= Solution
Define the sufficient data sums, all over $0\leq t\leq n-2$,
$$
A=1+\sum x_{t+1}^2,\quad D=1+\sum x_t^2,\quad C=\sum x_{t+1}x_t,\quad
r=\sum x_{t+1}x_{t+2},\quad s=\sum x_tx_{t+2}.
$$
Multiplying the <likelihood function> by the independent standard-normal priors and collecting the quadratic terms gives
$$
\pi(a,b)\propto\exp\left[-\frac12(Aa^2+2Cab+Db^2-2ra-2sb)\right].
$$
Let $\Lambda=\begin{pmatrix}A&C\\C&D\end{pmatrix}$ and $\Delta=AD-C^2$. This precision matrix is $I+\sum v_tv_t^T$, where $v_t=(x_{t+1},x_t)^T$, so it is positive definite even for a short or singular design. Completing the square proves
$$
\boxed{\begin{pmatrix}a\\b\end{pmatrix}\Bigm|x\sim N_2(\mu,\Sigma),\quad
\Sigma=\frac1\Delta\begin{pmatrix}D&-C\\-C&A\end{pmatrix},\quad
\mu=\frac1\Delta\begin{pmatrix}Dr-Cs\\As-Cr\end{pmatrix}.}
$$
The fully normalized <posterior> density is
$$
\boxed{\pi(a,b)=\frac{\sqrt\Delta}{2\pi}\exp\left[-\frac12\left(\begin{pmatrix}a\\b\end{pmatrix}-\mu\right)^T\Lambda\left(\begin{pmatrix}a\\b\end{pmatrix}-\mu\right)\right].}
$$
This is <Gaussian conjugacy for an initialized AR(2) regression>.
Completing each one-dimensional square, or using the supplied conditional-normal identity, gives
$$
\boxed{b\mid a,x\sim N\left(\frac{s-Ca}{D},\frac1D\right),\qquad
a\mid b,x\sim N\left(\frac{r-Cb}{A},\frac1A\right).}
$$
When $n=2$, all regressor sums vanish and the <posterior> remains the independent standard-normal prior. No stationary-parameter restriction is imposed: the specified prior is on all of $\mathbb R^2$, and the finite initialized chain is defined for all $(a,b)$.
Back to article page
