Probability in hidden Markov model

Question

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?

Answer 1

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.

Answer 2

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.