= Gaussian change-point Gibbs updates
{c}
For <independent random variables>[independent] <mean>[means] $\theta\sim N(\mu_\theta,t_\theta)$ and $\phi\sim N(\mu_\phi,t_\phi)$, the first <mean>'s conditional precision is $m/v+1/t_\theta$ and conditional precision-weighted <mean> is $\sum_{i\le m}x_i/v+\mu_\theta/t_\theta$; replace $m$ by $n-m$ for the second <mean>. With $v\sim\operatorname{IG}(a_0,b_0)$ its conditional law is $\operatorname{IG}(a_0+n/2,b_0+S_m/2)$. The discrete split has <conditional probabilities> proportional to $e^{-S_m/(2v)}$. These <full conditional distributions> give a <Gibbs sampler> including $m=n$. Averaging the sampled indicators of that configuration estimates its <probability>; averaging its <conditional probabilities> gives an estimate by <Rao-Blackwellization>.
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