I am trying to use this equation

enter image description here

to calculate the alpha (forward) probabilities for the EM/Baum-welch algorithm but I'm running into some confusion. I don't understand what the $h_t$ is. I know its a hidden state, I have 16 different hidden states but don't know which one is $h_1$ for instance. Can I arbitrarily assign them an index or are they related to $v_t$?

I current have a transition matrix, emission matrix and an initial vector as well as some series of observed emissions.


Perhaps the source of the confusion is your notation.

Usually, one defines the forward variables $α_j(t)$ as the probability of the partial observation sequence until time $t$, with state $S_j$ at time $t$ (reference: https://www.robots.ox.ac.uk/~vgg/rg/slides/hmm.pdf).

Hence, under your notation, you use the forward procedure to calculate inductively $\alpha (h_t)$. If you have $T=16$ hidden states then you start at $\alpha (h_1)$ and finish at $\alpha (h_{16})$; each $\alpha (h_i)$ is a vector whose length is the number of total possible values the hidden states can take.

  • $\begingroup$ I think I understand what your saying, so $h_1$ is just the initial vector (the probabilities of starting in each hidden state)? $\endgroup$ – R-4 Mar 29 '17 at 12:26
  • $\begingroup$ Yes, it's a vector of the same length as $\alpha (h_i)$. it's usually known as the prior; supposed to reflect your "belief" regarding the probability of a hidden state accepting a certain value, before any evidence is taken into account. $\endgroup$ – galoosh33 Mar 29 '17 at 12:46

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