 Gibbs updates for a Poisson count change point

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.