The maximum likelihood degree of sparse polynomial systems

We consider statistical models arising from the common set of solutions to a sparse polynomial system with general coefficients. The maximum likelihood degree counts the number of critical points of the likelihood function restricted to the model. We prove the maximum likelihood degree of a sparse polynomial system is determined by its Newton polytopes and equals the mixed volume of a related Lagrange system of equations. As a corollary, we find that the algebraic degree of several optimization problems is equal to a similar mixed volume.