  The maximum likelihood degree (ML degree) measures the algebraic complexity
of a fundamental optimization problem in statistics: maximum likelihood
estimation. In this problem, one maximizes the likelihood function over a
statistical model. The ML degree of a model is an upper bound to the number of
local extrema of the likelihood function and can be expressed as a weighted sum
of Euler characteristics. The independence model (i.e. rank one matrices over
the probability simplex) is well known to have an ML degree of one, meaning
their is a unique local maxima of the likelihood function. However, for
mixtures of independence models (i.e. rank two matrices over the probability
simplex), it was an open question as to how the ML degree behaved. In this
paper, we use Euler characteristics to prove an outstanding conjecture by
Hauenstein, the first author, and Sturmfels; we give recursions and closed form
expressions for the ML degree of mixtures of independence models.
