  We study the maximum likelihood degree (ML degree) of toric varieties, known
as discrete exponential models in statistics. By introducing scaling
coefficients to the monomial parameterization of the toric variety, one can
change the ML degree. We show that the ML degree is equal to the degree of the
toric variety for generic scalings, while it drops if and only if the scaling
vector is in the locus of the principal $A$-determinant. We also illustrate how
to compute the ML estimate of a toric variety numerically via homotopy
continuation from a scaled toric variety with low ML degree. Throughout, we
include examples motivated by algebraic geometry and statistics. We compute the
ML degree of rational normal scrolls and a large class of Veronese-type
varieties. In addition, we investigate the ML degree of scaled Segre varieties,
hierarchical loglinear models, and graphical models.
