= Gaussian autoregressive conditional precision
{c}
For a stationary unit-variance <normal distribution> autoregression with coefficient $a\in(-1,1)$, its vector of length $T\geq2$ has <precision matrix> $J$ that is tridiagonal: the two endpoint diagonal entries are $1/(1-a^2)$, interior entries are $(1+a^2)/(1-a^2)$, and adjacent off-diagonal entries are $-a/(1-a^2)$. An independent observation $r=\lambda Z+\eta$, with covariance $vI$, updates precision and information vector to
$$
Q=J+\lambda^2v^{-1}I,\qquad h=\lambda v^{-1}r.
$$
The conditional law is <multivariate normal distribution> $N(Q^{-1}h,Q^{-1})$.
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