The Maximum Likelihood Degree of Mixtures of Independence Models

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 an algebraic 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 $1$ matrices over the probability simplex) is well known to have an ML degree of one, meaning there is a unique local maximum of the likelihood function. However, for mixtures of independence models (i.e., rank $2$ 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.