## 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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