Calculating alpha in EM / Baum-Welch algorithm for Hidden Markov

I am trying to use this equation 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.

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.
• I think I understand what your saying, so $h_1$ is just the initial vector (the probabilities of starting in each hidden state)? – R-4 Mar 29 '17 at 12:26
• 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. – galoosh33 Mar 29 '17 at 12:46