Poisson-gamma conjugacy with unequal exposures ($\Theta \mid \mathbf{Y}\sim \mathrm{Gamma}(\alpha +{\sum }_j{Y}_j,\beta +{\sum }_j{m}_j)$)

Given $\Theta$, independent observations $Y_j$ have Poisson distribution with means $m_j\Theta$ for known exposures. A gamma distribution prior of shape $\alpha$ and rate $\beta$ yields a gamma posterior with shape $\alpha +{\sum }_j{Y}_j$ and rate $\beta +{\sum }_j{m}_j$. The likelihood kernel is ${\theta }^{{\sum }_j{Y}_j}{e}^{-\theta {\sum }_j{m}_j}$. Thus the posterior mean depends on total claims and total exposure, not the number of periods alone.