  For a multinomial distribution, suppose that we have prior knowledge of the
sum of the probabilities of some categories. This allows us to construct a
submodel in a full (i.e., no-restriction) model. Maximum likelihood estimation
(MLE) under this submodel is expected to have better estimation efficiency than
MLE under the full model. This article presents the asymptotic expansion of the
risk of MLE with respect to Kullback--Leibler divergence for both the full
model and submodel. The results reveal that, using the submodel, the reduction
of the risk is quite small in some cases. Furthermore, when the sample size is
small, the use of the subomodel can increase the risk.
