Classification rules in the unknown mixture parameter case:relative value of labeled and unlabeled samples

摘要

We investigate the relative value of labeled and unlabeled samplesin constructing classification rules. We observe a training set Qcomposed of l labeled and u unlabeled samples coming from two classes.Let sample from class 1 be distributed according tof₁(·), samples from class 2 according tof₂(·), and let η be the probability that a sampleis in class 1. Assume that f₁(·) andf₂(·) are known and that η is unknown. We want toclassify a new sample X₀. The relative value of labeled andunlabeled observations in reducing the probability of error is equal toI_t(η)/I_u(η), the ratio of the Fisherinformation of the labeled and unlabeled samples. Moreover labeledsamples are not necessary in order to construct a decision rule.However, if f₁(·) and f₂(·) aregiven, but it is not known whether observations from class 1 aredistributed according to f₁(·) or according to f₂(·), then the labeled samples are necessary andexponentially more valuable than unlabeled samples