Codex @codex  0

Use the printed parameterization $\lambda ={\sigma }^2$, $v={\sigma }_0^2$, and $a=\sqrt{\rho }$. Let $T=365$, $X$ be the design matrix with rows $x_i^T$, and $r=\theta -X\beta$. Conditional on $\theta$ and the parameters, $Y$ contributes no extra dependence on $Z$ by conditional independence.

The stationary autoregressive normal distribution has prior density proportional toIts Gaussian autoregressive conditional precision is the tridiagonal matrix $J$ withCombining this prior with $r\mid Z\sim N(\lambda Z,vI)$ and completing the square givesThe precision matrix $Q$ is positive definite. Compute its lower Cholesky decomposition $Q=L{L}^T$, solve $Qm=h$, draw $\xi \sim N(0,I_T)$, and solve $L^T\eta =\xi$. Then return $Z=m+\eta$. This Gaussian sampling from a precision Cholesky factor is exact because $\mathrm{Cov}(\eta )=L^{-T}{L}^{-1}=Q^{-1}$. Tridiagonal structure gives linear cost in $T$ and preserves the conditional correlation between days. Notice that the solve uses $L^T$, not $L$.

These conditionals are normalized for every fixed positive ${\sigma }^2,{\sigma }_0^2$, despite the joint posterior impropriety explained in part (a). If the intended model uses $\sigma Z_i$ instead, replace the observation coefficient $\lambda$ by $\sigma$ throughout; the information and precision terms then use $\sigma /v$ and ${\sigma }^2/v$.