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Per the Wikipedia on conjugate priors link, the conjugate prior for a Gamma of unknown $\alpha$ and $\beta$ is proportional to an expression involving both $\alpha$ and $\beta$ as well as a $\Gamma$ function and 4 parameters.

Previously, I have had experience using conjugate priors for multi-parameter distributions (namely Gaussian) by employing the technique described in this paper. With this method, I am able to set 2 different conjugate priors, one over $\mu$ and one over $\sigma^2$. The end result is that each parameter can be sampled from it's own posterior so I can make inferences about each separately.

Given the irregular form for the Gamma conjugate prior, and the fact that is described as proportional (so I suppose this isn't a proper posterior distribution?), is it possible to compare values of both $\alpha$ and $\beta$ across two different posteriors (using same prior but different data) in the same way I have done for the Gaussian, or even any simple one parameter distribution with conjugate priors that you can find in the same Wikipedia article? How would I actually sample from this posterior?

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If you take a Gamma Ga$(\alpha,\beta)$ likelihood $$\dfrac{\alpha^{n\beta}}{\Gamma(\beta)^n}\exp\left\{\beta\sum_{i=1}^n\log x_i -\alpha \sum_{i=1}^n x_i\right\}$$ the distribution with density $$\pi(\alpha,\beta) \propto \dfrac{\alpha^{\lambda\beta}}{\Gamma(\beta)^\lambda}\exp\left\{\beta\xi -\alpha \mu\right\}$$ is conjugate, if non-standard.

Furthermore, the conditional of $\alpha$ given $\beta$ is a Gamma, hence can be integrated out. A posteriori and a priori. The symmetric property for $\beta$ given $\alpha$ does not hold though.

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    $\begingroup$ This answers the question in the title but I think the real substance of the query is in the third paragraph, which this answer does not presently address. $\endgroup$ – Sycorax Feb 18 '17 at 17:57
  • $\begingroup$ @Sycorax you're correct. Would really love an answer to the third paragraph. $\endgroup$ – Frank P. Feb 19 '17 at 2:55
  • $\begingroup$ @FrankP.: your question is unclear as you do not describe in the text of the question what you have done for the Gaussian and what you want to do. I also do not understand in which sense you want to compare posteriors based on different datasets. Normalisation is not the issue. $\endgroup$ – Xi'an Feb 19 '17 at 3:04
  • $\begingroup$ @Xi'an in the case of the Gaussian distribution (and any distribution listed in the Wikipedia page on Conjugate Priors) you can fit posterior distributions on all of the parameters of the underlying model (2 in the case for Gaussian, but usually just 1 for most distributions with conjugate priors). After that I can make inferences about the parameters separately. For example, if I have 2 sets of data with the same priors, dataset 1 might imply a larger $\mu$ based on the posterior but dataset 2 might have a larger $\sigma^2$. I'm looking for something similar for the Gamma. $\endgroup$ – Frank P. Feb 19 '17 at 22:42

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