Log-gamma prior for a Poisson log-intercept ($p(\mu )=b^a\exp (a\mu -b{e}^{\mu })/\Gamma (a)$)

If the positive baseline rate has a gamma distribution of shape $a$ and rate $b$, its logarithm has this probability density function with respect to $d\mu$. The extra factor $e^{\mu }$ is the Jacobian determinant of the transformation. After a Gibbs sampler draws the conditional baseline rate, taking its logarithm is an exact update of the intercept. Omitting the Jacobian changes the prior.