Elkan et al claims and proves the probability that a sample is positive is dependent on the probability that a sample is labelled times a constant factor. Labelled means known positive, while unlabelled means could be a positive or negative.

In practice, we start with an imbalanced dataset with few positive samples. Then we take a subset of the positive, e.g. 70 % and assign them as positive/labelled, and sample equally as many of the remaining positive and negative sambles, assigning them as negative/unlabelled. Then train our probabilistic classifier. We repeat this process n times to build an ensemble, so we can cover most of our dataset. We then use the ensemble to predict on our test set.

Nb: Since the constant factor P(s=1|y=1) does not affect our AUC calculation, it can be disregarded if the threshold to set is known. Otherwise it can be set as c/2, as per [2].

Testing effects of positive and unlabelled training variables, inspired by [1] and [2]

Findings w/ PU training:

• Performance is largely the same training on ca 5 % to 100 % of positive examples
• PU training relies on relabelling unused positives as negatives.
• If training an ensemble, performance for different positive to negative ratio is best at 1:1 to 1:2
• If training an ensemble, training on all negatives leads to unstable performance when using n % of positives
• Training an ensemble stabilizes performance towards a mean

• [1] Semi-Supervised Classification of Unlabeled Data (PU Learning) https://towardsdatascience.com/semi-supervised-classification-of-unlabeled-data-pu-learning-81f96e96f7cb
• [2] Paper Elkan et al 2008: Learning classifiers from only positive and unlabeled data https://dl.acm.org/doi/10.1145/1401890.1401920