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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.

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I think we should not model only the sample mean. To me, I will employ a hierarchical model, as somehow you also mentioned, but with some modification. Specifically,

$x_{ij} \sim N(\mu_j, \sigma^2)$ 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.

P/S: See also one-way ANOVA.

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