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
Designing experiments for generalized linear models is difficult because optimal designs depend on unknown parameters. Here we investigate... Show moreDesigning experiments for generalized linear models is difficult because optimal designs depend on unknown parameters. Here we investigate local optimality. We propose to study for a given design its region of optimality in parameter space. Often these regions are semi-algebraic and feature interesting symmetries. We demonstrate this with the Rasch Poisson counts model. For any given interaction order between the explanatory variables we give a characterization of the regions of optimality of a special saturated design. This extends known results from the case of no interaction. We also give an algebraic and geometric perspective on optimality of experimental designs for the Rasch Poisson counts model using polyhedral and spectrahedral geometry. Show less
