Model-free multiclass conditional probability estimation via quantile regression

Conventional multiclass conditional probability estimation often requires distributional model assumptions such as Fisher's discriminate analysis and logistic regression. In this paper,a model-free estimation method is proposed to estimate multiclass conditional probability through a series of conditional quantile regression functions. Specifically, each quantile regression is constructed by optimizing a regularized formulation in a reproducing kernel Hilbert space, and then the conditional class probability can be well estimated by converting the quantile regression functions to the corresponding cumulative distribution functions. The primary advantage of the proposed estimation method is that it does not assume any distributional assumption and its computation cost does not increase with the number of classes. The theoretical and numerical studies demonstrate that the proposed estimation method is highly competitive against other alternatives, especially when the number of classes is relatively large.