Maximum likelihood degree of various toric varieties

Before we can speak of the maximum likelihood degree (ML degree), we should know the story of the maximum likelihood estimate (ML estimate). The ML estimate of given data (collected from sample realizations of the discrete random variable) with respect to a statistical model is a choice of parameters which maximizes the probability of observing these data. Here we define a statistical model for a discrete random variable as a subset of the probability simplex and a parametric statistical model is defined as the image of the parameter space. In this thesis we work in the realms of algebraic geometry and algebraic statistics. In algebraic statistics, the models are images of the parameter space under a polynomial map to the probability simplex. In general, finding the ML estimate with respect to a model is accomplished by maximizing the likelihood function. For an algebraic statistical model, this function is the product of powers of polynomials, where the polynomials are the coordinates of the polynomial map. One way of finding the ML estimate is to solve the critical equations of the likelihood function, subject to the constraint that the image of the map defining the model is indeed in the probability simplex. In this thesis we define the maximum likelihood degree of an algebraic statistical model as the number of complex critical points of the likelihood function. We mainly work with a large class of statistical models which are known as discrete exponential models and are defined by a monomial parametrization. In algebraic statistics, these models are known as toric varieties, and they have been studied extensively. Toric varieties form an important and rich class of examples in algebraic geometry, and our focus is the computation of the ML degree for some of them. In particular, we study rational normal curves, certain Veronese varieties, and some toric surfaces known as Hirzebruch surfaces. In our work we prove that for rational normal curves, the second Veronese embedding, and for the second hypersimplex varieties, the ML degree is equal to the degree of the respective varieties. In general, this result does not hold. For instance, in the case of Hirzebruch surfaces, we compute examples where the ML degree is smaller than the degree of the surface. Furthermore we also consider a family of rational normal curves obtained by scaling coordinates. We give a stratification of this family with respect to the ML degree by determining sets of scaling parameters for which the ML degree takes particular values. Finally, we present conjectures and computations for further study.