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Semi-Supervised Mixture Models under the Concept of Missing at Radom with Margin Confidence and Aranda Ordaz Function

Jinyang Liao, Ziyang Lyu

TL;DR

The paper tackles semi-supervised learning with missing labels under Missing at Random (MAR) within Gaussian mixture models. It introduces margin confidence to quantify classification uncertainty and the Aranda--Ordaz (AO) link to flexibly model how missingness depends on this uncertainty, enabling a MAR-aware mixture modeling approach. An Expectation-Conditional Maximization (ECM) algorithm jointly estimates the GMM parameters and MAR parameters, yielding a Bayesian classifier to impute missing labels. Across controlled simulations and a real-world MAGIC Gamma Telescope dataset, the proposed ECM--AO framework improves calibration and maintains robust discrimination under moderate label sparsity, outperforming a logistic baseline that ignores missingness. The AO link’s flexibility provides stability in uncertain settings, while limitations remain for extreme missingness and multi-component mixtures, pointing to future work on broader uncertainty measures and higher-order mixtures.

Abstract

This paper presents a semi-supervised learning framework for Gaussian mixture modelling under a Missing at Random (MAR) mechanism. The method explicitly parameterizes the missingness mechanism by modelling the probability of missingness as a function of classification uncertainty. To quantify classification uncertainty, we introduce margin confidence and incorporate the Aranda Ordaz (AO) link function to flexibly capture the asymmetric relationships between uncertainty and missing probability. Based on this formulation, we develop an efficient Expectation Conditional Maximization (ECM) algorithm that jointly estimates all parameters appearing in both the Gaussian mixture model (GMM) and the missingness mechanism, and subsequently imputes the missing labels by a Bayesian classifier derived from the fitted mixture model. This method effectively alleviates the bias induced by ignoring the missingness mechanism while enhancing the robustness of semi-supervised learning. The resulting uncertainty-aware framework delivers reliable classification performance in realistic MAR scenarios with substantial proportions of missing labels.

Semi-Supervised Mixture Models under the Concept of Missing at Radom with Margin Confidence and Aranda Ordaz Function

TL;DR

The paper tackles semi-supervised learning with missing labels under Missing at Random (MAR) within Gaussian mixture models. It introduces margin confidence to quantify classification uncertainty and the Aranda--Ordaz (AO) link to flexibly model how missingness depends on this uncertainty, enabling a MAR-aware mixture modeling approach. An Expectation-Conditional Maximization (ECM) algorithm jointly estimates the GMM parameters and MAR parameters, yielding a Bayesian classifier to impute missing labels. Across controlled simulations and a real-world MAGIC Gamma Telescope dataset, the proposed ECM--AO framework improves calibration and maintains robust discrimination under moderate label sparsity, outperforming a logistic baseline that ignores missingness. The AO link’s flexibility provides stability in uncertain settings, while limitations remain for extreme missingness and multi-component mixtures, pointing to future work on broader uncertainty measures and higher-order mixtures.

Abstract

This paper presents a semi-supervised learning framework for Gaussian mixture modelling under a Missing at Random (MAR) mechanism. The method explicitly parameterizes the missingness mechanism by modelling the probability of missingness as a function of classification uncertainty. To quantify classification uncertainty, we introduce margin confidence and incorporate the Aranda Ordaz (AO) link function to flexibly capture the asymmetric relationships between uncertainty and missing probability. Based on this formulation, we develop an efficient Expectation Conditional Maximization (ECM) algorithm that jointly estimates all parameters appearing in both the Gaussian mixture model (GMM) and the missingness mechanism, and subsequently imputes the missing labels by a Bayesian classifier derived from the fitted mixture model. This method effectively alleviates the bias induced by ignoring the missingness mechanism while enhancing the robustness of semi-supervised learning. The resulting uncertainty-aware framework delivers reliable classification performance in realistic MAR scenarios with substantial proportions of missing labels.
Paper Structure (18 sections, 17 equations, 7 figures, 2 tables, 1 algorithm)

This paper contains 18 sections, 17 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Gaussian mixture model
  4. Margin Confidence
  5. Missing at random mechanism
  6. Aranda--Ordaz Link Function
  7. log likelihood function under missing at random mechanism
  8. Approximation Validity
  9. Simulations
  10. Visualization of Simulations
  11. Assessment of Simulations
  12. Verification of Algorithmic Robustness
  13. MAGIC Gamma Telescope Dataset
  14. Data Description
  15. Model Comparison
  16. ...and 3 more sections

Figures (7)

  • Figure 1: The solid shows the exact Shannon entropy, while the dashed denotes its second-order Taylor approximation. The vertical indicates $m^2 = 0.36$.
  • Figure 2: Blue and red points represent the two components, while grey points indicate missing labels. The right strip shows their proportions.
  • Figure 3: ROC curves comparing the three estimators under MAR.
  • Figure 4: The AUCs of two estimators under different missing proportion
  • Figure 5: Pairwise scatterplot matrix of the four variables from MAGIC dataset. Blue and orange points represent gamma and hadron events, respectively.
  • ...and 2 more figures