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

We investigate the relative value of labeled and unlabeled samples in constructing classification rules. We observe a training set Q composed of l labeled and u unlabeled samples coming from two classes. Let sample from class 1 be distributed according to f₁(·), samples from class 2 according to f₂(·), and let η be the probability that a sample is in class 1. Assume that f₁(·) and f₂(·) are known and that η is unknown. We want to classify a new sample X₀. The relative value of labeled and unlabeled observations in reducing the probability of error is equal to I_t(η)/I_u(η), the ratio of the Fisher information of the labeled and unlabeled samples. Moreover labeled samples are not necessary in order to construct a decision rule. However, if f₁(·) and f₂(·) are given, but it is not known whether observations from class 1 are distributed according to f₁(·) or according to f₂ (·), then the labeled samples are necessary and exponentially more valuable than unlabeled samples