/*
  Beta-binomial model for litters data.
  Approximate uniform shrinkage prior
  with uniform prior for mu

  We need to truncate the beta distribution of p[i] to avoid a
  singularity at 1 when b[i] < 1.  This slows down the calculation
  considerably as it requires calculation of the incomplete beta
  function.

*/
model {
  for (i in 1:G)
  {
     for (j in 1:N)
     {
        r[i,j] ~ dbin(p[i,j], n[i,j]);
        p[i,j] ~ dbeta(a[i], b[i]) T(,0.9999); 
     }

     a[i] <- nbar[i] * mu[i] * phi[i];
     b[i] <- nbar[i] * (1 - mu[i]) * phi[i];

     mu[i] ~ dunif(0, 1);
     phi[i] <- nbar[i] * theta[i] / (1 - theta[i]);

     theta[i] ~ dunif(0, 1);
     nbar[i] <- mean(n[i,]);
  }
}