= Gibbs updates for a Poisson count change point
{c}
The <full conditional distributions> are $\lambda\mid m,x\sim\operatorname{Gamma}(\alpha+S_m,\beta+m)$ and $\phi\mid m,x\sim\operatorname{Gamma}(\gamma+S_N-S_m,\delta+N-m)$, in shape-rate notation. Conditional on the two rates, $m$ has weights $w_j=\exp[j(\phi-\lambda)+S_j\log(\lambda/\phi)]$. 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 $\mathbf1_{\{m=N\}}$ estimate the posterior probability of no observed change.
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