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

Matroid Stratification of ML Degrees of Independence Models