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Suppose that we have generated data from a Binomial distribution

$$ C_{1}, C_{2},..., C_{K} \sim Bin(N,p)$$

and our goal is to estimate both the parameters $N$ and $p$. I've see numerically (just plotting the likelihood function) that the model is not identifiable since I can find many combinations of $N$ and $p$ that can give rise to the data $C_{1}, C_{2},..., C_{K}.$ However, when I integrate out the parameter $p$ (assume that $p\sim Beta(1,1)$) the model seems to be identifiable for estimating $N,$ since the integrated likelihood function now has one distinct mode for $N$.

So, in the case where we integrate the $p$ we can reliably estimate the parameter $N$, is it as simple as that?

Fiodor1234
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  • See my answer at https://stats.stackexchange.com/questions/123367/estimating-parameters-for-a-binomial/123748#123748 Since this is a difficult problem (close to non-identifiable), see linked post for details) it is clear that including prior information on $p$, if you have it, will help a lot! – kjetil b halvorsen Aug 28 '22 at 20:08
  • @kjetilbhalvorsen From the literature I understand the following, the estimation of $N$ based on the likelihood function provided in this question is quite unstable, in the sense that we can find pairs of $(N,p)$ that give the same results. Now when integrating out the nuisance parameter $p$ we get more stable estimates for $N$, however, we introduce bias to the estimates because of the prior $p$ parameters $Beta(a,b).$ So, I guess that even though my integrated likelihoods are identifiable this is not supported by theory, it is just my cases. – Fiodor1234 Aug 30 '22 at 13:33
  • @kjetilbhalvorsen Lastly, if I add prior information to $N$ then we are going to have stable and estimable $N$. – Fiodor1234 Aug 30 '22 at 13:33
  • @kjetilbhalvorsen Side note, for small values of $p$ the integrated likelihood approach might not work well. And also basically everything is based on empirical results, not theoretical proofs – Fiodor1234 Aug 30 '22 at 13:34

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