I’m looking for help wrapping my head around how to model this particular problem, here’s a statement of it.

Suppose I have J groups, indexed by j. Group j has size n_j. Each group member votes either yes or no on some measure. The proportion of “Yes” votes, y_j, can be modeled as a binomial distribution with unknown parameter \theta_j. We can write:

We also assume that n_j and \theta_j are randomly drawn from distributions with hyperparameters. Just for simplicity, we’ll assume:

So far so good, but the catch is that we don’t directly observe either y_j or n_j, instead we observe the following quantities.

Here s_j is the quantity of Yes votes - No votes, censored for all values below 0. \pi_j is the percentage of total “Yes” votes.

This means that if s_j is positive, I can solve the system of equations with \pi_j to recover (y_j, n_j), but if it’s 0, then I don’t have enough information to do so. I’d like to estimate the hyperparameters (\alpha, \beta, \mu, \sigma) as well as estimate (y_j, n_j) for the cases where s_j = 0. Without the censoring, I have a good idea where to go, but those censored observations are throwing me for a loop.

Appreciate any help!