Binomial function and population model providing "real" population size

Modeling
4

Just to have breadcrumbs, I believe this sprang out from the discussion at Using beta binomial with size of the population provided as a "real" (where there are some links to papers proposing something related, which I unfortunately don’t understand).

One of the links follows through to Continuous binomial, negative binomial and Poisson where Mike warns that a proposed extension of the Binomial distribution is impractical.

I would however definitely try to reimplement the binomial distribution with real N directly, it might break badly, but it is IMHO worth trying, e.g.: (a sketch, didn’t even test if it compiles, could also be vectorized to have arrays as input instead)

functions {
real binomial_real_dangerous_lpmf(int k, real n, real p) {
  return  binomial_coefficient_log(n, k) + k * log(p) + (n - k) * log1m(p);
}
}

If that let’s you reliably recover the parameters from the simulation (and even better, if you can use SBC as a stronger check), I wouldn’t be too afraid to actually use it. For large-ish n, the difference between binomial_real_dangerous_lpmf(k, n, p) and binomial_lpmf(k, floor(n), p) should be pretty small.

A more difficult challenge would be to somehow ensure that ObsA[year] < pop[year,2] for all Mjuv values to make the likelihood well defined.

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