Poisson count change-point model with gamma priors ($p(\lambda ,\varphi ,m\mid x)\propto {\lambda }^{\alpha +S_m-1}{e}^{-(\beta +m)\lambda }{\varphi }^{\gamma +S_N-S_m-1}{e}^{-(\delta +N-m)\varphi }$)

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.

Integrating the two rates out of the Poisson count change-point model with gamma priors gives these weights up to a common factor, by the defining Gamma function integral. Normalizing across the finitely many possible indices gives the exact marginal posterior distribution of $m$, providing a check on Markov chain Monte Carlo estimates. One can also draw $m$ from these weights and then draw both rates from their conditional Gamma distributions, producing independent exact joint posterior samples.

The full conditional distributions are $\lambda \mid m,x\sim \mathrm{Gamma}(\alpha +S_m,\beta +m)$ and $\varphi \mid m,x\sim \mathrm{Gamma}(\gamma +S_N-S_m,\delta +N-m)$, in shape-rate notation. Conditional on the two rates, $m$ has weights $w_j=\exp [j(\varphi -\lambda )+S_j\log (\lambda /\varphi )]$. A Gibbs sampler cycles through these three draws. Each update preserves the joint posterior distribution; subtracting the largest log weight before exponentiation stabilizes the categorical draw. Averages of $1_{\{m=N\}}$ estimate the posterior probability of no observed change.