A modified Baum-Welch algorithm for hidden Markov models with multiple observation spaces

We derive an algorithm similar to the well-known Baum-Welch (1970) algorithm for estimating the parameters of a hidden Markov model (HMM). The new algorithm allows the observation PDF of each state to be defined and estimated using a different feature set. We show that estimating parameters in this manner is equivalent to maximizing the likelihood function for the standard parameterization of the HMM defined on the input data space. The processor becomes optimal if the state-dependent feature sets are sufficient statistics to distinguish each state individually from a common state.