I have a model to achieve Bayesian estimates the population size $N$ and probability of detection $\theta$ in a binomial distribution solely based on the observed number of observed objects $y$:
$$
p(N,\theta|y)\propto \frac{ \text{Bin}(y|N,\theta)}{N}
$$
for
$
\left\{N|N\in\mathbb{Z}\land N\ge \max(y)\right\}\times(0,1)
$. For simplicity, we assume that $N$ is fixed at the same, unknown value for each $y_i$.
In this example, $y=53,57,66,67,73$.
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I needed more space to clarify a point from Juho's answer. If I understand correctly, we can integrate $\theta$ out of the posterior to obtain the beta-binomial distribution:
$$
p(y|N,\alpha,\beta)={N\choose y} \frac{\text{Beta}(y+\alpha, N-y+\beta)}{\text{Beta}(\alpha,\beta)}
$$
In our case, $\alpha=1$ and $\beta=1$ because we have a uniform prior on $\theta$. I believe that the posterior should then be $p(N|y)\propto N^{-1}\prod_{i=1}^K p(y_i|N,\alpha=1,\beta=1)$ where $K=\#(y)$ because $p(N)=N^{-1}$. But this appears to wildly diverge from Juho's answer. Where have I gone wrong?
