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I have a hierarchical model that includes a normal distribution and a beta distribution.

For the normal distribution, it has two parameters: $\mu$ and $\tau^2$. However, I want to implement hierarchy such that the $\mu$ and $\tau^2$ parameters are sampled from group-level distributions. What are good distributions to use for each of these two parameters? I am assuming the $\mu$ parameter could be sampled from another normal distribution, and the $\tau^2$ parameter could be sampled from an inverse gamma distribution. Is this along the right lines?

For the beta distribution, it has two parameters: $\alpha$ and $\beta$. I want to also implement this hierarchically in the same manner, so I have the same question. What is a good distribution to use which would then sample values for $\alpha$ and $\beta$?

I am sorry if these are simple questions but I have been having a very difficult time finding answers online.

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