= Gaussian two-coordinate Gibbs recursion
{c}
{title2=$Y_{n+1}=\rho^2Y_n+\sqrt{1-\rho^2}(\rho Z_1+Z_2)$}
For a standard bivariate <normal distribution> with correlation $\rho$, each conditional has variance $1-\rho^2$ and mean $\rho$ times the other coordinate. A sweep draws $X_{n+1}=\rho Y_n+\sqrt{1-\rho^2}Z_1$, then $Y_{n+1}=\rho X_{n+1}+\sqrt{1-\rho^2}Z_2$. The resulting coordinate chain is an <AR(1)> process with coefficient $\rho^2$ and innovation variance $1-\rho^4$. It shows explicitly why near-unit correlation slows successive Gibbs sweeps.
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