Where information comes for Binomial Likelihood?

Question

Suppose that we have replicates from a Binomial distribution, i.e.

$$n_{1},n_{2},...,n_{R}\sim Bin(N,p)$$

Then the likelihood can be written as $L(p,N|n_{1},n_{2},...,n_{R})=\prod_{r=1}^{R}\binom{N}{n_{r}}p^{n_{r}}(1-p)^{N-n_{r}}$. Suppose that our interest is for $N$, then we can realize that the $n_{1},n_{2},...,n_{R}$ contain information about the parameter $N$. This comes from how much they vary (variance) the values $n_{1},n_{2},...,n_{R}$ and also what is there actual values (mean). Now, we can rewrite the likelihood function as

$$L(p,N|n_{1},n_{2},...,n_{R})=\frac{\prod_{r=1}^{R}\binom{N}{n_{r}}}{\binom{RN}{\sum_{r=1}^{R}n_{r}}}Bin(\sum_{r=1}^{R}n_{r};RN,p)$$

The second term $Bin(\sum_{r=1}^{R}n_{r};RN,p)$ doesn't give any information for $N$ since we sample a single number $\sum_{r=1}^{R}n_{r}$ from the $RN$. If we have many repetes of such single number $\sum_{r=1}^{R}n_{r}$ we would be able to say something but that only would be for $RN$ and not $N$.

So, all the information for $N$ comes from the term $\frac{\prod_{r=1}^{R}\binom{N}{n_{r}}}{\binom{RN}{\sum_{r=1}^{R}n_{r}}}$. Which corresponds to a multivariate hypergeometric distribution. The numerator is the number of different sampling combinations where one would have exactly $n_{r}$ from $N$ for $r=1,2,...,R$. The denominator is the total number of different combinations of individuals one could have in selecting $\sum_{r=1}^{R}n_{r}$ individuals from $RN$. Thus the equation is just the proportion of different possible scenarios, each of which has the same probability, that would give us $n_{r}$ from $N$.

All, this started by reading the paper On the Reliability of N-Mixture Models for Count Data and wanted to know if the reasoning that I stated holds.

Answer 1

I don't think your conclusion that the second factor in the likelihood (the binomial distribution) does not contain information on $N$ is true. First, we know for sure that $N\ge \max_r n_r$. Second, if you have any sort of prior information on $p$ that will help a lot, so this is a case where Bayes methods might be very helpful, if you have real prior information. For the case with a completely known $p$: Analytical expression of the log-likelihood of the Binomial model with unknown $n$ and known $y$ and $p$ and its conjugate prior

Without prior information on $p$, profile likelihood is helpful here. If we tentatively assume that $N$ is known, then the maximum likelihood estimator of $p$ is $$ \hat{p}_N=\frac{\sum_r n_r}{NR} $$ Substituting this for $p$ in the likelihood function gives the profile likelihood function for $N$. Taking the log gives the profile log likelihood: $$ \ell_P(N)= R\log\Gamma(N+1) - \sum_r \log\Gamma(n_r+1) - \sum_r\log\Gamma(N-n_r+1) + \sum_r n_r \log\left( \hat{p}_N \right) + (NR-\sum_r n_r)\log\left( 1-\hat{p}_N \right) $$ where we have used the Gamma function.

For some example data you can plot this, and also plot contribution from terms separately to see which terms contribute the most.

And, please tell us what is your application ...