Bayes Error Rate Estimation in Difficult Situations

TL;DR

This paper evaluates various estimators of the Bayes Error Rate (BER) to determine their accuracy and practicality in difficult, high-dimensional classification scenarios, highlighting kNN's superior performance under sample constraints.

Contribution

The study provides a comprehensive comparison of BER estimators using synthetic and real-world scenarios, establishing minimum sample requirements and demonstrating kNN's robustness.

Findings

kNN outperforms other estimators in accuracy

Minimum of 1000 samples per class needed for reliable BER estimation

More features require more samples to maintain estimation accuracy

Abstract

The Bayes Error Rate (BER) is the fundamental limit on the achievable generalizable classification accuracy of any machine learning model due to inherent uncertainty within the data. BER estimators offer insight into the difficulty of any classification problem and set expectations for optimal classification performance. In order to be useful, the estimators must also be accurate with a limited number of samples on multivariate problems with unknown class distributions. To determine which estimators meet the minimum requirements for "usefulness", an in-depth examination of their accuracy is conducted using Monte Carlo simulations with synthetic data in order to obtain their confidence bounds for binary classification. To examine the usability of the estimators for real-world applications, new non-linear multi-modal test scenarios are introduced. In each scenario, 2500 Monte Carlo simulations per scenario are run over a wide range of BER values. In a comparison of k-Nearest Neighbor (kNN), Generalized Henze-Penrose (GHP) divergence and Kernel Density Estimation (KDE) techniques, results show that kNN is overwhelmingly the more accurate non-parametric estimator. In order to reach the target of an under 5% range for the 95% confidence bounds, the minimum number of required samples per class is 1000. As more features are added, more samples are needed, so that 2500 samples per class are required at only 4 features. Other estimators do become more accurate than kNN as more features are added, but continuously fail to meet the target range.