Learning from a mixture of labeled and unlabeled examples with parametric side information

We investigate the tradeoff between labeled and unlabeled sample complexities in learning a classification rule for a parametric two-class problem. In the problem considered, a sample of m labeled examples and n unlabeled examples generated from a two-class, N-variate Gaussian mixture is provided together with side information specifying the parametric form of the probability densities. The class means and a priori class probabilities are, however, unknown parameters. In this framework we use the maximum likelihood estimation method to estimate the unknown parameters and utilize rates of convergence of uniform strong laws to determine the tradeoff between error rate and sample complexity. In particular, we show that for the algorithm used, the misclassification probability deviates from the minimal Bayes error rate by O(N^3/5n^-1/5) + O(e^-cm) where N is the dimension of the feature space, m is the number of labeled examples, n is the number of unlabeled examples, and c is a positive constant.