Gamma-integrated baseline Poisson likelihood ($p(y\mid \beta )\propto e^{{\sum }_k{y}_k{\eta }_k}/(b+{\sum }_k{e}^{{\eta }_k}{)}^{n+a}$)

For independent Poisson distributions with means $u{e}^{{\eta }_k}$ and a shape–rate Gamma distribution prior on $u$, integrating $u$ adds $a$ to the total-count exponent and $b$ to the sum of relative rates. This distinguishes gamma mixing from conditioning on the total count, which gives a multinomial distribution.