# Probability in hidden Markov model

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?

By the Markov property, the problem is the same as if $k=0$ and $N$ is replaced by $N-k$, so it is enough to understand the case $k=0$. Then the problem is the standard smoothing problem'' of filtering theory, which is solved by forward-backward algorithms. The wikipedia page https://mathoverflow.net/questions/156751/probability-in-hidden-markov-model has a reference to this, and classical filtering books (Liptser-Shiryayev, ...) have a chapter on that in a general context.
Are you sure you're not interested in $p(X_k=x_k|Y_{0:N})$ rather than the expression you gave? This would be an instance of the "smoothing problem" (see e.g. Hidden Markov model). These are solved with forward-backward algorithms as ofer mentioned above. If not I guess you could do some re-writes and still end up with a solution \begin{equation} p(X_k=x_k|X_{k-1}=x_{k-1},Y_{0:N}) = p(X_k=x_k|X_{k-1}=x_{k-1},Y_{0:k},Y_{k+1:N}) \propto p(X_k=x_k|X_{k-1}=x_{k-1},Y_{0:k}) p(Y_{k+1:N}|X_k=x_k,X_{k-1}=x_{k-1}) = p(X_k=x_k|X_{k-1}=x_{k-1})p(y_k|X_k=x_k) p(Y_{k+1:N}|X_k=x_k), \end{equation} where the last equality is due to Markov assumption. The first two probabilities are just the filtering distribution, the last one is (again) found using the forward backward algorithm.