= Continuous Bernoulli priors and certainty
{title2=$P(X_{n+1}=1\mid X_1=\cdots=X_n=1)=\frac{n+1}{n+2}$}
With a uniform <prior distribution> on a Bernoulli chance $\theta$, $n$ successes yield <posterior density> $(n+1)\theta^n$. Integration gives next-trial success <probability> $(n+1)/(n+2)$, but the singleton $\theta=1$ retains <probability> zero. More generally, ordinary conditioning with a finite integrable <likelihood> and positive finite <normalization constant> preserves absolute continuity, so a continuous prior does not acquire a new atom at certainty from a finite sample.
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