Incomplete categorical data often occur in the fields such as biomedicine, epidemiology, psychology and sports. In this paper, we first develop a novel minorization-maximization (MM) algorithm for calculating the MLE of parameters for general incomplete categorical data. How to develop the corresponding stochastic counterparts to existing MM algorithms is an important research topic. Up to now, we have not seen any papers that proposed a stochastic version of an MM algorithm. This is the first paper to propose a mode-sharing method in Bayesian computation for general incomplete categorical data by developing a new acceptance-rejection (AR) algorithm aided with the proposed MM algorithm. The key idea is to construct a class of envelope densities indexed by a working parameter and to identify a specific envelope density which can overcome the four drawbacks associated with the traditional AR algorithm.  Finally, four real data sets are used to illustrate the proposed methodologies.