that I am missing?

I tried looking into this myself but I was only able to find a derivation on Wikipedia that goes like this:

$$p(\theta,\phi|y) =\frac{p(y|\theta,\phi)p(\theta,\phi)}{p(y)} =\frac{p(y|\theta)p(\theta|\phi)p(\phi)}{p(y)}$$

Here it seems that they claim $p(y|\theta,\phi)p(\theta,\phi)=p(y|\theta)p(\theta|\phi)p(\phi)$. Is this true or is something else going on?

Answer 1

If $\phi$ are the parents of $\theta$, then the data $y$ are conditionally independent of $\phi$ given $\theta$. So, $p(y|\theta,\phi)=p(y|\theta)$.

There seems to be an assumption somewhere that $\theta$ are the sole parents of $y$ and $\phi$ are the sole parents of $\theta$. This is a standard setup in a hierarchical model.