Fractionally Supervised Classification with Maxima Nominated Samples

We consider semi-supervised classification from a partially classified sample arising from a two-component Weibull mixture. The feature is observed for all data, whereas some class labels are missing. The probability of a missing label is modelled as a function of classification uncertainty, giving a feature-dependent missing-at-random (MAR) mechanism that shares parameters with the Weibull-mixture classifier. The missing-label indicators can therefore provide information about the classifier in addition to the observed features and available class labels. Under a common Weibull shape, a Bayes' rule has at most one positive decision boundary, which is unique when the rule is nonconstant; under unequal shapes, it can have two. We characterise these decision regions, derive the Fisher information for the classifier after adjustment for nuisance parameters in the missingness model, and obtain a decision-boundary expansion of the expected error rate of the plug-in sample rule relative to the Bayes error. The expansion yields classification-specific asymptotic relative efficiency formulas for the one- and two-boundary cases and shows that a positive-definite increase in Fisher information is sufficient, but not necessary, for a smaller first-order expected error rate. Numerical studies and a semi-synthetic analysis based on hard-drive failure data illustrate potential reductions in expected error rate and improvements in decision-boundary estimation from modelling feature-dependent label missingness.

Conformalized selection has been widely applied to select high-quality candidates from large datasets with rigorous uncertainty quantification, such as reliable labeling, drug discovery, and the alignment of large language models. Nevertheless, existing methods assume clean responses on calibration data, an assumption that rarely holds in practice. In this paper, we formulate the above tasks as selecting candidates with true predicted labels or with responses exceeding certain values. We demonstrate that existing conformal selection methods fail to control the false discovery rate (FDR) or suffer from severe power loss under contaminated calibration data. To that end, we propose Robust Conformalized Selection (RCS), a unified framework for selective classification with valid FDR control under general label contamination. The key insight of RCS lies in a novel statistical reduction: by separately conditioning on different classes, we translate the intractable label noise into a localized covariate shift problem, which then enables a covariate-adjusted empirical-Bayes-type estimate of the number of false selections. Statistical properties such as the asymptotic FDR control, power optimality, and robustness of RCS are established. We further develop an instantiation of RCS under randomized response model, and also apply RCS to the task of selecting candidates with large response values. Extensive experiments on both simulated and real-world datasets demonstrate the effectiveness of RCS.

Factorizable joint shift (FJS) represents a type of distribution shift (or dataset shift) that comprises both covariate and label shift. Recently, it has been observed that FJS actually arises from consecutive label and covariate (or vice versa) shifts. Research into FJS so far has been confined mostly to the case of categorical labels. We propose a framework for analysing distribution shift in the case of a general label space, thus covering both classification and regression models. Based on the framework, we generalise existing results on FJS to general label spaces and present and analyse a related extension to label distribution estimation of the expectation maximisation (EM) algorithm for class prior probabilities. We also take a fresh look at generalized label shift (GLS) in the case of a general label space.