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SSLfmm: An R Package for Semi-Supervised Learning with Mixed Missingness

The SSLfmm R package introduces a likelihood-based Gaussian finite-mixture framework for semi-supervised learning that jointly models class distributions and label-missingness mechanisms (including MCAR, MAR, and mixed scenarios) to improve classification performance when label availability itself contains informative signals.

Original authors: Geoffrey J. McLachlan, Jinran Wu

Published 2026-08-20

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Geoffrey J. McLachlan, Jinran Wu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of data science, researchers often face a frustrating gap between what they know and what they can measure. Imagine a medical study where doctors have recorded the symptoms of every patient but only managed to confirm the final diagnosis for a portion of them. The rest remain a mystery, not because the data is lost, but because the confirmation process was too expensive, too slow, or simply impossible for certain cases. This is the realm of semi-supervised learning, a field dedicated to teaching computers to recognize patterns using a mix of known answers and unknown ones. Traditionally, statisticians have treated these missing answers as random accidents, assuming that the reason a label is missing has nothing to do with the data itself. However, in many real-world situations, the reason a label is missing is actually a clue. A patient might be harder to diagnose precisely because their symptoms are confusing, or a survey respondent might skip a question because the answer is uncomfortable. If the computer ignores this connection, it misses a vital piece of the puzzle.

A team of researchers at the University of Queensland has developed a new tool to help computers solve this specific problem. They created a software package called SSLfmm, which allows machines to learn from data where the labels are missing in two distinct ways: either completely at random, or because the data itself made the label difficult to obtain. The software is built on a framework that treats the missing label not just as a blank space, but as an outcome of a specific process. By modeling this process alongside the data, the system can distinguish between a missing label that happened by chance and one that happened because the data point was ambiguous or difficult to classify. This approach lets the computer use the very fact that a label is missing to improve its understanding of the groups it is trying to identify.

The researchers tested their new software by creating a simulated world of data where they knew exactly how the labels were missing. They generated a thousand data points representing two overlapping groups, then deliberately removed the labels for about thirty-two percent of them. They split these missing labels into two categories: some were removed randomly, while others were removed specifically because the data points were hard to distinguish from one another. They then asked their software to figure out the groups using two different strategies. In the first strategy, the software was told exactly which missing labels were random and which were difficult. In the second strategy, the software was only told that a label was missing, without knowing the reason. The results showed that even without knowing the specific reason for the missing labels, the software could still learn effectively. When tested on the data points whose labels were hidden, the version that knew the reasons achieved an accuracy of about eighty-one percent, while the version that had to guess the reasons achieved a nearly identical accuracy of about eighty-one percent. This demonstrated that the software's internal logic for handling missing information was robust, even when the source of the missingness was hidden.

To see how this works in a more realistic setting, the team applied the software to a dataset concerning blood transfusions. They took a complete record of seven hundred and forty-eight donors and artificially removed the labels for some of them. They kept the labels for donors who were easy to classify, removed the labels for a random selection of others, and removed the labels for a third group that was specifically chosen because they were difficult to classify. This created a scenario where the missing labels were a mix of random events and difficult cases. The researchers then ran the software under three different assumptions: that all missing labels were random, that all were due to difficulty, and that they were a mixture of both. The software that assumed a mixture of both causes performed the best. It correctly identified the blood donor groups with an accuracy of about sixty-one percent for the difficult cases, outperforming the models that assumed only one type of missingness. The software also estimated that about ten percent of the missing labels were random, a figure that closely matched the actual setup of the experiment.

Beyond just getting the right answer, the software provides a way to check its own work. It calculates a measure of uncertainty for every prediction, essentially asking the computer how confident it feels about each classification. When the researchers looked at these confidence scores, they found a clear pattern: the data points that were originally difficult to label had much higher uncertainty scores than the ones that were easy or random. This confirmed that the software had successfully learned to associate the missing labels with the difficulty of the data. The tool is now available as a free package for the R programming language, a standard environment for statistical computing. It allows researchers to fit models, make predictions, and run diagnostics without needing to write complex code from scratch. The package includes functions to simulate data, check how well the model is working, and visualize the relationship between missing labels and classification uncertainty.

The significance of this work lies in its ability to handle the messy reality of data collection. In many fields, from medicine to social science, the process of getting a label is rarely random. By acknowledging that the missingness itself carries information, the SSLfmm package offers a more honest and accurate way to learn from incomplete data. It does not claim to solve every problem with missing information, nor does it suggest that all missing data is informative. Instead, it provides a flexible framework where researchers can test different assumptions about why data is missing and see which assumption leads to the best results. The authors emphasize that while the software performed well in their tests, the specific numbers depend on the data being analyzed. The true value of the tool is that it gives scientists a way to ask, "Is the fact that I don't know this answer telling me something about the answer itself?" and to get a data-driven response.

Technical Summary of "SSLfmm: An R Package for Semi-Supervised Learning with Mixed Missingness"

Problem Formulation
The paper addresses semi-supervised learning (SSL) scenarios where feature vectors are fully observed for all training data, but class labels are available only for a subset. A critical, often unmodelled aspect of this setting is the mechanism governing label availability. Standard SSL procedures typically treat missing labels as unknown latent classes without modelling why they are missing. However, the authors argue that the missingness process itself may contain information relevant to classification. Specifically, the paper distinguishes between:

Missing Completely at Random (MCAR): Label availability is unrelated to observed features or class.
Missing at Random (MAR): The probability of a missing label depends on observed features (specifically, classification uncertainty/entropy) but not on the missing label itself.
Mixed Missingness: A scenario where a missing label arises from either an MCAR mechanism or an entropy-based MAR mechanism. Crucially, the source of the missingness (MCAR vs. MAR) may be either observed (recorded in the data) or latent (unobserved, with only the aggregate missingness indicator available).

Methodology
The authors propose a likelihood-based framework using Gaussian finite mixture models where the class distribution and the label-missingness process are modelled jointly. The core statistical formulation involves:

Class Model: Each class $i$ is associated with a multivariate Gaussian density $f_i(y; \omega_i)$ with prior probabilities $\pi_i$ .
Missingness Model:
- For the entropy-based MAR mechanism, the probability of a missing label is modelled via a logistic function of the Shannon entropy $e(y; \theta)$ of the observation: $q_j = \text{logit}^{-1}(\xi_0 + \xi_1 \log e(y_j; \theta))$ , where $\xi_1 > 0$ .
- For the mixed mechanism, the probability of a missing label is a mixture of the MCAR probability $\alpha$ and the MAR probability $(1-\alpha)q_j$ .
Estimation (ECM Algorithm): The parameters are estimated using an Expectation-Conditional Maximisation (ECM) algorithm.
- E-step: Computes posterior class probabilities and replaces missing class indicators with their conditional expectations.
- C-steps:
  - If the missingness source is observed, the algorithm updates the MCAR probability $\alpha$ and the MAR parameters $\xi$ separately based on the known source indicators, then updates the class parameters $\theta$ .
  - If the missingness source is latent, the algorithm treats the specific mechanism (MCAR vs. MAR) for each missing label as a latent variable. It computes the posterior probability that a missing label arose from the MCAR channel versus the MAR channel, then updates $\alpha$ , $\xi$ , and $\theta$ iteratively.

The SSLfmm R Package
The paper introduces SSLfmm, an R package implementing these methods. Key features include:

Unified Interface: A single function fit_sslfmm() handles complete-case, MCAR, entropy-based MAR, and mixed MCAR/MAR analyses.
Source Handling: Supports both indicator = "observed" (where the user provides the missingness source) and indicator = "latent" (where the source is inferred).
Functionality: Includes tools for model fitting, prediction (classes, posteriors, entropy), performance assessment (accuracy, F1, log loss, Brier score), simulation of mixed missingness data, and entropy-based diagnostics.
Robustness: Uses multiple starting values and bounded numerical optimisation to handle the non-convexity of the likelihood, particularly for latent-source models.

Results and Illustrations
The paper validates the package through two primary illustrations:

Reproducible Simulation: A simulation with 1,000 observations and 31.9% missing labels (generated via mixed MCAR/MAR) compares observed-source and latent-source fits.
- Both formulations achieved similar classification performance on the missing-label subset (Accuracy $\approx$ 0.806–0.809).
- The latent-source model successfully recovered the generating parameters ( $\alpha$ , $\xi_0$ , $\xi_1$ ) with reasonable accuracy, though estimates were slightly more variable than the observed-source fit.
- Entropy diagnostics confirmed that observations generated under the MAR mechanism exhibited higher classification entropy, consistent with the model's design.
Semi-Synthetic Application (Blood Transfusion Data): Using a real dataset where labels were artificially removed to create MCAR and MAR subsets, the authors compared MCAR, MAR, and Mixed models under equal and unequal covariance assumptions.
- The Mixed model consistently outperformed the pure MCAR and pure MAR models in terms of hard classification accuracy, balanced accuracy, and macro F1 score on the missing-label subset.
- The mixed model estimated an MCAR probability ( $\alpha$ ) close to the true construction value (0.10).
- Entropy diagnostics showed that the "MAR" subset (constructed from difficult-to-classify points) indeed had higher entropy than the "MCAR" and "observed" subsets, validating the interpretability of the fitted model.

Significance and Claims
The authors position SSLfmm not as a generic clustering engine, but as a specialized tool bridging model-based semi-supervised learning and likelihood-based missing-data modelling.

Distinct Contribution: Unlike existing packages (e.g., mclust, Rmixmod) that focus on component distributions or handle missing features, SSLfmm explicitly models the stochastic process of missing labels.
Novelty: It is the first implementation to support a mixed MCAR/MAR mechanism where the missingness source can be either observed or latent, utilizing distinct ECM algorithms for each case.
Practical Utility: The package allows researchers to fit, compare, and diagnose different assumptions about label missingness within a common framework.
Modest Scope: The authors acknowledge limitations: the current implementation assumes fully observed features and Gaussian class-conditional distributions. They caution that when the missingness source is latent, the data may contain limited information to separate MCAR and MAR contributions, necessitating careful interpretation of fitted parameters alongside diagnostics. They do not claim that the mixed model will always dominate, but rather that it provides a flexible framework to study how assumptions about label availability affect estimation and prediction.

Technical Summary of "SSLfmm: An R Package for Semi-Supervised Learning with Mixed Missingness"

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