= Proper gamma approximation to a Poisson Jeffreys prior
{title2=$\operatorname{Gamma}(1/2,\varepsilon),\quad\varepsilon\downarrow0$}
A <Poisson distribution> with mean $E\lambda$ and positive known exposure has <Fisher information> $E/\lambda$, so its <Jeffreys prior> kernel is $\lambda^{-1/2}$. The proper shape-rate <gamma distribution> $\operatorname{Gamma}(1/2,\varepsilon)$ has kernel $\lambda^{-1/2}e^{-\varepsilon\lambda}$, approximating the improper kernel where $\varepsilon\lambda$ is small. Its posterior is $\operatorname{Gamma}(y+1/2,E+\varepsilon)$, converging to $\operatorname{Gamma}(y+1/2,E)$. There is no normalized Jeffreys prior to which the proper priors converge weakly; this is kernel approximation and posterior convergence.
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