Likelihood of Gamma Distribution

I'm studying bayesian stats on my own and came across the following problem on Coursera. Can anyone help me understand how to work through this?

$$x_{i}\stackrel {iid}{\sim} Beta(\alpha,\beta), i=1....n$$

$$\alpha \sim Gamma(a,b)$$

$$\beta \sim Gamma(r,s)$$

Where $$\alpha$$ and $$\beta$$ are independent a priori. Give the full conditional density for $$\alpha$$ up to proportionality $$p(\alpha|\beta,x)$$?

Edited: I'm getting stuck with the setup for this.

I have assumed that $$p(\alpha|\beta,x)$$? = $$\frac{p(\alpha\cap\beta\cap x)}{p(\beta\cap x)}$$

I'm not sure how to proceed from here. Any help will be highly appreciated.

Thanks

• Thanks Xi'an. I'm probably going down the wrong path with this. I have edited the question to explain where I'm possibly getting confused. – Electromagnet Jan 4 at 7:26
• The notation $p(\alpha\cap\beta\cap x)$ is only appropriate for events and sets, not for random variables:$$p(\alpha|\beta,x)=p(\alpha,\beta,x)\big/p(\beta,x)$$ – Xi'an Jan 4 at 7:32
• Write down$$p(\alpha,\beta,x)=p(\alpha)p(\beta)\prod_i p(x_i|\alpha,\beta)$$and turn it into a density on $\alpha$. – Xi'an Jan 4 at 7:42