Fariborz Setoudehtazang

arXiv2026

Informative Label Missingness in Multiclass Classification Information Geometry and Excess Risk

Informative label missingness can change the usual efficiency ordering between completely and partially labelled classifiers because the pattern of missing labels may itself carry information about the classification model. We develop a general likelihood-based theory for this phenomenon in parametric multiclass classification. An efficient-information decomposition separates information lost through unavailable class memberships from information contributed by the missing-label mechanism. We then derive a quadratic expansion of plug-in excess risk over the active pairwise faces of the multiclass Bayes boundary, showing that classification efficiency depends on how information gains and losses align with directions that perturb the decision boundary. This yields a classification-weighted generalized-eigenvalue criterion under which informative partial classification may have smaller asymptotic classification risk without globally dominating complete classification in Fisher information. Near missing completely at random, with the marginal missing-label proportion fixed, redistribution of missing labels changes lost class-label information at first order, whereas efficient information from the missingness pattern appears only at second order. Three-class quadratic discriminant calculations, finite-sample experiments, and a semi-synthetic multiclass application illustrate the resulting regime-dependent behaviour.

stat.ML↗

arXiv2026

Favourable Missingness in Semi-Supervised Classification for Exponential Mixture Models

Semi-supervised classifiers are commonly trained from samples in which all features are observed but some class labels are missing. When label missingness is independent of the observed data, unavailable class memberships reduce Fisher information relative to a completely classified sample. We study a different regime in which the probability of label missingness depends on posterior classification uncertainty, so that the observed missing-label indicators can themselves carry information about the Bayes decision boundary. Building on the conditionally weighted information decomposition of Ahfock and McLachlan, we develop this phenomenon for a two-component exponential mixture. Although the exponential model is non-Gaussian, asymmetric, and supported on the positive half-line, its log-posterior odds remain linear in the feature. We derive Bayes' rule and its exact error rate, formulate entropy-logistic and squared-discriminant missingness mechanisms, and obtain the full partially classified likelihood. We then derive a decomposition of the Fisher information into the complete-data information, the conditionally weighted loss due to missing labels, and the information contributed by the missing labels. Numerical quadrature identifies regions in which the full likelihood classifier has asymptotic relative efficiency above or below one. Monte Carlo experiments with finite training samples broadly support the population calculations, with the largest departures from the asymptotic predictions occurring near the transition at which the relative efficiency crosses one.

stat.CO↗