One-dimensional Models of ML Degree One: Algebraic Statistics - Meets CR Geometry

A prominent success story in algebraic statistics is the study of maximum likelihood (ML) estimation for statistical models as an algebraic optimization problem. The algebraic complexity of this problem is measured by a model invariant known as the ML degree. When the ML degree is one, the model admits a rational maximum likelihood estimator as a function of the data. A classification of one-dimensional discrete models of ML degree one was given by Bik and Marigliano, where they conjectured that the degree of any such model is bounded above by a linear function in the size of its support. We prove this conjecture, therefore showing that there are only finitely many fundamental such models for any given number of states. We show how the enumeration of fundamental models is closely related to counting classes of monomial maps between unit spheres, a topic extensively studied by Cauchy-Riemann geometry.

This is joint work with Carlos Améndola and Viet Duc Nguyen.