Nonparametric “anti-Bayesian” quantile-based pattern classification

Abstract

Parametric and nonparametric pattern recognition have been studied for almost a century based on a Bayesian paradigm, which is, in turn, founded on the principles of Bayes theorem. It is well known that the accuracy of the Bayes classifier cannot be exceeded. Typically, this reduces to comparing the testing sample to mean or median of the respective distributions. Recently, Oommen and his co-authors have presented a pioneering and non-intuitive paradigm, namely that of achieving the classification by comparing the testing sample with another descriptor, which could also be quite distant from the mean. This paradigm has been termed as being “anti-Bayesian,” and it essentially uses the quantiles of the distributions to achieve the pattern recognition. Such classifiers attain the optimal Bayesian accuracy for symmetric distributions even though they operate with a non-intuitive philosophy. While this paradigm has been applied in a number of domains (briefly explained in the body of this paper), its application for nonparametric domains has been limited. This paper explains, in detail, how such quantile-based classification can be extended to the nonparametric world, using both traditional and kernel-based strategies. The paper analyzes the methodology of such nonparametric schemes and their robustness. From a fundamental perspective, the paper utilizes the so-called large sample theory to derive strong asymptotic results that pertain to the equivalence between the parametric and nonparametric schemes for large samples. Apart from the new theoretical results, the paper also presents experimental results demonstrating their power. These results pertain to artificial data sets and also involve a real-life breast cancer data set obtained from the University Hospital Centre of Coimbra. The experimental results clearly confirm the power of the proposed “anti-Bayesian” procedure, especially when approached from a nonparametric perspective.