# Where information comes for Binomial Likelihood?

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.

• Does this stats.stackexchange.com/questions/123367/… answer your question? Sep 15, 2022 at 19:39
• @kjetilbhalvorsen I've already been through this answer but doesn't fully tackle my question Sep 15, 2022 at 20:03

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.