Approximations to the MMI criterion and their effect on lattice-based MMI

Maximum mutual information (MMI) is a model selection criterion used for hidden Markov model (HMM) parameter estimation that was developed more than twenty years ago as a discriminative alternative to the maximum likelihood criterion for HMM-based speech recognition. It has been shown in the speech recognition literature that parameter estimation using the current MMI paradigm, lattice-based MMI, consistently outperforms maximum likelihood estimation, but this is at the expense of undesirable convergence properties. In particular, recognition performance is sensitive to the number of times that the iterative MMI estimation algorithm, extended Baum-Welch, is performed. In fact, too many iterations of extended Baum-Welch will lead to degraded performance, despite the fact that the MMI criterion improves at each iteration. This phenomenon is at variance with the analogous behavior of maximum likelihood estimation -- at least for the HMMs used in speech recognition -- and it has previously been attributed to `over fitting'. In this paper, we present an analysis of lattice-based MMI that demonstrates, first of all, that the asymptotic behavior of lattice-based MMI is much worse than was previously understood, i.e. it does not appear to converge at all, and, second of all, that this is not due to `over fitting'. Instead, we demonstrate that the `over fitting' phenomenon is the result of standard methodology that exacerbates the poor behavior of two key approximations in the lattice-based MMI machinery. We also demonstrate that if we modify the standard methodology to improve the validity of these approximations, then the convergence properties of lattice-based MMI become benign without sacrificing improvements to recognition accuracy.