Baum-Welch training is similar to the Viterbi training described
in the previous section except that the hard boundary implied
by the 
function is replaced by a soft boundary
function L which represents the probability of an observation being
associated any given Gaussian mixture component.
This occupation probability is computed from the forward
and backward probabilities.
For the isolated-unit style of training, the forward
probability 
for 1<j<N and
is calculated by the forward recursion
with initial conditions given by
for 1<j<N and final condition given by
The backward probability 
for 1<i<N and 
is
calculated by the backward recursion
with initial conditions given by
for 1<i<N and final condition given by
In the case of embedded training where the HMM spanning the observations
is a composite constructed by concatenating Q subword models, it is
assumed that at time t, the 
and 
values corresponding to the entry state and exit states of a HMM
represent the forward and backward probabilities at time 
and 
, respectively, where 
is small. The equations
for calculating and are then as follows.
For the forward probability, the initial conditions are established at time t=1 as follows
where the superscript in parentheses refers to the index of the model in
the sequence of concatenated models. All unspecified values of
are zero. For time t > 1,
For the backward probability, the initial conditions are set at time t=T as follows
where once again, all unspecified values are zero. For
time t<T,
The total probability 
can be computed
from either the forward or backward probabilities
