= Discrete uniform endpoint posterior
{title2=$p(N\mid y)\propto p(N)N^{-1}\mathbf1_{N\geq y}$}
For one uniformly sampled integer $Y\in\{1,\ldots,N\}$, the <likelihood function> for the unknown endpoint is $N^{-1}\mathbf1_{N\geq y}$. A flat <improper prior> on positive integers gives an improper <posterior distribution>, since the harmonic series diverges. A reciprocal prior instead gives a proper posterior proportional to $N^{-2}$ for $N\geq y$, with approximate survival probability $y/n$ and finite <quantiles>, but infinite <posterior mean>. A uniform prior cut off at $M$ gives posterior mean $(M-y+1)/(H_M-H_{y-1})\sim M/\log M$, demonstrating sensitivity to that arbitrary cutoff.
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