Higher-dimensional parameterization of ML-degree-one discrete models

Determine whether every r-dimensional discrete model of maximum likelihood degree one admits a parameterization of the form p:[0,1]^r→Δ_n with coordinates c_i t_1^{ν_{1i}}⋯t_r^{ν_{ri}}(1−t_1−⋯−t_r)^{ν_{(r+1)i}}, where the exponents are nonnegative and the coefficients are positive.

Background

The paper proves that discrete models with the displayed simplex-monomial parameterization have maximum likelihood degree one. It then asks whether this class exhausts all higher-dimensional discrete models of maximum likelihood degree one. The question is explicitly left unresolved and is presented as a generalization of the paper’s one-dimensional classification setting.

References

While the parameterization given in the following proposition yields higher-dimensional discrete models with rational ML estimator, it is open whether every discrete model of ML degree one has such a parameterization.

— One-dimensional Discrete Models of Maximum Likelihood Degree One  (2507.18686 - Améndola et al., 24 Jul 2025) in Section 6, “Outlook,” paragraph following Proposition 6.1