Applying the Bayesian Evidence Framework to \nu -Support Vector Regression

Abstract

Following previous successes on applying the Bayesian evidence framework to support vector classifiers and the ε-support vector regression algorithm, in this paper we extend the evidence framework also to the ν-support vector regression (ν-SVR) algorithm. We show that ν-SVR training implies a prior on the size of the ε-tube that is dependent on the number of training patterns. Besides, this prior has properties that are in line with the error-regulating behavior of ν. Under the evidence framework, standard ν-SVR training can then be regarded as performing level one inference, while levels two and three allow automatic adjustments of the regularization and kernel parameters respectively, without the need of a validation set. Furthermore, this Bayesian extension allows computation of the prediction intervals, taking uncertainties of both the weight parameter and the ε-tube width into account. Performance of this method is illustrated on both synthetic and real-world data sets.