 Solution

Define the sufficient data sums, all over $0\le t\le n-2$,Multiplying the likelihood function by the independent standard-normal priors and collecting the quadratic terms givesLet $\Lambda =\left(\begin{array}{cc}A& C\\ C& D\end{array}\right)$ and $\Delta =AD-C^2$. This precision matrix is $I+\sum v_t{v}_t^T$, where $v_t=(x_{t+1},x_t{)}^T$, so it is positive definite even for a short or singular design. Completing the square provesThe fully normalized posterior density isThis is Gaussian conjugacy for an initialized AR(2) regression.

Completing each one-dimensional square, or using the supplied conditional-normal identity, givesWhen $n=2$, 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 $(a,b)$.