Suppose that I have a random walk process defined by $\alpha_{t+1}$ ~ N$(\alpha_t, \omega^2)$.
Given $\alpha_t$ and $\alpha_{t+2}$, I understand why the conditional formula for $(\alpha_{t+1}|\alpha_{t},\alpha_{t+2})$ is N$(\frac{\alpha_t + \alpha_{t+2}}{2},\frac{\omega^2}{2})$.
I am trying to derive $(\alpha_{t+1}|\alpha_{t},\alpha_{t+3})$.
Now $(\alpha_{t+1}|\alpha_{t},\alpha_{t+3})$ = $\int (\alpha_{t+1}, \alpha_{t+2}|\alpha_{t},\alpha_{t+3})\, d\alpha_{t+2}$
But I don't see where to go from here. One option is $\int (\alpha_{t+1}|\alpha_{t},\alpha_{t+2}, \alpha_{t+3}) \cdot (\alpha_{t+2}|\alpha_t, \alpha_{t+3})\, d\alpha_{t+2}$ but the second part of this integral isn't known (indeed, this is the mirror of what you're trying to find)
How can you break up $(\alpha_{t+1}, \alpha_{t+2}|\alpha_{t},\alpha_{t+3})\, d\alpha_{t+2}$ to only involve terms in $(\alpha_{t+1}|\alpha_{t},\alpha_{t+2})$ and $(\alpha_{t+2}|\alpha_{t+1},\alpha_{t+3})$ so that you can do the integral?
Any help greatly appreciated! My motivation for doing this is trying to speed up part of a Gibbs sampler.
Alternatively (or as well if possible), I'd be happy with something showing what $(\alpha_{t+1},\alpha_{t+2} | \alpha_t, \alpha_{t+3})$ is so that I can sample the points simultaneously. I'm ultimately looking to generalise this to $(\alpha_{t+1},\alpha_{t+2}, \ldots, \alpha_{t+n-1} | \alpha_t, \alpha_{n})$ so that I can do joint draws of the random walk.