The Maximum Likelihood Degree of Toric Models is Monotonic

We settle a conjecture by Coons and Sullivant stating that the maximum likelihood (ML) degree of a facial submodel of a toric model is at most the ML degree of the model itself. We discuss the impact on the ML degree from observing zeros in the data. Moreover, we connect this problem to tropical likelihood degenerations, and show how the results can be applied to discrete graphical and quasi-independence models.