Conjugate prior for gamma distribution

**malaguena** · February 13th 2011, 02:23 PM

Conjugate prior for the gamma distribution (with parameters $\alpha,\beta$ ) is supposed to be:

$\frac{1}{Z}\frac{p^{\alpha -1}e^{-\beta q}}{{\gamma(\alpha)}^r \beta^{-\alpha s}}$

with the hyperparameters, p, q, r, s when both shape parameter and the scale parameter are known.

Here I have a gamma distributed variable x, with parameters $\alpha$ and $\beta$ . And I have a conjugate prior with hyperparemeters p,q,r and s. Can anybody help me solving the integration over the parameters $\alpha,\beta$ ?

$\int_{a,b} \frac{1}{\gamma(\alpha)\beta^\alpha} x^{\alpha-1} e^{-x/\beta} \frac{1}{Z}\frac{p^{\alpha -1}e^{-\beta q}}{{\gamma(\alpha)}^r \beta^{-\alpha s}} d{a,b}$

It has been a long time since I took a calculus class in the university, so I don't remember much about integration solving. Any assistance will be appreciated!

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