Contrastive pessimistic likelihood estimation

As explained at the beginning of this chapter, in many real life problems, it's cheaper to retrieve unlabeled samples, rather than correctly labeled ones. For this reason, many researchers worked to find out the best strategies to carry out a semi-supervised classification that could outperform the supervised counterpart. The idea is to train a classifier with a few labeled samples and then improve its accuracy after adding weighted unlabeled samples. One of the best results is the Contrastive Pessimistic Likelihood Estimation (CPLE) algorithm, proposed by M. Loog (in Loog M., Contrastive Pessimistic Likelihood Estimation for Semi-Supervised Classification, arXiv:1503.00269).

Before explaining this algorithm, an introduction is necessary. If we have a labeled dataset (X, Y) containing N samples, it's possible to define the log-likelihood cost function of a generic estimator, as follows:

After training the model, it should be possible to determine...