I'm considering a Hidden Markov Model as follows:
$X_{n+1} = F_n(X_n,\Theta)$
$Y_n = G_n(X_n,\Theta)$
where the $Y_n$ are the observations, the $X_n$ the hidden states and $\Theta$ the parameters. At some point, I have to deal with this probability
$p(X_k = x_k | X_{k-1} = x_{k-1}, Y_{0:N} = y_{0:N})$
where $0 < k < N$ and $Y_{0:N}$ denotes the $Y_0, \dots, Y_N$. If it were
$p(X_k = x_k | X_{k-1} = x_{k-1}, Y_{0:k} = y_{0:k})$
I would know how to deal with it (I think) but I wonder what it changes to add the observations for all future times. Any idea how I could handle this probability, maybe express it with respect to usual and known probabilities?
Is there something I'm completely missing here?