  Given a statistical model, the maximum likelihood degree is the number of
complex solutions to the likelihood equations for generic data. We consider
discrete algebraic statistical models and study the solutions to the likelihood
equations when the data contain zeros and are no longer generic. Focusing on
sampling and model zeros, we show that, in these cases, the solutions to the
likelihood equations are contained in a previously studied variety, the
likelihood correspondence. The number of these solutions give a lower bound on
the ML degree, and the problem of finding critical points to the likelihood
function can be partitioned into smaller and computationally easier problems
involving sampling and model zeros. We use this technique to compute a lower
bound on the ML degree for $2 \times 2 \times 2 \times 2$ tensors of border
rank $\leq 2$ and $3 \times n$ tables of rank $\leq 2$ for $n=11, 12, 13, 14$,
the first four values of $n$ for which the ML degree was previously unknown.
