Supervised learning via smoothed Polya trees

Abstract

We propose a generative classification model that extends Quadratic Discriminant Analysis (QDA) (Cox in J R Stat Soc Ser B (Methodol) 20:215–242, 1958) and Linear Discriminant Analysis (LDA) (Fisher in Ann Eugen 7:179–188, 1936; Rao in J R Stat Soc Ser B 10:159–203, 1948) to the Bayesian nonparametric setting, providing a competitor to MclustDA (Fraley and Raftery in Am Stat Assoc 97:611–631, 2002). This approach models the data distribution for each class using a multivariate Polya tree and realizes impressive results in simulations and real data analyses. The flexibility gained from further relaxing the distributional assumptions of QDA can greatly improve the ability to correctly classify new observations for models with severe deviations from parametric distributional assumptions, while still performing well when the assumptions hold. The proposed method is quite fast compared to other supervised classifiers and very simple to implement as there are no kernel tricks or initialization steps perhaps making it one of the more user-friendly approaches to supervised learning. This highlights a significant feature of the proposed methodology as suboptimal tuning can greatly hamper classification performance; e.g., SVMs fit with non-optimal kernels perform significantly worse.