We study the problem of maximum likelihood (ML) estimation for statistical
models defined by reflexive polytopes. Our focus is on the maximum
likelihood degree of these models as an algebraic measure of complexity of
the corresponding optimization problem. We compute the ML degrees of
all 4319 classes of three-dimensional reflexive polytopes, and observe some
surprising behavior in terms of the presence of gaps between ML degrees and
degrees of the associated toric varieties. We interpret these drops in the
context of discriminants and prove formulas for the ML degree for families of
reflexive polytopes, including the hypercube and its dual, the cross polytope, in
arbitrary dimension. In particular, we determine a family of embeddings for the
-cube
that implies ML degree one. Finally, we discuss generalized constructions of families
of reflexive polytopes in terms of their ML degrees.
Keywords
likelihood geometry, reflexive polytopes, maximum
likelihood estimation
