= Gaussian-prior Poisson log-effect conditional
{title2=$\log\pi(\alpha\mid\cdots)=r\alpha-ce^\alpha-\alpha^2/(2s^2)+\mathrm{constant}$}
For $c>0$ and a centered <normal distribution> prior, this conditional <log-posterior> has second <derivative> $-ce^\alpha-s^{-2}<0$. It is strictly <concave> but does not belong to a <normal distribution> or <gamma distribution> family. A <Random-walk Metropolis algorithm>, or a locally scaled proposal around its unique mode with the full proposal correction, can update it. Gaussian conjugacy is unavailable because the <Poisson distribution> <likelihood> contains an exponential rate.
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