The goal of positive-unlabeled (PU) learning is to train a binary classifier
on the basis of training data containing positive and unlabeled instances,
where unlabeled observations can belong either to the positive class or to the
negative class. Modeling PU data requires certain assumptions on the labeling
mechanism that describes which positive observations are assigned a label. The
simplest assumption, considered in early works, is SCAR (Selected Completely at
Random Assumption), according to which the propensity score function, defined
as the probability of assigning a label to a positive observation, is constant.
On the other hand, a much more realistic assumption is SAR (Selected at
Random), which states that the propensity function solely depends on the
observed feature vector. SCAR-based algorithms are much simpler and
computationally much faster compared to SAR-based algorithms, which usually
require challenging estimation of the propensity score. In this work, we
propose a relatively simple and computationally fast test that can be used to
determine whether the observed data meet the SCAR assumption. Our test is based
on generating artificial labels conforming to the SCAR case, which in turn
allows to mimic the distribution of the test statistic under the null
hypothesis of SCAR. We justify our method theoretically. In experiments, we
demonstrate that the test successfully detects various deviations from SCAR
scenario and at the same time it is possible to effectively control the type I
error. The proposed test can be recommended as a pre-processing step to decide
which final PU algorithm to choose in cases when nature of labeling mechanism
is not known.

基于正无标（PU）学习目标是基于包含正例和无标实例的训练数据训练一个二元分类器，其中无标观察可以属于正类或负类。本文提出了一种简单且计算速度较快的测试方法，用于确定观察数据是否符合 SCAR 假设，并在实验中成功地检测了多种与 SCAR 场景不符的偏差。这种测试方法可以用作预处理步骤，用于在未知标签机制的情况下决定选择哪个最终的 PU 算法。

验证正无标签学习中随机选择完全合理假设

Verifying the Selected Completely at Random Assumption in  Positive-Unlabeled Learning

Learning from positive and unlabeled data (PU learning) is actively
researched machine learning task. The goal is to train a binary classification
model based on a training dataset containing part of positives which are
labeled, and unlabeled instances. Unlabeled set includes remaining part of
positives and all negative observations. An important element in PU learning is
modeling of the labeling mechanism, i.e. labels' assignment to positive
observations. Unlike in many prior works, we consider a realistic setting for
which probability of label assignment, i.e. propensity score, is
instance-dependent. In our approach we investigate minimizer of an empirical
counterpart of a joint risk which depends on both posterior probability of
inclusion in a positive class as well as on a propensity score. The non-convex
empirical risk is alternately optimised with respect to parameters of both
functions. In the theoretical analysis we establish risk consistency of the
minimisers using recently derived methods from the theory of empirical
processes. Besides, the important development here is a proposed novel
implementation of an optimisation algorithm, for which sequential approximation
of a set of positive observations among unlabeled ones is crucial. This relies
on modified technique of 'spies' as well as on a thresholding rule based on
conditional probabilities. Experiments conducted on 20 data sets for various
labeling scenarios show that the proposed method works on par or more
effectively than state-of-the-art methods based on propensity function
estimation.

基于正样本和未标记数据的学习（PU 学习）是一种积极研究的机器学习任务，目标是基于包含部分标记的正样本和未标记实例的训练数据集训练一个二元分类模型。本研究提出了一个优化算法，通过建立标签机制、估计倾向性分值和进行参数优化，以实现风险一致性和更有效的方法。