What is the posterior over $\alpha$ in this scheme?

Consider the following probabilistic scheme:

$\alpha \sim \mbox{Gamma}(a,b)$

$\rho \sim \mbox{Beta}(\alpha, \beta)$

$z_i \sim \mbox{Bernoully}(\rho) \quad$ for $i=1..N$

The variables $z_i, a, b, \beta$ are all known. What is the posterior distribution over $\alpha$?

That is, how is $\alpha | \mathbf{z}, a, b, \beta$ distributed?

Yemon Choi
8,608●1●20●45

asked Feb 6 2010 at 20:06

	This may be a naive question or misunderstanding: but what goes wrong with just attempting to do the calculation by force? – Yemon Choi Feb 6 2010 at 23:10
	Well, I would hope to get some conjugacy to the Gamma prior. I was hoping some one here can see how to read that from the formula you get for the posterior. – Jonathan Feb 8 2010 at 8:40