A Probability Model for Errors of Classification. I. General Considerations

Abstract

This paper seeks to meet the need for a general treatment of the problem of error in classification. Within an m-attribute classificatory system, an object's typical subclass is that subclass to which it is most often allocated under repeated experimentally independent applications of the classificatory criteria. In these terms, an error of classification is an atypical subclass allocation. This leads to definition of probabilities O of occasional subclass membership, probabilities T of typical subclass membership, and probabilities E of error or, more generally, occasional subclass membership conditional upon typical subclass membership. In the relationship f: (O, T, E) the relative incidence of independent O, T, and E values is such that generally one can specify O values given T and E, but one cannot generally specify T and E values given O. Under the restrictions of homogeneity of E values for all members of a given typical subclass, mutual stochastic independence of errors of classification, and suitable conditions of replication, one can find particular systems O = f (T, E) which are solvable for T and E given O. A minimum of three replications of occasional classification is necessary for a solution of systems for marginal attributes, and a minimum of two replications is needed with any cross-classification. Although for such systems one can always specify T and E values given O values, the solution is unique for dichotomous systems only.