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Learning Criteria

 

In many applications of HMMs such as speech recognition, there are actually two sequences of interest: the observation (e.g., acoustic) sequence, , and the classification (e.g. correct word) sequence, . The traditional approach to HMM speech recognition is to consider independently a different HMM for each word (or speech unit) and to learn the distribution of acoustic subsequences associated with each word or speech unit. By concatenating the speech units associated with the correct word sequence , one can represent the conditional acoustic probability . An HMM that is constrained by the knowledge of the correct word sequence is called a constrained model, and is illustrated in Figure 8. On the other hand, during speech recognition, the correct word sequence is unknown, and all the word sequences allowed by the language model must be taken into account. An HMM that allows all these word sequences is called a recognition model (and it is generally much larger than a constrained model). It represents the joint distribution (or, when summing over all possible state paths, the observed sequence probability ).

  
Figure 8: Example of a constrained HMM, representing the conditional distribution . Here is an acoustic sequence and is the sequence of two words /the/ and /dog/.

A maximum likelihood criterion can then be applied to estimate the parameters of the speech units that maximize the constrained acoustic likelihood, , over all the training sequences (indexed by p above). For this purpose, the EM or GEM algorithms described in section 5.2 are often used.

The above approach is called non-discriminant because it is based on learning the acoustic distribution associated with each speech unit (i.e., class-conditional density functions), rather than learning how the various linguistic classes differ acoustically. When trying to discriminate between different interpretations (i.e., different classes), it is sufficient to know about these differences, e.g, using or even directly describing the decision surface in the space of acoustic sequences. The non-discriminant models contain the additional information . Furthermore, they strongly rely on the assumptions made on the form of the probability density of the acoustic data . Since the models chosen to represent the data are generally imperfect, it has been found that better classification results can often be obtained when the objective of learning is closer to the reduction of the number of classification errors. Several approaches have been proposed to train HMMs with a discriminant criterion. The most common are the maximum a posteriori criterion, to maximize , and the maximum mutual information criterion [&make_named_href('', "node31.html#BrownPhD","[67]"), &make_named_href('', "node31.html#Bahl87","[68]"), &make_named_href('', "node31.html#Nadas88","[69]")], to maximize . The maximum mutual information criterion is therefore obtained by comparing the log-probability of the constrained model, , with the log-probability of the unconstrained recognition model, allowing all the possible interpretations (word sequences). Maximizing this criterion attempts to increase the likelihood of the correct (i.e., constrained) model while decreasing the likelihood of all the other models. Other criteria have been proposed to approximate the minimization of the number of classification errors [&make_named_href('', "node31.html#Juang92","[70]"), &make_named_href('', "node31.html#Leprieur95","[71]")].

A gradient-based numerical optimization method is generally used with these criteria: the EM algorithm cannot be used in general (an exception is the synchronous or asynchronous Input-Output HMM with discrete observations, described in section 5).


next up previous
Next: Imbalance Between Emission and Up: Speech Recognition with HMMs Previous: Performance

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