I have to solve a bayesian statistics problem like a follows
$y_i$ distr as $Bin(n_i, \theta_i)$ conditional on $\theta_i$
I observe several $n_i$ and $y_i$ and I’m interested in $\pi(\theta_i|y)$ for any $i$
Moreover I know that $logit(\theta_i)= \alpha_i$ and
$\alpha_i$ distr $N(\mu, \sigma^2)$
Moreover $\mu$ and $\sigma^2$ are not constant but r.vs. themselves (hierarchical model). The prior $\pi(\mu, \sigma^2)$ is proportional to $1/ \sigma^2$
It seem me that the main trick here is to discover the distribution of $\alpha_i$ given $y_i$ that probably have no usual form. So we have to simulate from it with MCMC procedure. So we have a chain for $\alpha_i$ and is easy to compute the related chain for $\theta_i$, so we have the posterior distribution of interest.
Is the correct procedure? What is the distribution/kernel for $\alpha_i$ given $y_i$ ?