Editing Baum–Welch algorithm (section)

====Multiple sequences====

The algorithm described thus far assumes a single observed sequence <math>Y = y_1, \ldots, y_T</math>. However, in many situations, there are several sequences observed: <math>Y_1, \ldots, Y_R</math>. In this case, the information from all of the observed sequences must be used in the update of the parameters <math>A</math>, <math>\pi</math>, and <math>b</math>. Assuming that you have computed <math>\gamma_{ir}(t)</math> and <math>\xi_{ijr}(t)</math> for each sequence <math>y_{1,r},\ldots,y_{N_r,r}</math>, the parameters can now be updated:
*<math>\pi_i^* = \frac{\sum_{r=1}^{R}\gamma_{ir}(1)}{R}</math>
*<math>a_{ij}^*=\frac{\sum_{r=1}^{R} \sum^{T-1}_{t=1}\xi_{ijr}(t)}{\sum_{r=1}^{R} \sum^{T-1}_{t=1}\gamma_{ir}(t)},</math>
*<math>b_i^*(v_k)=\frac{\sum_{r=1}^{R} \sum^T_{t=1} 1_{y_{tr}=v_k} \gamma_{ir}(t)}{\sum_{r=1}^{R} \sum^T_{t=1} \gamma_{ir}(t)},</math>
where
:<math>
1_{y_{tr}=v_k}=
\begin{cases}
1 & \text{if } y_{t,r}=v_k,\\
0 & \text{otherwise}
\end{cases}
</math>
is an indicator function