This is almost certainly a fatal misunderstanding of mine / knowledge gap but I am confused as to how to interpret the population parameters of a Bayesian Hierarchical model.

This is incredibly artificial but to demonstrate the point let's say we wanted to fit the following model + priors:

$$
\begin{align}
X_{ij} &\sim N(\mu_j, \sigma) \\
\mu_j &\sim \text{LogNormal}(\mu, \tau) \\
\mu &\sim \text{N}(2, 1) \\
\sigma &\sim \text{LogNormal}(0.5, 0.5) \\
\tau &\sim \text{LogNormal}(0.5, 0.5) \\
\end{align}
$$

Now lets say a question we want to answer is "what is the population-mean of the $\mu_j$'s ?".

Naively I would assume that because they follow a Log-Normal distribution that the mean of the distribution is then $exp(\mu + \tau^2/2)$ ([wiki](https://en.wikipedia.org/wiki/Log-normal_distribution)) where we just sub in the posterior samples of $\mu$ and $\tau$ to get an uncertainty distribution.  

But this doesn't feel right to me... 

For example the posterior of $\tau$ is not necesarily a Log-Normal distribution anymore (that was just our prior), so why would the posterior population distribution of $\mu_j$ still be a Log-Normal distribution ? But if its not a Log-Normal distribution what do the values of $\mu$ and $\tau$ even mean anymore then ? 

Any guidance as to where my mental model has gone wrong would be greatly appreciated.