A priori knowledge about an application (such as speech and language) may be used to impose a structure on an HMM and a meaning for the values of the state variable. We have already seen that each state may be associated with a certain label. Furthermore, the topology of the HMM can be strongly constrained: most transition probabilities are forced to be zero. Since the number of free parameters and the amount of computation are directly dependent on the number of non-zero transition probabilities, imposing such structure is very useful when it is appropriate. Furthermore, imposing such structure can almost completely answer one of the most difficult questions in constructing an HMM (including not only its parameters but also its structure): what should the hidden state represent? The most basic structure that is often imposed on HMMs is the left-to-right structure: states (e.g. within a word HMM) are ordered sequentially and transitions go from the ``left'' to the ``right'', or from a state to itself, as in Figures 3, 4 and 6.
The set of states is generally partitioned in subsets to which a particular
linguistic meaning is attached (e.g., phoneme or word, in a particular
context). An example of the topology of a part of an HMM for speech
recognition is shown in Figure 6. To reduce the number of
free parameters and help generalize, designers of speech recognition HMMs use
the notion of a speech unit [&make_named_href('',
"node31.html#Lee89","[49]")] (representing a particular
linguistic meaning and the associated distribution on acoustic subsequences)
which can be re-used (or shared) in many different places of an HMM. The
simplest set of speech units one can think of is simply a set phonemes. For
example, a speech unit for phoneme /a/ may be used in several higher-level
units such as words that contain an /a/ in their linguistic definition. More
complex speech units are context-dependent: they represent the acoustic
realization of a linguistic unit in the context (left or right) of other
linguistic units. For example, a triphone denoted /
/, could
represent the phoneme /o/ when it is preceded by the phoneme /d/ and followed
by the phoneme /g/. In that case, there could be transitions from the
triphone /
/ to any of the triphones of the form /
/ (where x
represent any phonetic label).
Figure 6 illustrates the simpler case of a context-independent model, in which each word model is simply represented by a sequence of phonemes: each word model can be built as a concatenation of corresponding speech units. If there are multiple pronunciations for a word, then a word model would be made of several of these concatenations in parallel.
By imposing such a meaning on the values of the state variable, very strong probabilistic assumptions on the relation between the speech signal and the word sequence are made, and these assumptions are known to be wrong. These assumptions, however, have been found very useful in practice, in order to build the current state-of-the-art speech recognition systems. The focus of much current research in this field is therefore to build more faithful models (while keeping them tractable), or make sure that the imperfections of the model do not hurt too much the final decision taking. See section 3.7 for a discussion of these unrealistic assumptions.