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

Condition on initial observations $x_0,x_1$ and modelwith known ${\sigma }^2>0$ and independent standard-normal prior distributions for $a,b$. Put $v_t=(x_{t+1},x_t{)}^{\mathsf{T}}$ and $\beta =(a,b{)}^{\mathsf{T}}$. Gaussian conjugacy for a normal linear model gives posterior precision matrix and posterior meanThe prior distribution makes $\Lambda$ positive-definite even with a deficient design matrix. The unscaled sums occur specifically when ${\sigma }^2=1$.

If $\Lambda =\left(\begin{array}{cc}A& C\\ C& D\end{array}\right)$ and $h=(r,s{)}^{\mathsf{T}}$, the two full conditional distributions have means $(r-Cb)/A$ and $(s-Ca)/D$ and variances $1/A$ and $1/D$. These formulas use the conditional autoregressive model of order two likelihood function. Giving the initial pair its stationary law, or imposing a stationary-parameter restriction, changes that posterior distribution.