Equivalence between LC-CRF and HMM, and Discriminative Computing of HMM-Based MPM and MAP

Practitioners have used hidden Markov models (HMMs) in different problems for about sixty years. Moreover, conditional random fields (CRFs) are an alternative to HMMs and appear in the literature as different and somewhat concurrent models. We propose two contributions: First, we show that the basic linear-chain CRFs (LC-CRFs), considered as different from HMMs, are in fact equivalent to HMMs in the sense that for each LC-CRF there exists an HMM—that we specify—whose posterior distribution is identical to the given LC-CRF. Second, we show that it is possible to reformulate the generative Bayesian classifiers maximum posterior mode (MPM) and maximum a posteriori (MAP), used in HMMs, as discriminative ones. The last point is of importance in many fields, especially in natural language processing (NLP), as it shows that in some situations dropping HMMs in favor of CRFs is not necessary.