= Posterior mean from adjacent mixed Poisson probabilities
{title2=$\mathbb E[\Theta\mid S=s]=\frac{(s+1)p_n(s+1)}{np_n(s)}$}
= Poisson mixture posterior mean identity
{c}
{synonym}
For a <mixed Poisson distribution> with known exposure $n>0$, multiply the unnormalized <posterior density> $\theta^s e^{-n\theta}\pi(\theta)$ by $\theta$ and compare its integral with the next <probability> mass. The factors $n^s/s!$ give the displayed identity whenever $p_n(s)>0$. Both mass <probabilities> have the same exposure $n$. For a future <conditionally independent> unit-exposure <Poisson distribution> count, this is also its <posterior predictive> <expected value>. It holds for arbitrary proper mixing distributions, with integrals taken against the <prior> measure when there is no density.
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