Gaussian-prior Poisson log-effect conditional ($\log \pi (\alpha \mid \cdots )=r\alpha -c{e}^{\alpha }-{\alpha }^2/(2s^2)+\mathrm{constant}$)

For $c>0$ and a centered normal distribution prior, this conditional log-posterior has second derivative $-c{e}^{\alpha }-s^{-2}<0$. 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.