, the likelihood is given by
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The decoding criterion specified in (1)
and (2) require the computation of the likelihood of the
data given a phone (state) sequence. Using the notation , the likelihood is given by
In the interest of computational tractability and ease of training, standard
HMMs make the assumptions of observation independence
and that the Markov process
is first order, i.e., 
. The recurrent hybrid approach, however, makes the less
severe assumption that 
which maintains the acoustic context in the
local observation model. Manipulation of this results in an expression for
the observation likelihood given by
The computation of (9) is straightforward.
The recurrent network is used to estimate 
. Because
is independent of the phone sequence, it
has no effect on the decoding process and is ignored. The one remaining
issue in computing the scaled local likelihood is computation of
. The simplest solution is to assume
where 
is determined from
the relative frequency of the phone 
in the training data*. Although this works well in
practice, it is obviously a wrong assumption and this area deserves
further investigation.
