Probability in Hidden Markov Model

#1
Hello,

I'm considering a Hidden Markov Model as follows:

\( X_{n + 1} = F_n(X_n,\Theta,\eta_n) \) (that's X_{n+1} here, I don't know why the + is removed)
\( Y_n = G_n(X_n,\Theta,\xi_n) \)

where the \( Y_n \) are the observations and the \( X_n \) the hidden states. 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?

Thanks a lot,
 
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