For all the above distributions, the EM (Expectation-Maximization)
algorithm [&make_named_href('',
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can be used to estimate the parameters of the HMM
in order to maximize the likelihood function
over the set
of training sequences (indexed by the letter p).
The EM algorithm itself is discussed in section 5.2.
In this application of EM, the state variable
is viewed
as the missing variable. It was shown early on [&make_named_href('',
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that the maximum likelihood estimates of the parameters of
a hidden Markov process are consistent. More recently,
it has been shown [&make_named_href('',
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for HMMs, maximum likelihood estimates are
asymptotically normal as expected and consistent estimates of their variance
can be constructed, so that the estimation procedure
is asymptotically valid.
Other emission distributions of the exponential
family (or mixtures thereof) could also be used.
In speech and other sequence recognition applications, the EM algorithm
can also be used when the state sequence is constrained, which corresponds
to modeling the conditional distribution
,
e.g., with
a sequence of ``correct''
labels which should be associated with the observed sequence
.
When the states are associated to labels, conditioning
on
corresponds to a restriction on the graph
representing topology of the HMM: see for example how
the unconstrained recognition model in Figure 4
is constrained with the correct word sequence ``the dog''
in Figure 8.
It should be noted that other criteria than the maximum likelihood criterion can be used to train HMMs, for example to incorporate a prior on parameters, to make the training more discriminant (focus more on doing the classification correctly), or to make it more robust (penalized maximum likelihood [&make_named_href('', "node31.html#Silverman-encyc86","[55]")]). For more complex distributions than those described above or for several learning criteria other than maximum likelihood, numerical optimization methods other than the EM algorithm are often used, usually based on the gradient of the learning criterion with respect to the free numerical parameters. When possible, the EM algorithm is generally preferred because of its faster convergence properties. These topics will be further discussed in sections 5.2 and 3.5.