Gamma-integrated baseline Poisson likelihood
= Gamma-integrated baseline Poisson likelihood
{title2=$p(y\mid\beta)\propto e^{\sum_k y_k\eta_k}/(b+\sum_ke^{\eta_k})^{n+a}$}
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>.