title: On optimal Bayesian classification and risk estimation under multiple classes.
abstract: A recently proposed optimal Bayesian classification paradigm addresses optimal error rate analysis for small-sample discrimination, including optimal classifiers, optimal error estimators, and error estimation analysis tools with respect to the probability of misclassification under binary classes. Here, we address multi-class problems and optimal expected risk with respect to a given risk function, which are common settings in bioinformatics. We present Bayesian risk estimators (BRE) under arbitrary classifiers, the mean-square error (MSE) of arbitrary risk estimators under arbitrary classifiers, and optimal Bayesian risk classifiers (OBRC). We provide analytic expressions for these tools under several discrete and Gaussian models and present a new methodology to approximate the BRE and MSE when analytic expressions are not available. Of particular note, we present analytic forms for the MSE under Gaussian models with homoscedastic covariances, which are new even in binary classification.
authors.affiliation: Department of Electrical and Computer Engineering, The Ohio State University, Columbus, 43210 OH USA ; Department of Biomedical Informatics, The Ohio State University, Columbus, 43210 OH USA.
authors.forename: Lori A
authors.initials: LA
authors.lastname: Dalton
authors.affiliation: Department of Electrical and Computer Engineering, The Ohio State University, Columbus, 43210 OH USA.
authors.forename: Mohammadmahdi R
authors.initials: MR
authors.lastname: Yousefi
keywords: Bayesian estimation; Genomics; Minimum mean-square error; Multi-class classification; Risk estimation; Small samples