  Classification is an important statistical learning tool. In real
application, besides high prediction accuracy, it is often desirable to
estimate class conditional probabilities for new observations. For traditional
problems where the number of observations is large, there exist many well
developed approaches. Recently, high dimensional low sample size problems are
becoming increasingly popular. Margin-based classifiers, such as logistic
regression, are well established methods in the literature. On the other hand,
in terms of probability estimation, it is known that for binary classifiers,
the commonly used methods tend to under-estimate the norm of the classification
function. This can lead to biased probability estimation. Remedy approaches
have been proposed in the literature. However, for the simultaneous
multicategory classification framework, much less work has been done. We fill
the gap in this paper. In particular, we give theoretical insights on why heavy
regularization terms are often needed in high dimensional applications, and how
this can lead to bias in probability estimation. To overcome this difficulty,
we propose a new refit strategy for multicategory angle-based classifiers. Our
new method only adds a small computation cost to the problem, and is able to
attain prediction accuracy that is as good as the regular margin-based
classifiers. On the other hand, the improvement of probability estimation can
be very significant. Numerical results suggest that the new refit approach is
highly competitive.
