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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

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  • $\begingroup$ 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. $\endgroup$ – Electromagnet Jan 4 at 7:26
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    $\begingroup$ The notation $p(\alpha\cap\beta\cap x)$ is only appropriate for events and sets, not for random variables:$$p(\alpha|\beta,x)=p(\alpha,\beta,x)\big/p(\beta,x)$$ $\endgroup$ – Xi'an Jan 4 at 7:32
  • $\begingroup$ Write down$$p(\alpha,\beta,x)=p(\alpha)p(\beta)\prod_i p(x_i|\alpha,\beta)$$and turn it into a density on $\alpha$. $\endgroup$ – Xi'an Jan 4 at 7:42

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