Classification on convex sets in the presence of missing covariates

TL;DR

This paper studies statistical classification on convex sets with missing covariates, deriving optimal classifiers, and demonstrating the consistency of convex hull estimators and plug-in classifiers in such scenarios.

Contribution

It derives the form of the optimal classifier for convex sets with missing data and proves the Bayes consistency of convex hull estimators and plug-in classifiers.

Findings

Optimal classifier form for convex sets with missing covariates

Convex hulls are consistent estimators of underlying convex sets

Plug-in classifiers are Bayes consistent in this setting

Abstract

A number of results related to statistical classification on convex sets are presented. In particular, the focus is on the case where some of the covariates in the data and observation being classified can be missing. The form of the optimal classifier is derived when the class-conditional densities are uniform over convex regions. In practice, the underlying convex sets are often unknown and must be estimated with a set of data. In this case, the convex hull of a set of points is shown to be a consistent estimator of the underlying convex set. The problem of estimation is further complicated since the number of points in each convex hull is itself a random variable. The corresponding plug-in version of the optimal classifier is derived and shown to be Bayes consistent.