In the gradient based method, any parameter 
of the HMM
is updated according to the standard formula,
where J is a quantity to be minimized. We define in this case,
Since the minimization of 
is equivalent to the maximization of
, eqn.1.19 yields the required optimization criterion,
ML. But the problem is to find the derivative 
for any parameter of the model. This can
be easily done by relating J to model parameters via . As a
key step to do so, using the eqns.1.7 and 1.9
we can obtain,
Differentiating the last equality in eqn. 1.20 wrt an arbitrary
parameter ,
Eqn.1.22 gives , if we
know 
which can be found
using eqn.1.21. However this derivative is specific to the
actual parameter concerned. Since there are two main parameter sets in
the HMM, namely transition probabilities 
and observation probabilities 
, we can find the derivative
for each of the parameter
sets and hence the gradient, .
