How can I write the unnormalized posterior
$ f(p_1, p_2 | Y) = (z_1-1)*log(p_1) + (n_1-z_1-1)*log(1-p_1) + (z_2-1)*log(p_2) + (n_2-z_2-1)*log(1-p_2) $
in terms of the log odds-ratio $\alpha$ and the log odds-product $\eta$?
$\alpha = log \left(\frac{p_2/(1-p_2)}{p_1/(1-p_1}\right) $ and $\eta = log \left(\frac{p_2}{1-p_2} \times \frac{p_1}{1-p_1}\right)$?
where z_1, n_1, z_2, and n_2 are constants.