The existence of the maximum likelihood estimate in multinomial logistic regression for mixed-membership models

It is known that the maximum likelihood estimate of a logistic regression model may not exist. Specifically, the maximum likelihood estimate only exists if the dataset is non separable. We show that one method of ensuring its existence is by assigning positive probability to every class in the sample dataset, a type of regularization. This is equivalent to treating logistic regression as a mixed-membership (grade-of-membership) model.