In this style of model training, a set of training observations is used to estimate the parameters of a single HMM. The basic formula for the reestimation of the transition probabilities is
where 1<i<N and 1<j<N and is the total probability of the r'th observation. The transitions from the non-emitting entry state are reestimated by
where 1<j<N and the transitions from the emitting states to the final non-emitting exit state are reestimated by
For a HMM with mixture components in stream s, the means, covariances and mixture weights for that stream are reestimated as follows. Firstly, the probability of occupying the m'th mixture component in stream s at time t for the r'th observation is
For single Gaussian streams, the probability of mixture component occupancy is equal to the probability of state occupancy and hence it is more efficient in this case to use
Given the above definitions, the re-estimation formulae may now be expressed in terms of as follows.