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For $i=1, \ldots, K$ and $j=1, \ldots,n$, assume the following model. \begin{align} X_{ij} \mid \mu_i & \sim N(\mu_i, \sigma^2) \nonumber \\ P(\mu_i, \sigma^2) & \propto 1/\sigma^2 \end{align} We assume everything is independent. Then the posterior is as below \begin{align*} \dfrac{\mu-\bar X_i}{s/ \sqrt{n}} \sim T_{n-1} \end{align*} With this noninformative prior, is it possible to derive the marginal distribution of $\bar X_i$?

Thank you very much!

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The short answer is that no, with improper prior distributions, the marginal distribution of the data will not be proper.

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