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Estimating the parameters from $k$ independent Bin$(n,p)$ random variables,when both parameters $n$ and $p$ are unknown, is relevant to a variety ofapplications. It is particularly difficult if $n$ is large and $p$ is small.Over the past decades, several articles have proposed Bayesian approaches toestimate $n$ in this setting, but asymptotic results could only be establishedrecently in \cite{Schneider}. There, posterior contraction for $n$ is proven inthe problematic parameter regime where $n\rightarrow\infty$ and $p\rightarrow0$at certain rates. In this article, we study numerically how far the theoreticalupper bound on $n$ can be relaxed in simulations without losing posteriorconsistency.

Posterior Consistency in the Binomial $(n,p)$ Model with Unknown $n$ and $p$: A Numerical Study