Abstract: Let $X$ be a smooth subvariety of a complex torus. For general data vectors the MLE problem on $X$ has exactly $|\chi(X)|$ solutions. We investigate what happens to these solutions when the data vector approaches a non-general value. Assuming that $X$ is schön we relate this behaviour to special rays in the tropical variety of $X$. We also explain how the non-general data vectors are related a Bernstein-Sato ideal associated to X. Based on joint work with Anna-Laura Sattelberger.