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.
Best Answer
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
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.