Abstract
This paper closely examines HMMs in which all the hidden random variables arebinary. Its main contributions are (1) a birational parametrization for every such HMM, with anexplicit inverse for recovering the hidden parameters in terms of observables, (2) a semialgebraicmodel membership test for every such HMM, and (3) minimal dening equations for the 4-nodefully binary model, comprising 21 quadrics and 29 cubics, which were computed using Grobnerbases in the cumulant coordinates of Sturmfels and Zwiernik. The new model parameters in (1) arerationally identiable in the sense of Sullivant, Garcia-Puente, and Spielvogel, and each model'sZariski closure is therefore a rational projective variety of dimension 5. Grobner basis computationsfor the model and its graph are found to be considerably faster using these parameters. In thecase of two hidden states, item (2) supersedes a previous algorithm of Schonhuth which is onlygenerically dened, and the dening equations (3) yield new invariants for HMMs of all lengths 4. Such invariants have been used successfully in model selection problems in phylogenetics, andone can hope for similar applications in the case of HMMs.
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