Trying to understand if a model is an empirical Bayes or not I am trying to understand if the model published in section 4.1.1 here is or is not an Empirical Bayes model (which the author claims it is). Or, maybe, if it is a valid one or not. The model looks as follows:
$$
X_i|\theta_i\sim N(\theta_i,\sigma),i\in\{1,2,\ldots,n\},\text{ and }\sigma \text{ known}\\
~\theta_i|\eta_i\sim N(\eta,\tau^2),i\in\{1,2,\ldots,n\},\text{ and }\tau^2 \text{ known}\\
\eta_i|\bar{X}\sim N\left(\bar{X},\frac{\sigma+\tau^2}{n}\right)
$$
I thought that using the data itself is not allowed when setting the prior. Is this sometimes okay, and others not?
I was also wondering if changing the model such that if we know that some $\theta_i=\theta_j ~ (i\neq j)$ and swapping $\bar{X}$ for $\bar{X}_j$ for the $\eta_i$s where we know them to be the same makes any difference?
Something like this:
$$
X_{ij}|\theta_{j}\sim N(\theta_j,\sigma),i\in\{1,2,\ldots,n\},j\in\{1,2,\ldots,m\},\text{ and }\sigma \text{ known}\\
~\theta_{j}|\eta_{j}\sim N(\eta_j,\tau^2),i\in\{1,2,\ldots,n\},j\in\{1,2,\ldots,m\},\text{ and }\tau^2 \text{ known}\\
\eta_{j}|\bar{X}_j\sim N\left(\bar{X}_j,\frac{\sigma+\tau^2}{n}\right)
$$
For simplicity assuming $\sigma,\tau$ to be the same for all data. $i$ are repeated measurements for the $j$:th mean.

The empirical Bayesian approach estimates the parameters from the data. In the case of the model described, it is an empirical Bayesian model because $\eta$ is estimated from the data. It is also discussed in the passage below the model definition

The choice of $\alpha(n) = n$ corresponds to an empirical Bayes model
that attempts to account for the uncertainty in estimating the “fixed”
parameter $\eta$ with  $\bar X$. This is justified by noting that
(4.3) with $\alpha(n) = n$ arises naturally by swapping $\eta$ and $\bar X$
in the sampling distribution of $\bar X$. However, in the interest of
remaining as general as possible, we permit $\alpha(n)$ to be any
positive function of $n$.

To learn more when it is ok, check the How is empirical Bayes valid? thread. Your re-parametrization just defines another model, that still has parameters calculated from the data, so is an empirical Bayesian model.
