 Poisson count change-point model with gamma priors

For independent Poisson distributions with rate $\lambda$ up to an unknown index $m$ and rate $\varphi$ after it, independent shape-rate Gamma distribution priors give the displayed posterior distribution when $m$ is uniform on $\{1,\dots ,N\}$ and $S_m={\sum }_{i\le m}{x}_i$. The model differs from a continuous-time Poisson change-point posterior. The endpoint $m=N$ represents no observed change: the second segment is empty and its rate keeps its prior law.