  Relations between categorical variables can be analyzed conveniently by
multiple correspondence analysis (MCA). %It is well suited to discover
relations that may exist between categories of different variables. The
graphical representation of MCA results in so-called biplots makes it easy to
interpret the most important associations. However, a major drawback of MCA is
that it does not have an underlying probability model for an individual
selecting a category on a variable. In this paper, we propose such probability
model called multinomial multiple correspondence analysis (MMCA) that combines
the underlying low-rank representation of MCA with maximum likelihood. An
efficient majorization algorithm that uses an elegant bound for the second
derivative is derived to estimate the parameters. The proposed model can easily
lead to overfitting causing some of the parameters to wander of to infinity. We
add the nuclear norm penalty to counter this issue and discuss ways of
selecting regularization parameters. The proposed approach is well suited to
study and vizualise the dependences for high dimensional data.
