Effects of unlabeled data on classification error in normal discriminant analysis

• We propose a framework of normal discriminant analysis for partially labeled data.
• We differentiate between feature independent and dependent labeling mechanisms.
• We define a criterion ARE to compare partially labeled data and labeled data alone.
• We derive and compute the ARE for (non)normal data under two mechanisms.
• Noninformative unlabeled data can increase classification error.

Semi-supervised learning, i.e., the estimation of parameters based on both labeled and unlabeled data, is widely believed to be effective in constructing a boundary in classification problems. The present paper investigates whether this belief is true in the case of normal discrimination in terms of the classification error for normal and nonnormal data. For this investigation, we use the framework of missing-data analysis because data consisting of labeled and unlabeled individuals can be regarded as missing data. Based on this framework, we introduce two labeling mechanisms: feature-independent labeling and feature-dependent labeling. For each of these labeling mechanisms, we analytically derive the asymptotic relative efficiency based on the labeled data alone and based on both the labeled and unlabeled data. Numerical computations reveal that (i) under the feature-independent labeling mechanism, unlabeled data tend to contribute to the improvement of the classification error even for nonnormal data and (ii) under the feature-dependent labeling mechanism, unlabeled data from both normal and nonnormal distributions are helpful when the labeled data are informative, but unlabeled data can augment the classification error when the labeled data are not informative. Finally, we describe some future areas of research.