# How to jointly model $N$ groups where data in each group is i.i.d. Normal and infer the posterior distribution?

I am given the following data of income scores of individuals from $$N$$ groups:

$$(\textbf{x}_1, \textbf{x}_2 \ldots \textbf{x}_N),$$

where $$\textbf{x}_j = (x_j^1, x_j^2 \ldots x_j^{N_j}),\quad j = 1, 2 \ldots N,$$ i.e, from the $$j$$-th group, we have income scores of $$N_j$$ individuals.

It is also given that $$x_j^i \overset{i.i.d.}{\sim} N(\mu_j, \sigma^2), \quad i = 1, 2\ldots N_j,$$ where $$\sigma^2$$ is known

Assuming prior on $$\mu_j$$ as $$\mu_j \sim N(a. b^2)$$ where $$a$$ and $$b$$ are known, I want to jointly model this data and infer the posterior of $$\mu_j$$. To do so, it is suggested that I may consider the income scores coming from a particular $$j$$-th group as a single observation (sample mean) $$\overline{\textbf{x}_j} \sim N(\mu_j, \frac{\sigma^2}{N_j})$$.

This suggestion is confusing me. Why is it justified to consider the sample mean of the test scores coming from a particular group rather than going about it the usual way? The usual way according to me is, for each $$mu_j$$, I consider $$N_j$$ i.i.d observations and write down the data likelihood and calculate the posterior.

Basically, it doesn't seem clear to me why the data of groups other than the $$j$$'th group would play any role in deciding the posterior of $$mu_j$$. So what exactly is being asked to be jointly modelled here?

Also, if using sample mean makes sense, then can I jointly go about inferring the posterior of $$\mu_j$$?

Any help will be much appreciated. I think if I can understand how we can go about jointly modelling this, then I will be able to write down all the necessary expressions.

$$x_{ij} \sim N(\mu_j, \sigma^2)$$ and $$\mu_j \sim N(\mu, \sigma_g^2)$$,
where we assume that $$\mu_j$$ share a common mean $$\mu$$. We then need specify priors for parameters.