Editing Baum–Welch algorithm (section)

====Update====
We can now calculate the temporary variables, according to Bayes' theorem:
:<math>\gamma_i(t)=P(X_t=i\mid Y,\theta) = \frac{P(X_t=i,Y\mid\theta)}{P(Y\mid\theta)} = \frac{\alpha_i(t)\beta_i(t)}{\sum_{j=1}^N \alpha_j(t)\beta_j(t)},</math>
which is the probability of being in state <math>i</math> at time <math>t</math> given the observed sequence <math>Y</math> and the parameters <math>\theta</math>
:<math>\xi_{ij}(t)=P(X_t=i,X_{t+1}=j\mid Y,\theta) = \frac{P(X_t=i,X_{t+1}=j,Y\mid\theta)}{P(Y\mid\theta)} = \frac{\alpha_i(t) a_{ij} \beta_j(t+1) b_j(y_{t+1})}{\sum_{k=1}^N \sum_{w=1}^N \alpha_k(t) a_{kw} \beta_w(t+1) b_w(y_{t+1}) }, </math>
which is the probability of being in state <math>i</math> and <math>j</math> at times <math>t</math> and <math>t+1</math> respectively given the observed sequence <math>Y</math> and parameters <math>\theta</math>.

The denominators of <math>\gamma_i(t)</math> and <math>\xi_{ij}(t)</math> are the same ; they represent the probability of making the observation <math>Y</math> given the parameters <math>\theta</math>.

The parameters of the  hidden Markov model <math>\theta</math> can now be updated:
*<math>\pi_i^* = \gamma_i(1),</math>
which is the expected frequency spent in state <math>i</math> at time <math>1</math>.
*<math>a_{ij}^*=\frac{\sum^{T-1}_{t=1}\xi_{ij}(t)}{\sum^{T-1}_{t=1}\gamma_i(t)},</math>
which is the expected number of transitions from state ''i'' to state ''j'' compared to the expected total number of transitions away from state ''i''. To clarify, the number of transitions away from state ''i'' does not mean transitions to a different state ''j'', but to any state including itself. This is equivalent to the number of times state ''i'' is observed in the sequence from ''t''&nbsp;=&nbsp;1 to ''t''&nbsp;=&nbsp;''T''&nbsp;−&nbsp;1.
*<math>b_i^*(v_k)=\frac{\sum^T_{t=1} 1_{y_t=v_k} \gamma_i(t)}{\sum^T_{t=1} \gamma_i(t)},</math>
where
:<math>
1_{y_t=v_k}=
\begin{cases}
1 & \text{if } y_t=v_k,\\
0 & \text{otherwise}
\end{cases}
</math>
is an indicator function, and <math>b_i^*(v_k)</math> is the expected number of times the output observations have been equal to <math>v_k</math> while in state <math>i</math> over the expected total number of times in state <math>i</math>.

These steps are now repeated iteratively until a desired level of convergence.

'''Note:''' It is possible to over-fit a particular data set. That is, <math>P(Y\mid\theta_\text{final}) > P(Y \mid \theta_\text{true}) </math>. The algorithm also does '''not''' guarantee a global maximum.