  High-dimensional multinomial regression models are very useful in practice
but receive less research attention than logistic regression models, especially
from the perspective of statistical inference. In this work, we analyze the
estimation and prediction error of the contrast-based $\ell_1$-penalized
multinomial regression model and extend the debiasing method to the multinomial
case, which provides a valid confidence interval for each coefficient and
$p$-value of the individual hypothesis test. We apply the debiasing method to
identify some important predictors in the progression into dementia of
different subtypes. Results of intensive simulations show the superiority of
the debiasing method compared to some other inference methods.
