Finite mixture distributions arise naturally in many applications including clustering and classification. Since they usually do not yield closed forms for maximum likelihood estimates (MLEs), numerical methods using the well known Fisher Scoring or Expectation-Maximization algorithms are considered. In this work, an approximation to the Fisher Information Matrix of an arbitrary mixture of multinomial distributions is introduced. This leads to an Approximate Fisher Scoring algorithm (AFSA), which turns out to be closely related to Expectation-Maximization, and is more robust to the choice of initial value than Fisher Scoring iterations. A combination of AFSA and the classical Fisher Scoring iterations provides the best of both computational efficiency and stable convergence properties.