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Continuous Emissions

For continuous HMMs, the two most commonly used emission distributions are the Gaussian distribution, and the Gaussian mixture,

where , , and is the probability of observing the vector x under the density function for the Gaussian distribution with mean vector and covariance matrix . A variant is the continuous HMM with shared Gaussian mixtures [&make_named_href('', "node31.html#Huang90","[50]"), &make_named_href('', "node31.html#HAJ90","[51]")], in which the Gaussian parameters are shared across states, and the parameters specific to each state are only the mixture weights:

Here, the mixture weights play a role that is similar to the multinomial coefficients of the discrete emission HMMs described above.

As in many modeling approaches, there are many characteristics of an HMM which have to be selected by the modeler, based on prior knowledge and/or the data, e.g., the number of values of the state variable, the topology of the HMM (forcing some transition probabilities to zero), the type of distribution for the emissions, which includes such choices as the number of Gaussians in a mixture, etc... In this paper we will basically not address this model selection question and restrict the discussion to the general use of prior knowledge in the topology of speech recognition HMMs, and to the free numerical parameters, i.e., those that are chosen numerically with a learning or parameter estimation algorithm.



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