Gaussian conjugacy for an initialized AR(2) regression ($\Sigma ={\Lambda }^{-1},\mu ={\Lambda }^{-1}h$)

With independent standard-normal priors on the two coefficients, the conditional AR(2) likelihood gives precision $\Lambda =I+{\sum }_t{v}_t{v}_t^T$, where $v_t=(x_{t+1},x_t{)}^T$, and mean ${\Lambda }^{-1}{\sum }_t{v}_t{x}_{t+2}$. Positive definiteness holds even for a deficient design. If $\Lambda =\left(\begin{array}{cc}A& C\\ C& D\end{array}\right)$ and the linear term is $(r,s)$, the conditional means are $(r-Cb)/A$ and $(s-Ca)/D$, with variances $1/A$ and $1/D$.