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Thread: Probability in Hidden Markov Model

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    Probability in Hidden Markov Model


    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,
    Last edited by Dason; 03-16-2014 at 04:27 PM.

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