Bounds on the Bayes Error Given Moments

TL;DR

This paper develops methods to compute bounds on the maximum possible Bayes error under moment constraints, revealing limitations of Gaussian assumptions and providing bounds with linear decision boundaries.

Contribution

It introduces a novel approach using truncated moment problem solutions to bound the supremum Bayes error under moment constraints.

Findings

Gaussian assumption is not robust for Bayes error bounds

Provides a method to compute lower bounds on Bayes error

Constructs an upper bound with linear decision boundary

Abstract

We show how to compute lower bounds for the supremum Bayes error if the class-conditional distributions must satisfy moment constraints, where the supremum is with respect to the unknown class-conditional distributions. Our approach makes use of Curto and Fialkow's solutions for the truncated moment problem. The lower bound shows that the popular Gaussian assumption is not robust in this regard. We also construct an upper bound for the supremum Bayes error by constraining the decision boundary to be linear.