In the field of algebraic statistics, we view statistical models as part of an algebraic variety and use tools from algebra, geometry, and combinatorics to learn statistically relevant information about these models. In this talk, we discuss the algebraic interpretation of likelihood inference for discrete statistical models. We present recent work on the iterative proportional scaling (IPS) algorithm, which is used to compute the maximum likelihood estimate (MLE), and give algebraic conditions under which this algorithm outputs the exact MLE in one cycle. Next, we introduce quasi-independence models, which describe the joint distribution of two random variables where some combinations of their states cannot co-occur, but they are otherwise independent. We combinatorially classify the quasi-independence models whose MLEs are rational functions of the data. We show that each of these has a parametrization which satisfies the conditions that guarantee one-cycle convergence of the IPS algorithm.