= Linear contraction of a two-coordinate Gaussian Gibbs sweep
{title2=$b_{k+1}-\mu_b=\frac{C^2}{AD}(b_k-\mu_b)+\eta_k$}
For Gaussian precision $\left(\begin{smallmatrix}A&C\\C&D\end{smallmatrix}\right)$, a systematic update of the first coordinate followed by the second makes the centered second coordinate an AR(1) chain with coefficient $C^2/(AD)<1$. Independent conditional Gaussian noise supplies its stable innovation. This proves convergence of the Gaussian Gibbs sampler and explains slower mixing when posterior correlation has magnitude near one.
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