Favourable Missingness in Semi-Supervised Classification for Exponential Mixture Models

Abstract

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.