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This question is related to a previous question on this site. Assume some data is generated from Gamma distribution $p(x\mid\alpha,\beta) \sim \operatorname{Gamma} (\alpha,\beta)$, and both parameters $\alpha$ and $\beta$ are unknown. Wikepedia presents that the conjugate prior of $\alpha$ and $\beta$ (link) when these two parameters are known is $p(\alpha,\beta)\propto\frac{p^{\alpha - 1} \exp(\beta q)}{\Gamma(\alpha)^r \beta^{-\alpha s}}$ where $p, q, r, s$ are hyperparameters.

My questions are:

  1. Is there any known formula to compute the partition function of $p(\alpha,\beta)$?

  2. How to interpret the meaning of the four hyperparameters $p,q,r,s$?

  3. Is it possible to derive the posterior distribution for a single parameter, e.g., $p(\beta\mid X,\alpha)$?

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  • $\begingroup$ 1. I do not think so - 2. Not likely. - 3. $\beta$ given $\alpha$ is from a Gamma, a priori and a posteriori. $\endgroup$ – Xi'an Sep 26 '19 at 18:03

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