The relative value of labeled and unlabeled samples in pattern recognition with an unknown mixing parameter

We observe a training set Q composed of l labeled samples {(X(1).theta(1)),...(X(l),theta(l))} and u unlabeled samples {X'(1),...X'(u)}. The labels theta(i) are independent random variables satisfying Pr {theta 2--- = 1} = eta, Pr {theta(i) = 2} = 1 - eta. The labeled observations X(2) are independently distributed with conditional density f(theta i)(.) given theta(2). Let (X(0), theta(0)) be a new sample, independently distributed as the samples in the training set. We observe X(0) and we wish to infer the classification theta(0). In this paper we first assume that the distributions f(1)(.) and f(2)(.) are given and that the mixing parameter eta is unknown, We show that the relative value of labeled and unlabeled samples in reducing the risk of optimal classifiers is the ratio of the Fisher informations they carry about the parameter eta. We then assume that two densities g(1)(.) and g(2)(.) are given, but we do not know whether g(1)(.) = f(1)(.) and g(2)(.) = f(2)(.) or if the opposite holds, nor do we know eta. Thus the learning problem consists of both estimating the optimum partition of the observation space and assigning the classifications to the decision regions, Here, we show that labeled samples are necessary to construct a classification rule and that they are exponentially more valuable than unlabeled samples.