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I am learning to use HMM and I am trying to solve the following problem. There is a robot moving around the nodes in graph. The robot can move to adjacent nodes with certain probabilities. Each time the robot steps into a new "node", a (noisy) information about the node is generated. That is, I do not know the exact node. I have the following data:

  • Each node node is a hidden state (finite number)

  • A transition matrix that defines the probabilies for the transitions between nodes. $A$

  • Emission probabilities for each hidden node

Using HMM standard functions I should be able to predict the "hidden state" at time t if I have "t observations" (observations from $1,\dots,t$) $P(X_t|O_{1,\dots,t})$. Is there a way to predict the next move (hidden state at t+1)? If I have "$t$-observations" (all observations at time $t$), is it possible to predict the most probable "hidden state" at time "$t+1$"? Which HMM principle should I use?

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    $\begingroup$ To get P(t+1) simply multiply your distribution over the hidden states at time t by the transition probabilities. $\endgroup$
    – jerad
    Oct 4, 2013 at 22:51
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    $\begingroup$ Rabiner's review paper on HHMs is an amazing source for starting to work with HMMs. All the important algorithms are described (forward algo, backward algo, and EM algo) Take a look (cs.cornell.edu/Courses/cs4758/2012sp/materials/…) $\endgroup$
    – bdeonovic
    Oct 9, 2013 at 19:59
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    $\begingroup$ My current interest is same kind of HMM model. I will not give detailed explanation, because there is a better alternative. I have learned whatever I know thanks to this paper: www1.se.cuhk.edu.hk/~hcheng/paper/sdm2013.pdf Especially focus on section 3.2 (3.2.1 and 3.2.2). $\endgroup$
    – Dorukhan Arslan
    Jan 3, 2016 at 4:18

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You use the forward algorithm to predict $P(X_{t+1})$.

$P(X_{t+1}|X_t, Y_{1:t} ) = \sum_{X} P(X_{t+1}|X_t) \cdot P(X_t|Y_{1:t}) $

So, you use the same principle for predicting $P(X_{t})$, but without being able to incorporate $Y_{t+1}$, since it is not observed yet.

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    $\begingroup$ How would we calculate P(Xt|Y1:t) ? it is opposite of emission probabilities. I have P(Y(t)/X(t)) which is emission probabilities. $\endgroup$
    – Arpit Sisodia
    Dec 23, 2016 at 5:34
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To get the probability over hidden states at t_2, just multiply your posterior over t_1 by your transition matrix.

https://stackoverflow.com/questions/15554923/how-to-perform-a-prediction-with-matlabs-hidden-markov-model-statistics-toolbo

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