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We study the maximum likelihood degree (ML degree) of toric varieties, known as discrete exponential models in statistics. By introducing scaling coefficients in the monomial parametrization of the toric variety, one can change the ML degree. We show that the ML degree is equal to the degree of the toric variety for generic scalings, while it drops if and only if the scaling vector is in the locus of the principal A-determinant. We compute the ML degree of rational normal scrolls and a large class of Veronese type varieties. In addition we investigate the ML degree of scaled Segre varieties, hierarchical loglinear models, and graphical models.