Gaussian AR1 bridge
= Gaussian AR1 bridge
{c}
For a stationary <autoregressive process of order one> with mean $\mu$, autoregressive coefficient $\phi$ and innovation <variance> $\sigma^2$, an interior observation conditional on its neighbors has
$$
X_t\mid X_{t-1},X_{t+1}\sim N\left(\mu+\frac{\phi\{X_{t-1}+X_{t+1}-2\mu\}}{1+\phi^2},\frac{\sigma^2}{1+\phi^2}\right).
$$
The <Markov property> makes the same law valid when all other observations are also conditioned on.