= Poisson credibility with a shape-three Pareto intensity
{c}
{title2=$\widehat m=(k+3)/(n+2)$}
Let a fixed policy intensity have <prior density> $3\theta^{-4}$ on $\theta>1$, and let $n$ annual <claim counts> be conditionally independent with <Poisson distribution> mean $\theta$. The <expected process variance> is $3/2$ and the <variance of hypothetical means> is $3/4$, so the <Bühlmann credibility factor> is $n/(n+2)$. A total count $k$ gives the displayed <credibility premium>. By contrast, the <Bayesian posterior> density is proportional to $\theta^{k-4}e^{-n\theta}$ on $\theta>1$. Its mean is the ratio $\int_1^\infty\theta^{k-3}e^{-n\theta}\,d\theta/\int_1^\infty\theta^{k-4}e^{-n\theta}\,d\theta$. For $n=2,k=5$, that <Bayesian credibility> estimate is $5/3$, whereas the affine estimate is $2$.
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