Fitting mixture of binomials

You can build hierarchical mixtures, but it’s not clear to me that it works here because you have continuous parameters generating discrete observations. You might be able to model a binomial mixtures of Bernoulli likelihoods, but that would require you to model every single observation (rather than N and p you would just have p and every observations would be a single “flip”). Maybe someone more clever than I would be able to work out a better solution.