  When learning from positive and unlabelled data, it is a strong assumption
that the positive observations are randomly sampled from the distribution of
$X$ conditional on $Y = 1$, where X stands for the feature and Y the label.
Most existing algorithms are optimally designed under the assumption. However,
for many real-world applications, the observed positive examples are dependent
on the conditional probability $P(Y = 1|X)$ and should be sampled biasedly. In
this paper, we assume that a positive example with a higher $P(Y = 1|X)$ is
more likely to be labelled and propose a probabilistic-gap based PU learning
algorithms. Specifically, by treating the unlabelled data as noisy negative
examples, we could automatically label a group positive and negative examples
whose labels are identical to the ones assigned by a Bayesian optimal
classifier with a consistency guarantee. The relabelled examples have a biased
domain, which is remedied by the kernel mean matching technique. The proposed
algorithm is model-free and thus do not have any parameters to tune.
Experimental results demonstrate that our method works well on both generated
and real-world datasets.
