Bayesian MMSE estimation of classification error and performance on real genomic data

Abstract

Small sample classifier design has become a major issue in the biological and medical communities, owing to the recent development of high-throughput genomic and proteomic technologies. And as the problem of estimating classifier error is already handicapped by limited available information, it is further compounded by the necessity of reusing training-data for error estimation. Due to the difficulty of error estimation, all currently popular techniques have been heuristically devised, rather than rigorously designed based on statistical inference and optimization. However, a recently proposed error estimator has placed the problem into an optimal mean-square error (MSE) signal estimation framework in the presence of uncertainty. This results in a Bayesian approach to error estimation based on a parameterized family of feature-label distributions. These Bayesian error estimators are optimal when averaged over a given family of distributions, unbiased when averaged over a given family and all samples, and analytically address a trade-off between robustness (modeling assumptions) and accuracy (minimum mean-square error). Closed form solutions have been provided for two important examples: the discrete classification problem and linear classification of Gaussian distributions. Here we discuss the Bayesian minimum mean-square error (MMSE) error estimator and demonstrate performance on real biological data under Gaussian modeling assumptions.