Abstract
A straightforward ''collision'' (quadratic) entropy estimator is used to give support, in a very pragmatic approach, to data analysis based on the concept of effective cardinality of sets. Thus, by using basic concepts of probability and set theories, a method is proposed to estimate the minimum probability of classification error, in two-class problems, without the deployment of any particular classifier. The usefulness as well as some limitations of the analysis based on effective cardinality are exemplified throughout the text.
Highlights
A fundamental problem in pattern recognition is the optimization of classifiers
We propose an approach to estimate the minimum probability of classification error through cardinality of sets and simple probability concepts, in the manner of early works on information theory
These results strongly corroborates the intuition that the quadratic entropy, being related to collision detection – which turns out to be a matter of fundamental comparison between instances of random variables – would be more closely related to classification error
Summary
A fundamental problem in pattern recognition is the optimization of classifiers. Whenever the criterion is the minimization of probability of misclassification, it is well known that the optimum performance is reached by the Bayesian classifier [1]. Timate of Pr(error) for classes with uniform distribution, whereas the estimator based on order one cardinalities (related to the Shannon entropy) clearly fails1 These results strongly corroborates the intuition that the quadratic entropy, being related to collision detection – which turns out to be a matter of fundamental comparison between instances of random variables – would be more closely related to classification error. It should be noticed that the computation of Pr(error) is prone to strong deviations in practical situations where an experimenter has a limited amount of instances of each random variable In this case, a straightforward approach would be that of approximating the Bayesian classifier with explicit estimates of fX and fY which, in turn, yields empirically approximated optimum classification boundaries between (possibly disjoint) regions RX or RY. As a matter of fact, coincidence is probably the most fundamental concept in cognition, and the proposed method can be applied wherever coincidence is defined – nothing else is required
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