# Equivalence between LC-CRF and HMM, and Discriminative Computing of HMM-Based MPM and MAP
## 5 Discussion and Conclusions  
We have proposed two results. First, we have shown that the basic LC-CRF (Equation (3)) is equivalent to a classical HMM (Equation (1)) in that one can find an HMM whose posterior distribution is exactly the given LC-CRF. More precisely, we specified the way to calculate the parameters \(p(x_{1})\), \(p(x_{n+1}|x_{n})\), and \(p(y_{n}|x_{n})\) which define Equation (1) from the parameters \(V_{n}(x_{n},x_{n+1})\), \(U_{n}(x_{n},y_{n})\) which define Equation (3). Second, noting that all Bayesian classifiers are discriminative in the sense that they do not depend on the observation distribution, we showed that classifiers based on HMMs, usually considered as \"generative\", can also be considered as discriminative classifiers. More specifically, we have proposed discriminative methods for computing classic maximum posterior mode (MPM) and maximum a posteriori (MAP) classifiers based on HMMs. The first result shows that LC-CRFs are as general as classical HMMs. The second shows that at the application level, HMMs offer strictly the same processing power, at least with regard to MPM and MAP, as LC-CRFs.  
The practical interest of our contributions is as follows. Until now, some authors considered LC-CRFs and HMMs to be equivalent without presenting rigorous proof, and others considered LC-CRFs to be more general [19]. Partly because of this uncertainty, CRFs have often been preferred over HMMs. We have proved, at least in the particular framework considered, that the two models were equivalent, and the abandonment of HMMs in favor of CRFs was not always justified. In other words, faced with a particular application, there is no reason to choose CRFs systematically. However, we also cannot say that HMMs should be chosen. Finally, our contribution is likely to encourage practitioners to consider both models, or rather, according to Remark 3, both parametrizations of the same model, on an equal footing.  
We considered basic LC-CRFs and HMMs, which are a limited, yet widely used framework. LC-CRFs and HMMs can be extended in different directions. The general question to study is to search an extension of HMM which would be equivalent to the general CRF (Equation (2)). This seems to be a hard problem. However, one can study some existing extensions of HMMs and wonder what kind of CRFs they would be equivalent to. For example, HMMs have been extended to \"pairwise Markov models\" (PMMs [23]), and the question is therefore what kind of CRFs would be equivalent to PMMs? Another kind of extension consists of adding a latent process \(U_{1:N}\) to the pair \((X_{1:N},Y_{1:N})\). In the case of CRFs this leads to hidden CRFs (HCRFs [24]), and in the case of HMMs this leads to triplet Markov models (TMMs [25]). Finally, CRFs were generalized to semi-Markov CRFs [26], and HMMs were generalized to hidden semi-Markov models, with explicit distribution of the exact sojourn time in a given state [27], or with explicit distribution of the minimal sojourn time in a given state [28]. The study of the relations between these different extensions of CRFs and HMMs is an interesting perspective for further investigations.  
Conceptualization, E.A., E.M., and W.P.; methodology, E.A., E.M., and W.P.; validation, E.A., E.M., and W.P.; investigation, E.A., and W.P.; writing--original draft preparation, W.P.; writing--review and editing, E.A., E.M., and W.P; supervision, W.P.; project administration, W.P. All authors have read and agreed to the published version of the manuscript.  
This research was partly funded by the French Government Agency ASSOCIATION NATIONAIC RECHERCHE TECHNOLOGIES (ANRT), as part of the CIFRE (\"Conventions industrielles de formation par la recherche\") scholarship.