  We study the maximum likelihood (ML) degree of discrete exponential
independence models and models defined by the second hypersimplex. For models
with two independent variables, we show that the ML degree is an invariant of a
matroid associated to the model. We use this description to explore ML degrees
via hyperplane arrangements. For independence models with more variables, we
investigate the connection between the vanishing of factors of its principal
$A$-determinant and its ML degree. Similarly, for models defined by the second
hypersimplex, we determine its principal $A$-determinant and give computational
evidence towards a conjectured lower bound of its ML degree.
