Latent Dirichlet Allocation with Binomial response

The standard LDA example given in the STAN manual assumes purely a bernoulli presence/absence response. I’m trying to edit this example for a binomial response where I have a proportion given by the number of species | number of total species hits.

I have two more variables added to the data for this response: the numerator (specieshits[N]) and the denominator (tothits[M]). I cannot figure out how to modify gamma in order to incorporate this type of response instead of a purely binary one.

data {
  int<lower=2> K;               // num communities
  int<lower=2> V;               // num species
  int<lower=1> M;               // num sites
  int<lower=1> N;               // total species instances
  int<lower=1,upper=V> w[N];    // species n
  int<lower=1,upper=M> doc[N];  // site ID for species n
  int specieshits[N];           //number of species hits at site M for species n
  int<lower=1> tothits[M];      // total number of all species hits at site M
  vector<lower=0>[K] alpha;     // community prior
  vector<lower=0>[V] beta;      // species prior
parameters {
  simplex[K] theta[M];   // community dist for site m
  simplex[V] phi[K];     // species dist for community k
model {
  for (m in 1:M)
    theta[m] ~ dirichlet(alpha);  // prior, proportion of each community at each site
  for (k in 1:K)
    phi[k] ~ dirichlet(beta);     // prior, proportion of each species within each community
  for (n in 1:N) {
    real gamma[K];
    for (k in 1:K)
      gamma[k] = log(theta[doc[n], k]) + log(phi[k, w[n]]);
    target += log_sum_exp(gamma);  // likelihood;