I am trying to build the following model:
For 1\leq i \leq k
p_i \sim Beta(\alpha_i, \beta_i)
Where the \{ p_i\} are correlated through a gaussian copula with a given covariance \Sigma
B_i \sim Bernoulli(p_i)
Q=\prod_i p_i
I have observations of \{B_i\} with some missing data.
How do i sample from the posterior distribution of Q?
Also, how do i do it with a custom distribution instead of beta?