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Thread: Probability of next observation in Hidden Markov Model

  1. #1
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    Probability of next observation in Hidden Markov Model

    I am learning Hidden Markov Models and trying to solve this problem. A Hidden Markov Model has 2 states \{1,2\} and 3 observations \{a,b,c\}. The state transition matrix (A), observation matrix (B)and initial matrix ( \pi) of the model is given below.

    A  = \begin{bmatrix}0.9 & 0.1\\ 0.3 & 0.7 \end{bmatrix}

    B = \begin{bmatrix} 0.3 & 0.4 & 0.3 \\ 0.2 & 0.6 & 0.2\end{bmatrix}

    \pi =\begin{bmatrix} 0.7 & 0.3 \end{bmatrix}

    (a) if the observation sequence is \{a,a,b\} and the hidden sequence is \{1,1,2\} , calculate the probablity that the next observation is ``b"
    (b) For question (a), if the next observation is b, what is the probability that the hidden sequence is \{1,1,2,2\}
    Now I have some questions here. For part (a), using the conditional probability laws, I would have

    P(b | \{a,a,b\}) = \frac{P(\{a,a,b,b\}}{P(\{a,a,b\}}

    Is this correct ?
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  2. #2
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    Re: Probability of next observation in Hidden Markov Model

    Hey issacnewton.

    Do you understand how to use the transition matrix to get a conditional probability?
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  3. #3
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    Re: Probability of next observation in Hidden Markov Model

    Chiro, I think what is being asked in part (a) is the probability of getting next observation b given that the earlier observation sequence is \{a,a,b\}. So we have been asked the conditional probability here. And I am just using the definition of conditional probability. But on second thoughts I think this might be not correct for Markov processes. In Markov processes, we only consider the previous state. So with this in mind, we should have

    P(b) = P(b|b)P(b) + P(b|a)P(a)

    Would this be correct ?
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  4. #4
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    Re: Probability of next observation in Hidden Markov Model

    There is a reason for the transition matrix in a markov model.

    If you are studying this you should read about how to get the probability for a given state [i.e. in terms of the offset of the state] in terms of the transition matrix.

    As an example - in a normal model you have A^n to get the matrix for the nth observation and you multiply that with some initial probability to get the probabilities for the states at the nth time.
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