This paper closely examines HMMs in which all the hidden random variables are... Show moreThis paper closely examines HMMs in which all the hidden random variables are
binary. Its main contributions are (1) a birational parametrization for every such HMM, with an
explicit inverse for recovering the hidden parameters in terms of observables, (2) a semialgebraic
model membership test for every such HMM, and (3) minimal dening equations for the 4-node
fully binary model, comprising 21 quadrics and 29 cubics, which were computed using Grobner
bases in the cumulant coordinates of Sturmfels and Zwiernik. The new model parameters in (1) are
rationally identiable in the sense of Sullivant, Garcia-Puente, and Spielvogel, and each model's
Zariski closure is therefore a rational projective variety of dimension 5. Grobner basis computations
for the model and its graph are found to be considerably faster using these parameters. In the
case of two hidden states, item (2) supersedes a previous algorithm of Schonhuth which is only
generically 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, and
one can hope for similar applications in the case of HMMs. Show less