  Multiclass probability estimation is the problem of estimating conditional
probabilities of a data point belonging to a class given its covariate
information. It has broad applications in statistical analysis and data
science. Recently a class of weighted Support Vector Machines (wSVMs) has been
developed to estimate class probabilities through ensemble learning for
$K$-class problems (Wu, Zhang and Liu, 2010; Wang, Zhang and Wu, 2019), where
$K$ is the number of classes. The estimators are robust and achieve high
accuracy for probability estimation, but their learning is implemented through
pairwise coupling, which demands polynomial time in $K$. In this paper, we
propose two new learning schemes, the baseline learning and the One-vs-All
(OVA) learning, to further improve wSVMs in terms of computational efficiency
and estimation accuracy. In particular, the baseline learning has optimal
computational complexity in the sense that it is linear in $K$. Though not
being most efficient in computation, the OVA offers the best estimation
accuracy among all the procedures under comparison. The resulting estimators
are distribution-free and shown to be consistent. We further conduct extensive
numerical experiments to demonstrate finite sample performance.
