# Posterior (conjugate) prior of two parameter Gamma likelihood

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., $$p(\beta\mid X,\alpha)$$?

• 1. I do not think so - 2. Not likely. - 3. $\beta$ given $\alpha$ is from a Gamma, a priori and a posteriori. Sep 26, 2019 at 18:03