Fast and Stable Maximum Likelihood Estimation for Incomplete Multinomial Models

We propose a fixed-point iteration approach to the maximum likelihood estimation of the incomplete multinomial model, which provides a unified framework for analyzing ranking data. Incomplete observations cannot be distinguished as belonging to a unique category, but instead they fall in a subset of categories. We develop an minorize-maximization (MM) type of algorithm, which requires relatively fewer iterations and better time efficiency to achieve convergence. Under such a general framework, incomplete multinomial models can be reformulated to include several well-known ranking models as special cases, such as the Bradley--Terry, Plackett--Luce models and their variants. Experimental results show that our algorithm runs faster than existing methods on synthetic data and real data. The simple form of iteratively updating equations in our algorithm involves only basic matrix operations, which makes it easy to implement with large data.