We study the maximum likelihood estimation problem for several classes of toric
Fano models. We start by exploring the maximum likelihood degree for all
-dimensional
Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of
the surface in every case except for the quintic del Pezzo surface with two ordinary
double points and provide explicit expressions that allow one to compute
the maximum likelihood estimate in closed form whenever the ML degree
is less than 5. We then explore the reasons for the ML degree drop using
-discriminants
and intersection theory. Finally, we show that toric Fano varieties associated to
3-valent phylogenetic trees have ML degree one and provide a formula for the
maximum likelihood estimate. We prove it as a corollary to a more general result
about the multiplicativity of ML degrees of codimension zero toric fiber
products, and it also follows from a connection to a recent result about staged
trees.
Keywords
algebraic statistics, maximum likelihood estimation,
maximum likelihood degree, Fano varieties, toric varieties,
toric fiber product