  Multinomial mixtures are widely used in the information engineering field,
however, their mathematical properties are not yet clarified because they are
singular learning models. In fact, the models are non-identifiable and their
Fisher information matrices are not positive definite. In recent years, the
mathematical foundation of singular statistical models are clarified by using
algebraic geometric methods. In this paper, we clarify the real log canonical
thresholds and multiplicities of the multinomial mixtures and elucidate their
asymptotic behaviors of generalization error and free energy.
