Modelling poisson data with selection bias.

truncated_poisson_data.stan

functions {
    real normalisation(real lam, int lb, int ub) {
        real result = 0;
        for (x in lb:ub) {
            result += (lam^x)/tgamma(x+1);
        }
        result *= exp(-lam);
        return result;
    }
}
data {
    int N;
    int L;
    int U;
    int<lower=L, upper=U> y[N];
}

parameters {
    real<lower=0> beta;
}

model {
    for (val in y) {
        target += poisson_lpmf(val | beta) - log(normalisation(beta, L, U));
    }
}