Semi-Supervised Classification with Informative Missing Labels in Weibull Mixture Models

This paper investigates semi-supervised classification for positive-valued data modeled by a two-component Weibull mixture. Specifically, the authors examine scenarios where some class labels are missing, and the probability of this missingness depends on the observed feature. They aim to determine how this informative missing-at-random (MAR) mechanism interacts with the unique decision geometry of Weibull mixtures to affect classification error.

The authors model the missing-label probability as a function of classification uncertainty, using the squared log-posterior odds as a proxy for how difficult an observation is to classify. By sharing parameters between the classifier and the missingness mechanism, the missing-label indicators become informative. The researchers characterize the Bayes decision boundaries for both common-shape and unequal-shape Weibull mixtures, derive the Fisher information for the classifier after adjusting for nuisance parameters in the missingness model, and develop an asymptotic expansion of the expected error rate to evaluate classification efficiency.

The study reveals that the missing-label pattern can be leveraged to improve classification performance. When the missingness is concentrated near the Bayes decision boundary, the full-likelihood estimator—which jointly estimates the classifier and the missingness mechanism—consistently outperforms the estimator that ignores the missingness mechanism. The authors show that a positive-definite increase in Fisher information is a sufficient condition for reducing the first-order expected error rate. Numerical simulations and a semi-synthetic analysis of hard-drive failure data confirm that this approach leads to more accurate decision-boundary estimation and lower error rates compared to standard semi-supervised methods that treat missing labels as non-informative.