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.

翻译：我们考虑由一套共同的解决方案产生的统计模型,这些模型来自具有一般系数的稀疏多元海洋系统的共同解决方案。最大可能性度计为该模型限值的概率函数临界点数。我们证明稀疏多元海洋系统的最大可能性由它的牛顿多面体决定,等于相关拉格朗格方程的混合体积。作为必然结果,我们发现若干优化问题的代数度等于相似的混合体积。