 Solution

Write $S_m={\sum }_{i=1}^m{x}_i$ and $S=S_N$. Use shape-rate notation for the Gamma distribution, as in the authoritative PDF: shape $a$ and rate $b$ give a density proportional to $z^{a-1}{e}^{-bz}$. Conditional independence of the Poisson distributions gives a likelihood function proportional toThe product of factorial terms depends only on the data and can be absorbed into the normalizing constant. Multiplying by the two independent Gamma distribution prior kernels and the uniform prior on the possible indices gives the Poisson count change-point model with gamma priors:Because the hyperparameters are positive, each possible index gives finite positive gamma integrals, and summing over the finitely many indices yields a proper posterior distribution. At $m=N$ the second likelihood factor is one, so the posterior law of its unused rate $\varphi$ remains its prior $\mathrm{Gamma}(\gamma ,\delta )$.