A note on the binomial model with simplex constraints

Liu (2000) considered maximum likelihood estimation and Bayesian estimation in a binomial model with simplex constraints using the expectation–maximization (EM) and data augmentation   (DA) algorithms. By introducing latent variables {Zij}{Zij} and {Yij}{Yij} (to be defined later), he formulated the constrained parameter problem into a missing data problem. However, the derived DA algorithm does not work because he actually assumed that the {Yij}{Yij} are known. Furthermore, although the final results from the derived EM algorithm are correct, his findings are based on the assumption that the {Yij}{Yij} are observable. This note provides a correct DA algorithm. In addition, we obtained the same E-step and M-step under the assumption that the {Yij}{Yij} are unobservable. A real example is used for illustration.

► We considered maximum likelihood estimation and Bayesian estimation in a binomial model with simplex constraints.
► We corrected some errors in Liu (2000, JASA), who formulated the constrained parameter problem as a missing data problem.
► We provided correct expectation–maximization and data augmentation algorithms.