  Mixtures of $r$ independent distributions for two discrete random variables
can be represented by matrices of nonnegative rank $r$. Likelihood inference
for the model of such joint distributions leads to problems in real algebraic
geometry that are addressed here for the first time. We characterize the set of
fixed points of the Expectation-Maximization algorithm, and we study the
boundary of the space of matrices with nonnegative rank at most $3$. Both of
these sets correspond to algebraic varieties with many irreducible components.
