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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 ...

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This can be a challenging problem and one has to think quite carefully about it. There's multiple frameworks for how to elicit expert judgments such as SHELF or Delphi. Asking non-statistical experts (and for that matter statisticians) about some quantity of interest in a way that is not a natural way of thinking for them is not a good idea. And, people ...

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The Gamma pdf is $f(\lambda\mid \tilde\alpha, \tilde\beta) \propto \lambda^{\tilde\alpha -1}\exp(-\tilde\beta \lambda)$. So if you plug the quantities $\tilde\alpha = \alpha + n\bar{x}$ and $\tilde\beta = \beta + n$ you do obtain that  p(\lambda\mid x) \propto L(\lambda)p(\lambda) \propto \lambda^{\alpha + n\bar{x} -1}\exp\left(-\lambda(\beta + n)\right) = ...

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I think what I was looking for is Eq. 234 in this document: I believe that this marginal likelihood is also often called the "model evidence" or the "prior predictive density". For example, this document uses the term "prior predictive distribution" to refer to the 1-D version of Eq. 234: https://www.statlect.com/fundamentals-...

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