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Parameter Estimation

For all the above distributions, the EM (Expectation-Maximization) algorithm [&make_named_href('', "node31.html#Dempster77","[52]"), &make_named_href('', "node31.html#Baum67","[16]"), &make_named_href('', "node31.html#Baum70","[17]"), &make_named_href('', "node31.html#Baum72","[18]")] 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('', "node31.html#Baum66","[53]")] that the maximum likelihood estimates of the parameters of a hidden Markov process are consistent. More recently, it has been shown [&make_named_href('', "node31.html#bickel+ritov95","[54]")] that 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.


next up previous
Next: The Viterbi Algorithm Up: Parametrization: Choice of Distributions Previous: Continuous Emissions

Yoshua Bengio
Tue Oct 7 08:34:36 EDT 1997