## Kategoriearchiv

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• ## Hidden Markov Model training using the Baum-Welch Algorithm

The Baum-Welch algorithm determines the (locally) optimal parameters for a Hidden Markov Model by essentially using three equations.

One for the initial probabilities:

\begin{align} \pi_i &= \frac{E\left(\text{Number of times a sequence started with state}\, s_i\right)}{E\left(\text{Number of times a sequence started with any state}\right)} \end{align}

Another for the transition probabilities:

\begin{align} a_{ij} &= \frac{E\left(\text{Number of times the state changed from}\, s_i \, \text{to}\,s_j\right)}{E\left(\text{Number of times the state changed from}\, s_i \, \text{to any state}\right)} \end{align}

And the last one for the emission probabilities:

\begin{align} b_{ik} &= \frac{E\left(\text{Number of times the state was}\, s_i \, \text{and the observation was}\,v_k\right)}{E\left(\text{Number of times the state was}\, s_i\right)} \end{align}

If one had a fully labeled training corpus representing all possible outcomes, this would be exactly the optimal solution: Count each occurrence, normalize and you’re good. If, however, no such labeled training corpus is available – i.e. only observations are given, no according state sequences – the expected values $E(c)$ of these counts would have to be estimated. This can be done (and is done) using the forward and backward probabilities $\alpha_t(i)$ and $\beta_t(i)$, as described below.
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